Academic literature on the topic 'Wave Packet Dynamic'

Through the wave-packet propagation diagnosis (WPD), the essay discusses and analyzes the characteristics of wave packet distribution and dynamic system of wave energy propagation during the extended period of the massive drought in southwestern region in the country by using the reanalysis grid data of the daily NCEP/NCAR geopotential height field and 10×10 horizontal resolution from September 2009 to April 2010. The results show that: the characteristics of dynamic system of wave energy propagation at the middle and high height fields can better reflect the stage features of this drought during the period. The wave energy time-variation chart shows that the wave disturbance during the drought is weak and the wave packet value is small. However, the wave packet value during the drought still has some changes, reflecting the characteristics of the drought change in weakness. At the maintenance period of the wave packet value, the drought is often severe, and when the wave packet value changes greatly, the magnitude of the drought will change. Seen from the characteristics of longitudinal and altitudinal propagation of the wave energy, when the wav packet strength increases, the energy will accumulate and the drought tends to be weakened; on the contrary, when the wave packet strength decreases, energy will discharge and the drought will become more severe—particularly when the wave packet weakens quickly, the drought will become more drastic. This study provides a meaningful idea for discussing the forecast of the scale weather process during the extended period.

Geometric, kinematic and dynamic properties of focusing deep-water surface gravity wave packets are examined in a simplified model with the intent of deriving a wave breaking threshold parameter. The model is based on the spatial modified nonlinear Schrödinger equation of Dysthe (Proc. R. Soc. Lond. A, vol. 369 (1736), 1979, pp. 105–114). The evolution of initially narrow-banded and weakly nonlinear chirped Gaussian wave packets are examined, by means of a trial function and a variational procedure, yielding analytic solutions describing the approximate evolution of the packet width, amplitude, asymmetry and phase during focusing. A model for the maximum free surface gradient, as a function of $\unicode[STIX]{x1D716}$ and $\unicode[STIX]{x1D6E5}$, for $\unicode[STIX]{x1D716}$ the linear prediction of the maximum slope at focusing and $\unicode[STIX]{x1D6E5}$ the non-dimensional packet bandwidth, is proposed and numerically examined, indicating a quasi-self-similarity of these focusing events. The equations of motion for the fully nonlinear potential flow equations are then integrated to further investigate these predictions. It is found that a model of this form can characterize the bulk partitioning of $\unicode[STIX]{x1D716}-\unicode[STIX]{x1D6E5}$ phase space, between non-breaking and breaking waves, serving as a breaking criterion. Application of this result to better understanding air–sea interaction processes is discussed.

In this paper, we obtain the quantum dynamics in the framework of the general theory of relativity, where a quantum partide is described by a distribution of matter, with amplitude functions of the matter density, in the two conjugate spaces of the spațial coordinates and of the momentum, called wave functions. For a free partide, these wave functions are conjugate wave packets in the coordinate and momentum spaces, with time dependent phases proporțional to the relativistic lagrangian, as the wave velocities in the coordinate space are equal to the distribution velocity described by the wave packet in this space. From the wave velocities of the partide wave functions, we obtain lorentz’s force and the maxwell equations. For a quantum partide in electromagnetic field, we obtain dynamic equations in the coordinate and momentum spaces, and the partide and antiparticle wave functions. We obtain the scattering or tunneling rate in an electromagnetic field, for the two possible cases, with the spin conservation, or inversion

Breaking waves on the ocean surface transfer energy and momentum into currents and turbulence. What is less well understood, however, is the associated total loss of wave energy and momentum flux. Further, finding a robust and universal diagnostic parameter that determines the onset of breaking and its strength is still an open question. Derakhti & Kirby (J. Fluid Mech., vol. 761, 2014, pp. 464–506) have recently studied bubble entrainment and turbulence modulation by dispersed bubbles in isolated unsteady breaking waves using large-eddy simulation. In this paper, a new diagnostic parameter ${\it\xi}(t)$ is defined based on that originally proposed by Song & Banner (J. Phys. Oceanogr., vol. 32, 2002, pp. 2541–2558), and it is shown that using a threshold value of ${\it\xi}_{th}=0.05$, the new dynamic criteria is capable of detecting single and multiple breaking events in the considered packets. In addition, the spatial variation of the total energy and momentum flux in intermediate- and deep-water unsteady breaking waves generated by dispersive focusing is investigated. The accuracy of estimating these integral measures based on free surface measurements and using a characteristic wave group velocity is addressed. It is found that the new diagnostic parameter just before breaking, ${\it\xi}_{b}$, has a strong linear correlation with the commonly used breaking strength parameter $b$, suggesting that ${\it\xi}_{b}$ can be used to parameterize the averaged breaking-induced dissipation rate and its associated energy flux loss. It is found that the global wave packet time and length scales based on the spectrally weighted packet frequency proposed by Tian et al. (J. Fluid Mech., vol. 655, 2010, pp. 217–257), are the reasonable estimations of the time and length scales of the carrier wave in the packet close to the focal/break point. A global wave steepness, $S_{s}$, is defined based on these spectrally weighted scales, and its spatial variation across the breaking region is examined. It is shown that the corresponding values of $S_{s}$ far upstream of breaking, $S_{s0}$, have a strong linear correlation with respect to $b$ for the considered focused wave packets. The linear relation, however, cannot provide accurate estimations of $b$ in the range $b<5\times 10^{-3}$. A new scaling law given by $b=0.3(S_{s0}-0.07)^{5/2}$, which is consistent with inertial wave dissipation scaling of Drazen et al. (J. Fluid Mech., vol. 611, 2008, pp. 307–332), is shown to be capable of providing accurate estimates of $b$ in the full range of breaking intensities, where the scatter of data in the new formulation is significantly decreased compared with that proposed by Romero et al. (J. Phys. Oceanogr., vol. 42, 2012, pp. 1421–1444). Furthermore, we examine nonlinear interactions of different components in a focused wave packet, noting interactive effect on a characteristic wave group velocity in both non-breaking and breaking packets. Phase locking between spectral components is observed in the breaking region as well, and subsequently illustrated by calculating the wavelet bispectrum.

We recently introduced a model of incoherent quasielastic neutron scattering (QENS) that treats the neutrons as wave packets of finite length and the protein as a random walker in the free energy landscape. We call the model ELM for “energy landscape model.” In ELM, the interaction of the wave packet with a proton in a protein provides the dynamic information. During the scattering event, the momentum Q(t) is transferred by the wave packet to the struck proton and its moiety, exerting the force F(t)=dQ(t)/dt. The resultant energy E⋆ is stored elastically and returned to the neutron as it exits. The energy is given by E⋆=kB(T0+χQ), where T0 is the ambient temperature and χ (≈ 91 K Å) is a new elastobaric coefficient. Experiments yield the scattering intensity (dynamic structure factor) S(Q;T) as a function of Q and T. To test our model, we use published data on proteins where only thermal vibrations are active. ELM competes with the currently accepted theory, here called the spatial motion model (SMM), which explains S(Q,T) by motions in real space. ELM is superior to SMM: It can explain the experimental angular and temperature dependence, whereas SMM cannot do so.

Abstract. We examine the group dynamic of African easterly waves (AEWs) generated in a realistic, spatially non-homogeneous African easterly jet (AEJ) using an idealized general circulation model. Our objective is to investigate whether the limited zonal extent of the AEJ is an impediment to AEW development. We construct a series of basic states using global reanalysis fields and initialize waves via transient heating over West Africa. The dominant response is a localized, near-stationary wave packet that disperses upstream and downstream. The inclusion of a crude representation of boundary layer damping stabilizes the waves in most cases, consistent with other studies in the past. In some basic states, however, exponential growth occurs even in the presence of damping. This shows that AEWs can occasionally emerge spontaneously. The key result is that, whether triggered by an external forcing or generated internally, the wave packet can remain within the AEJ for multiple wave periods instead of being swept away. Drawing from other studies, this also suggests that even the damped waves can grow if coupled with additional sources of energy such as moist convection and dust radiative feedback. The wave packet in the localized AEJ appears to satisfy a condition for absolute instability, a form of spatial hydrodynamic instability. However, this needs to be verified more rigorously. We conclude that the limited zonal extent of the AEJ is not an impediment. Our results also suggest that the intermittent nature of AEWs is mediated, not by transitions between convective and absolute instability, but likely by external sources such as propagating equatorial wave modes.

Abstract This work presents a simulation of the response of packets of microbubbles in an ultrasonic pulse-echo scan line. Rayleigh-Plesset equation has been used to predict the echo from numerically obtained radial dynamics of microbubbles. Varying the number of scattering microbubbles on the pulse wave form has been discussed. To improve microbubble-specific imaging at high frequencies, the subharmonic and second harmonic signals from individual microbubbles as well as microbubbles packets were simulated as a function of size and pressure. Two different modes of harmonic generation have been distinguished. The strength and bandwidth of the subharmonic component in the scattering spectrum of microbubbles is greater than that of the second harmonic. The pressure spectra provide quantitative and detailed information on the dynamic behaviour of ultrasound contrast agent microbubbles packet.

In this work, we investigate the structural and dynamic magnetic properties of yttrium iron garnet (YIG) films grown onto gadolinium gallium garnet (GGG) substrates with thin platinum, iridium, and gold spacer layers. Separation of the YIG film from the GGG substrate by a metal film strongly affects the crystalline structure of YIG and its magnetic damping. Despite the presence of structural defects, however, the YIG films exhibit a clear ferromagnetic resonance response. The ability to tune the magnetic damping without substantial changes to magnetization offers attractive prospects for the design of complex spin-wave conduits. We show that the insertion of a 1-nm-thick metal layer between YIG and GGG already increases the effective damping parameter enough to efficiently absorb spin waves. This bilayer structure can therefore be utilized for magnonic waveguide termination. Investigating the dispersionless propagation of spin-wave packets, we demonstrate that a damping unit consisting of the YIG/metal bilayers can dissipate incident spin-wave signals with reflection coefficient R < 0.1 at a distance comparable to the spatial width of the wave packet.

A gas-phase molecular ensemble coherently excited to have an oriented rotational angular momentum has recently emerged as an appropriate microscopic system to illustrate quantum mechanical behavior directly linked to classical rotational motion, which has a definite direction. To realize an intuitive visualization of such a unidirectional molecular rotation, we report high-resolution direct imaging of direction-controlled rotational wave packets in nitrogen molecules. The rotational direction was regulated by a pair of time-delayed, polarization-skewed laser pulses, introducing the dynamic chirality to the system. The subsequent spatiotemporal propagation was tracked by a newly developed Coulomb explosion imaging setup. From the observed molecular movie, time-dependent detailed nodal structures, instantaneous alignment, angular dispersion, and fractional revivals of the wave packet are fully characterized while the ensemble keeps rotating in one direction. The present approach, providing an accurate view on unidirectional rotation in quantum regime, will guide more sophisticated molecular manipulations by utilizing its capability in capturing highly structured spatiotemporal evolution of molecular wave packets.

The present work is divided in two part. The first is dedicated to the investigation of the gas-metal interactions, an interesting area in the basic surface science but also in applied one, since it could provide a more efficient way to design corrosion-resistant structural metals. In particular, we concentrate our attention on the study H2S on Fe surface. Experimental studies, of adsorption of H2S on Fe, and first-principle calculations were carried out for these systems, clarifying some important questions, such as adsorption geometry and dissociation pathways for H2S, on the above close-packed metal surfaces. However, real samples will also include a number of defects, in particular step edges where bonding of adsorbates is usually stronger than at facets. It is therefore interesting to investigate adsorption of H2S on a stepped Fe surface, a task which has not been considered yet to the best of our knowledge. In the present work we study the H2S interaction with Fe(310) surfaces by DFT calculations in order to understand the role of step defects in the adsorption properties. We recall that the (310) surface is relatively stable, and its surface energy predicted to be even smaller than that of Fe(110). We do not only obtain the binding sites and adsorption energies of H2S and its components, but we also relate bonding to the detailed features of the localdensity of states (LDOS). The second part of the present thesis is devoted to the dynamics of scattering. Scattering underlies various physical processes in different field of physics, mainly in solid state, as for example in thermoelectricity, about the filtering of hot electrons by defects, or adsorption and desorption by a surface, or in charge injection and field emission trough interface, usually associated with tunneling mechanisms. The recent developments of nanotechnology and the advent of modern high-speed high-density MOS devices, have revived the technological and theoretical interest of the scientific community on the scattering problem and in particular on quantun tunneling mechanism usually associated. Ultrascaled nanometric CMOS compatible single electron transistors (SETs) and single atom trasistors has lead the emergence of density of states graining and fluctuations in the contacts which may determine discretization of energy levels, charge localization at intradopant length scale and selection rules on quantum states in tunnelling. Consequently, the understanding of dependence of charge dynamics, across a barrier, from the initial position constitutes a relevant aspect in such systems. In this work we study the scattering process in the non stationary framework using Gaussian wave packet (GWP) to describe the particle wave function of the system so as to consider the dependence of scattering dynamics from the initial conditions. Through a numerical solution of the Schr¨odinger equation we analyse the evolution of the system calculating the transmission of the scattering GWP as a function of the initial spread and position x(0), and comparing simulated data with theoretical results. By our analysis a new important issue emerges: the time spent by the particle to reach its asymptotic probability to be observed beyond the barrier ( that we call formation time), strongly depends on initial conditions, and in particular on x(0). Finally, to analytically express such a dependence, we propose a semi-classical approximated model in which tf is described as the time spent by a finite support (accounting for the 0.99 of the probability) of the incident wave packet to cross the barrier, namely the time required to locate, in coordinate space, the greatest amount of the GWP’s probability distribution beyond the barrier interface.

Electron dynamics in atoms and molecules occurs on a time-scale of attoseconds (10⁻¹⁸s). With the availability of strong field (∼ 10¹²- 10¹³ W cm⁻²) femtosecond (10⁻¹⁵s) laser pulses with electric fields that can reach and exceed the Coulomb field strength experienced by an electron in the ground state of an atom, it is now possible to generate even shorter pulses with durations on the order of attoseconds by the process of high-harmonic generation (HHG). In this dissertation, experiments to study electron dynamics on attosecond time-scales in a helium atom using attosecond pulses generated by HHG will be described. We use extreme-ultraviolet (XUV) attosecond pulse trains and strong femtosecond near-infrared (IR) laser pulses to excite and ionize helium atoms. We first discuss an experimental technique that allows us to quantify and reduce the detrimental effects of Gouy phase slip on attosecond XUV-IR experiments. We then discuss our experiments to study the dynamic behavior of electronic states in a strong field modified helium atom where we use attosecond pulses to explore the strong-field modified atomic landscape. Using the Floquet theory to interpret our experimental observations we measure the variation in quantum phase of interferences between different fourier components of Floquet states as the IR intensity is varied and as different ionization channels dominate, in real-time. Next, we briefly discuss quantum interferences between photo-electrons ionized from XUV excited states in helium using an IR field which is polarized orthogonal to the XUV polarization. We observe variation in angular distribution of photo-electrons as a function of XUV-IR time-delay. We then discuss a new technique to measure the time-of-birth of attosecond pulses using XUV+IR photo-ionization in helium as a measurement probe. Finally, experiments to study the evolution of XUV excited wave-packets in helium on a time-scale of 100's of femtoseconds with attosecond resolution will be described.

Wave-mechanical phenomena such as resonance and interference, in both light and matter, are central to the principles of quantum coherent control over molecular processes. Focusing on the dynamical aspects, this dissertation is a compilation of studies on the interaction physics involving wave packets in molecules, the driving light field, and the underlying coherence and control. In each work, we will demonstrate interesting correlations between the properties of a carefully designed excitation light field and desirable outcomes of the molecules quantum dynamics. We will analyze the dynamical effect of a Feshbach resonance in the adiabatic Raman photoassociation for ultracold diatomic molecule formation from ultracold atoms. A narrow resonance is shown to be able to increase the effective number of collisions, in an ultracold atomic gas, that are available for photoassociation. This results in an optimal resonance width much smaller than the atomic collision energy bandwidth, due to the balance between the effective collision rate and single-collision transfer probability. Next, we demonstrate the linear molecular response to high-intensity, broadband, shaped optical fields. We show that this originates from interferences based on intra-pulse Raman excitations, and thus response linearity is not unique to the first-order perturbative limit and can not be used to infer the strength of the field. In the last study, we simulate the stochastic vibrational wave packet and dissociation-flux dynamics in a molecule excited by light with temporal and spectral incoherent properties. Between this case and that using a coherent pulse with the same spectral profile, we compare the vibrational wave functions and the loss of electronic and vibrational coherence, and demonstrate the qualitative difference between coherently and incoherently driven dynamics in molecules.<br>Science, Faculty of<br>Physics and Astronomy, Department of<br>Graduate

In this thesis, we present a theoretical investigation of reactive processes involving the HeH$^+$ molecular ion, with applications in laboratory and astrophysical plasma physics. We consider in particular two processes, which are the charge transfer in H + He$^+$ collisions at low energy from a molecular approach and the photodissociation of HeH$^+$.<p><p>At the molecular level, the cross section is the basic quantity that has to be determined in order to achieve an understanding of reactive processes. Its calculation will be based on the description of the reactions using an emph{ab initio}, quantum mechanical approach. In this work, we will rely on the Born-Oppenheimer approximation, which allows the molecular motion to be separated into an electronic and a nuclear motion. The evaluation of cross sections then follows two steps.<p>The first is the determination of the electronic structure of the molecule. We will calculate the adiabatic potential energy curves of the excited electronic states as well as the dipole matrix elements between these states. The non-adiabatic radial and rotational couplings, which result from the breakdown of the Born-Oppenheimer approximation, are also estimated. The second step is to solve the nuclear motion, which we achieve using a time-dependent method based on the propagation of wave packets on the coupled electronic states. <p><p>A particular emphasis will be put on the importance of the excited states and of the non-adiabatic couplings in the description of reactive processes. <p>In the treatment of the charge transfer reaction between H and He$^+$ in excited states, it is well known that the non-adiabatic radial couplings cannot be neglected. However, we will show that the inclusion of the non-adiabatic rotational couplings is also necessary in order to obtain accurate state-to-state cross sections.<p>In the description of the photodissociation of HeH$^+$ from its ground state, we will show the influence of the excited states on the rate constant and the role of the non-adiabatic radial couplings in the determination of partial cross sections.<p>We will also consider the possible astrophysical applications of the first triplet state of HeH$^+$. We will show that this state is metastable by evaluating its lifetime, and calculate the cross sections and rate constants for the photodissociation and radiative association of HeH$^+$ in this state.<p><br>Doctorat en Sciences<br>info:eu-repo/semantics/nonPublished

This thesis presents the first study of the damping of a Bloch oscillating wave packet by Zener tunneling to above-barrier states [1]. We investigate the time evolution of an below-barrier subband Wannier-Stark wave packet in a strongly coupled GaAs/AlGaAs superlattice (SL) with shallow quantum well barriers by optical interband spectroscopy. We use a sub-100 fs homodyne pump-probe technique which is sensitive to the intraband polarization. The presented experimental data unambiguously show an electric field-dependent continuous decrease of the intraband coherence time. Besides the continuous field-induced damping of the intraband polarization, we observe the signature of resonant Zener tunneling of a Bloch oscillating wave packet between discrete states belonging to below and above-barrier bands. This coupling manifests itself as a revival of the intraband polarization [2]. The experiment is modelled in two aspects. First, in a 1D single-particle calculation the wave functions the BO wave packet is composed of are derived. Here, the inter-subband dynamics are found to be given by the energetic splitting between nearly-degenerate below and above-barrier states. The wave packet tunnels from the below-barrier band to the above-barrier band while remaining coherently oscillating. At this time, it is spatially spread over more than 100 nm...

This textbook details the basic theory of ultrafast molecular spectroscopy starting from time-dependent quantum mechanical perturbation theory in Hilbert space. The emphasis is on the dynamics of nuclear and electronic motion initiated and monitored by femtosecond laser pulses that underlies nonlinear optical signal formation and interpretation. Topics include short-pulse optical absorption, the molecular adiabatic approximation, transient-absorption spectroscopy, vibrational adiabaticity during conformational change, femtosecond stimulated Raman spectroscopy, multi-dimensional electronic spectroscopy and wave-packet interferometry, and two-dimensional wave-packet interferometry of electronic excitation-transfer systems. Numerous exercises embedded in the text explore and expand upon the physical concepts encountered in this important research field.

The dynamics of vibrational wave packets in molecules that are generated by an ultrashort light pulse have revealed effects that are clearly attributable to the quantum mechanical nature of the system. Several nonclassical aspects of the dynamics have been reported, notably the decay and revival of a single wave packet1, and quantum interference between two wave packets2.

Abstract Introducing conductive nanofillers into polymeric, cementitious, and ceramic composites can impart multifunctional properties such as self-sensing capabilities via the piezoresistive effect. Much work has been done to utilize this multifunctionality for conductivity-based structural health monitoring (SHM) and condition monitoring. To date, the majority of such investigations concern static and quasi-static loading conditions. Much less work has been done with regard to general dynamic loading conditions such as transient wave propagation. This is an important gap in state of the art for two reasons: First, the self-sensing nature of these materials potentially allows for full-field (i.e. sub-surface) dynamics monitoring which cannot be achieved via traditional surface-mounted dynamic sensors. And second, conductivity-based and vibratory-based SHM are both independently well researched areas. Combined into a single, piezoresistive elastodynamic formulation, however, they may give rise to unprecedented new diagnostic capabilities. Therefore, the initial results presented in this manuscript seek to address this gap in the state of the art by experimentally exploring the role of dynamic excitation on transient piezoresistive behavior in nanocomposite structures. Specifically, an electromagnetic shaker is used to inject highly-controlled planar strain wave packets into a slender prismatic carbon nanofiber (CNF)-modified epoxy rod. Resistance measurements are then taken as the wave packets travel along the length of the rod. It was found that resistance changes taken from the rod are able to accurately reconstruct the injected strain wave and can be used to discern dynamic properties of CNF-modified epoxy. An external laser vibrometry (LV) system was used as extrinsic validation. Results from this preliminary investigation may lay the foundation for a new exciting field of fully coupled piezoresistive elastodynamics.

Wave packet systems in semiconductor superlattices are discussed. The wave packet approach is an intuitive method for the study of coherent carrier dynamics in superlattices that have been excited by using a short laser pulse. The theoretical formalism1 can be applied to a variety of III–V and II–VI systems. In this discussion, only GaAs/AlGaAs superlattices are studied. Significant results are seen. Coupled wells are shown to exhibit quasi-periodic dynamics for certain superlattice and excitation parameters. Phenomena that have been observed and predicted in atomic wave packet systems are seen in these systems. These include the breakup of wave packets when energy levels are not evenly spaced and the revival of the wave packet after long times when energy spacings are nearly equal. These superlattices also exhibit semiconductor features, such as excitonic and hole effects. A discussion of comparisons with experimental results is also presented.

We study the dynamics of a circular-orbit wave packet1 and find that it has both classical and quantum features. A circular-orbit wave packet consists of aligned, standard hydrogenic eigenstates of maximum angular momentum (l = in = n − 1), superimposed with a Gaussian weighting function. We calculate the uncertainty products in all three degrees of freedom and find that, as the quantum numbers of the constituent eigenstates increase, the uncertainty products tend toward h/2. The classical aspect of the dynamics displays itself during revivals, when a well-localized wave packet travels along a Kepler orbit, obeying the classical equations of motion. Spreading and revivals (including fractional revivals2) of the wave packet, which are manifestations of its quantum nature, are related to the nonlinearity of the hydrogen atom Hamiltonian. We study fractional revivals of the wavepacket in detail.

An atomic Rydberg electron wave packet above the saddle point of the combined Coulomb and static electric field potential [1], created by a short laser pulse, shows a few oscillations in angular momentum (l) and radial co-ordinate before escaping from the atomic potential [2]. Up to now, experimental studies on the dynamics of the wave packet focused on recurrences of wave packets to the atomic core. These studies have been carried out both in the (1) time and (2) frequency domain. (1) In optical pump-probe studies the overlap near the atomic core of the evolved wave packet with the initial wave packet is directly measured as a function of time. (2) These recurrences of the launched wave packet to the atomic core lead to resonant structure in the photoionization spectrum. We introduce a new device, the atomic streak camera, which measures directly the escape over the saddle point of the Rydberg electron in a static electric field. By measuring the time dependent leaking of the wave packet over the saddle point, instead of measuring recurrences to the atomic core we found that the life time of the Rydberg electron as measured by an optical technique is not the same as the time it takes the electron to leave the atom. Optical techniques measure how many recurrences to the core region the electron makes, and have no access to other parts of the atomic potential. Hence, while the electron might no longer be visible for optical techniques it can still be in the atomic potential, and ionization is not yet seen by the atomic streak camera.

Ultrafast pump-probe experiments have been applied to the study of vibrational wave-packet dynamics in the excited states of the cesium dimer. Pumping the C(1П u )←X(1Σg+) band at several wavelengths between 627 and 650 nm produces a vibrational wave packet with an oscillation period of 600 fs. The wave packet is detected by photoionizing the excited molecule with a time delayed probe pulse. The wave packet oscillations persist for approximately 5 ps, and their decay is attributed to predissociation of the C state. Numerical quantum calculations using the FFT spectral method have been performed on the excited state surface to simulate the time evolution of the wave packet density. Experimental results and simulations for these processes will be presented.

Pipe is a very important tool for long-distance transportation of nature gas. In the long-term running, there will be inevitably an appearance of a rupture, leak or damage usually caused by manmade event or by nature disaster. Leaks may generate dangerous clouds of gas escaping from the high-pressure pipe and produce serious incidences involving fire and explosion endangering the life and property safety of people in and around the area. Monitoring of natural gas pipeline leaks will timely find out and locate these dangerous occurrences and reduce loss. Within the leak monitoring, the core contents are the accurate location of leaks as well as the rapid identification of different signal sources reducing false alarm ratio. Once a leak occurs, the supersonic jet of escaping gas can generate a non-linear &amp; chaotic negative pressure wave signal based on static pressure measurement and an acoustic signal based on dynamic pressure measurement [1]. By properly interpreting these two kinds of signals together, it is possible to detect and locate the leak along the pipe. However, useful signals usually mix in the powerful backdrop signals and noises. In order to resolve the problem, the wavelet packet decomposition technique [2] is used to reduce the noises and get the feature signals of negative pressure wave and acoustic wave. Furthermore, a lot of different condition regulating signals for instance compressor start-stop, valve adjusting and gas turbulence can interfere with the accurate identification of leaks and result in false alarm. It is quite required to classify these similar signals. Thus, BP neural network [3] is used to quickly recognize the different pressure fluctuation signals. Finally, an integrated system developed by LabView is introduced to timely monitor the operation condition and locate the leak. Field tests indicate this system using negative pressure wave method, acoustic wave method, wavelet packet decomposition technique as well as BP network has a good effect.

Modulation instability (MI) is one possible mechanism to explain the formation of extreme waves in uni-directional and narrow-banded seas. It can be triggered, when side-bands around the main frequency are excited and subsequently follow an exponential growth. In physical domain this dynamics translates to periodic pulsations of wave groups that can reach heights up to three times the initial amplitude of the wave train. It is well-known that these periodic wave groups propagate with approximately half the waves phase speed in deep-water. We report an experimental study on modulationally unstable wave groups that propagate with a velocity that is higher than the group velocity since the modulation frequency is complex. It is shown that when this additional velocity to the wave groups is small a good agreement with exact nonlinear Schrödinger (NLS) models, that describe the nonlinear stage of MI, is reached. Otherwise a significant deviation is observed that could be compensated when increasing accuracy of the water wave modeling beyond NLS.

1.1 Macroeconomic summary Economic recovery has consistently outperformed the technical staff’s expectations following a steep decline in activity in the second quarter of 2020. At the same time, total and core inflation rates have fallen and remain at low levels, suggesting that a significant element of the reactivation of Colombia’s economy has been related to recovery in potential GDP. This would support the technical staff’s diagnosis of weak aggregate demand and ample excess capacity. The most recently available data on 2020 growth suggests a contraction in economic activity of 6.8%, lower than estimates from January’s Monetary Policy Report (-7.2%). High-frequency indicators suggest that economic performance was significantly more dynamic than expected in January, despite mobility restrictions and quarantine measures. This has also come amid declines in total and core inflation, the latter of which was below January projections if controlling for certain relative price changes. This suggests that the unexpected strength of recent growth contains elements of demand, and that excess capacity, while significant, could be lower than previously estimated. Nevertheless, uncertainty over the measurement of excess capacity continues to be unusually high and marked both by variations in the way different economic sectors and spending components have been affected by the pandemic, and by uneven price behavior. The size of excess capacity, and in particular the evolution of the pandemic in forthcoming quarters, constitute substantial risks to the macroeconomic forecast presented in this report. Despite the unexpected strength of the recovery, the technical staff continues to project ample excess capacity that is expected to remain on the forecast horizon, alongside core inflation that will likely remain below the target. Domestic demand remains below 2019 levels amid unusually significant uncertainty over the size of excess capacity in the economy. High national unemployment (14.6% for February 2021) reflects a loose labor market, while observed total and core inflation continue to be below 2%. Inflationary pressures from the exchange rate are expected to continue to be low, with relatively little pass-through on inflation. This would be compatible with a negative output gap. Excess productive capacity and the expectation of core inflation below the 3% target on the forecast horizon provide a basis for an expansive monetary policy posture. The technical staff’s assessment of certain shocks and their expected effects on the economy, as well as the presence of several sources of uncertainty and related assumptions about their potential macroeconomic impacts, remain a feature of this report. The coronavirus pandemic, in particular, continues to affect the public health environment, and the reopening of Colombia’s economy remains incomplete. The technical staff’s assessment is that the COVID-19 shock has affected both aggregate demand and supply, but that the impact on demand has been deeper and more persistent. Given this persistence, the central forecast accounts for a gradual tightening of the output gap in the absence of new waves of contagion, and as vaccination campaigns progress. The central forecast continues to include an expected increase of total and core inflation rates in the second quarter of 2021, alongside the lapse of the temporary price relief measures put in place in 2020. Additional COVID-19 outbreaks (of uncertain duration and intensity) represent a significant risk factor that could affect these projections. Additionally, the forecast continues to include an upward trend in sovereign risk premiums, reflected by higher levels of public debt that in the wake of the pandemic are likely to persist on the forecast horizon, even in the context of a fiscal adjustment. At the same time, the projection accounts for the shortterm effects on private domestic demand from a fiscal adjustment along the lines of the one currently being proposed by the national government. This would be compatible with a gradual recovery of private domestic demand in 2022. The size and characteristics of the fiscal adjustment that is ultimately implemented, as well as the corresponding market response, represent another source of forecast uncertainty. Newly available information offers evidence of the potential for significant changes to the macroeconomic scenario, though without altering the general diagnosis described above. The most recent data on inflation, growth, fiscal policy, and international financial conditions suggests a more dynamic economy than previously expected. However, a third wave of the pandemic has delayed the re-opening of Colombia’s economy and brought with it a deceleration in economic activity. Detailed descriptions of these considerations and subsequent changes to the macroeconomic forecast are presented below. The expected annual decline in GDP (-0.3%) in the first quarter of 2021 appears to have been less pronounced than projected in January (-4.8%). Partial closures in January to address a second wave of COVID-19 appear to have had a less significant negative impact on the economy than previously estimated. This is reflected in figures related to mobility, energy demand, industry and retail sales, foreign trade, commercial transactions from selected banks, and the national statistics agency’s (DANE) economic tracking indicator (ISE). Output is now expected to have declined annually in the first quarter by 0.3%. Private consumption likely continued to recover, registering levels somewhat above those from the previous year, while public consumption likely increased significantly. While a recovery in investment in both housing and in other buildings and structures is expected, overall investment levels in this case likely continued to be low, and gross fixed capital formation is expected to continue to show significant annual declines. Imports likely recovered to again outpace exports, though both are expected to register significant annual declines. Economic activity that outpaced projections, an increase in oil prices and other export products, and an expected increase in public spending this year account for the upward revision to the 2021 growth forecast (from 4.6% with a range between 2% and 6% in January, to 6.0% with a range between 3% and 7% in April). As a result, the output gap is expected to be smaller and to tighten more rapidly than projected in the previous report, though it is still expected to remain in negative territory on the forecast horizon. Wide forecast intervals reflect the fact that the future evolution of the COVID-19 pandemic remains a significant source of uncertainty on these projections. The delay in the recovery of economic activity as a result of the resurgence of COVID-19 in the first quarter appears to have been less significant than projected in the January report. The central forecast scenario expects this improved performance to continue in 2021 alongside increased consumer and business confidence. Low real interest rates and an active credit supply would also support this dynamic, and the overall conditions would be expected to spur a recovery in consumption and investment. Increased growth in public spending and public works based on the national government’s spending plan (Plan Financiero del Gobierno) are other factors to consider. Additionally, an expected recovery in global demand and higher projected prices for oil and coffee would further contribute to improved external revenues and would favor investment, in particular in the oil sector. Given the above, the technical staff’s 2021 growth forecast has been revised upward from 4.6% in January (range from 2% to 6%) to 6.0% in April (range from 3% to 7%). These projections account for the potential for the third wave of COVID-19 to have a larger and more persistent effect on the economy than the previous wave, while also supposing that there will not be any additional significant waves of the pandemic and that mobility restrictions will be relaxed as a result. Economic growth in 2022 is expected to be 3%, with a range between 1% and 5%. This figure would be lower than projected in the January report (3.6% with a range between 2% and 6%), due to a higher base of comparison given the upward revision to expected GDP in 2021. This forecast also takes into account the likely effects on private demand of a fiscal adjustment of the size currently being proposed by the national government, and which would come into effect in 2022. Excess in productive capacity is now expected to be lower than estimated in January but continues to be significant and affected by high levels of uncertainty, as reflected in the wide forecast intervals. The possibility of new waves of the virus (of uncertain intensity and duration) represents a significant downward risk to projected GDP growth, and is signaled by the lower limits of the ranges provided in this report. Inflation (1.51%) and inflation excluding food and regulated items (0.94%) declined in March compared to December, continuing below the 3% target. The decline in inflation in this period was below projections, explained in large part by unanticipated increases in the costs of certain foods (3.92%) and regulated items (1.52%). An increase in international food and shipping prices, increased foreign demand for beef, and specific upward pressures on perishable food supplies appear to explain a lower-than-expected deceleration in the consumer price index (CPI) for foods. An unexpected increase in regulated items prices came amid unanticipated increases in international fuel prices, on some utilities rates, and for regulated education prices. The decline in annual inflation excluding food and regulated items between December and March was in line with projections from January, though this included downward pressure from a significant reduction in telecommunications rates due to the imminent entry of a new operator. When controlling for the effects of this relative price change, inflation excluding food and regulated items exceeds levels forecast in the previous report. Within this indicator of core inflation, the CPI for goods (1.05%) accelerated due to a reversion of the effects of the VAT-free day in November, which was largely accounted for in February, and possibly by the transmission of a recent depreciation of the peso on domestic prices for certain items (electric and household appliances). For their part, services prices decelerated and showed the lowest rate of annual growth (0.89%) among the large consumer baskets in the CPI. Within the services basket, the annual change in rental prices continued to decline, while those services that continue to experience the most significant restrictions on returning to normal operations (tourism, cinemas, nightlife, etc.) continued to register significant price declines. As previously mentioned, telephone rates also fell significantly due to increased competition in the market. Total inflation is expected to continue to be affected by ample excesses in productive capacity for the remainder of 2021 and 2022, though less so than projected in January. As a result, convergence to the inflation target is now expected to be somewhat faster than estimated in the previous report, assuming the absence of significant additional outbreaks of COVID-19. The technical staff’s year-end inflation projections for 2021 and 2022 have increased, suggesting figures around 3% due largely to variation in food and regulated items prices. The projection for inflation excluding food and regulated items also increased, but remains below 3%. Price relief measures on indirect taxes implemented in 2020 are expected to lapse in the second quarter of 2021, generating a one-off effect on prices and temporarily affecting inflation excluding food and regulated items. However, indexation to low levels of past inflation, weak demand, and ample excess productive capacity are expected to keep core inflation below the target, near 2.3% at the end of 2021 (previously 2.1%). The reversion in 2021 of the effects of some price relief measures on utility rates from 2020 should lead to an increase in the CPI for regulated items in the second half of this year. Annual price changes are now expected to be higher than estimated in the January report due to an increased expected path for fuel prices and unanticipated increases in regulated education prices. The projection for the CPI for foods has increased compared to the previous report, taking into account certain factors that were not anticipated in January (a less favorable agricultural cycle, increased pressure from international prices, and transport costs). Given the above, year-end annual inflation for 2021 and 2022 is now expected to be 3% and 2.8%, respectively, which would be above projections from January (2.3% and 2,7%). For its part, expected inflation based on analyst surveys suggests year-end inflation in 2021 and 2022 of 2.8% and 3.1%, respectively. There remains significant uncertainty surrounding the inflation forecasts included in this report due to several factors: 1) the evolution of the pandemic; 2) the difficulty in evaluating the size and persistence of excess productive capacity; 3) the timing and manner in which price relief measures will lapse; and 4) the future behavior of food prices. Projected 2021 growth in foreign demand (4.4% to 5.2%) and the supposed average oil price (USD 53 to USD 61 per Brent benchmark barrel) were both revised upward. An increase in long-term international interest rates has been reflected in a depreciation of the peso and could result in relatively tighter external financial conditions for emerging market economies, including Colombia. Average growth among Colombia’s trade partners was greater than expected in the fourth quarter of 2020. This, together with a sizable fiscal stimulus approved in the United States and the onset of a massive global vaccination campaign, largely explains the projected increase in foreign demand growth in 2021. The resilience of the goods market in the face of global crisis and an expected normalization in international trade are additional factors. These considerations and the expected continuation of a gradual reduction of mobility restrictions abroad suggest that Colombia’s trade partners could grow on average by 5.2% in 2021 and around 3.4% in 2022. The improved prospects for global economic growth have led to an increase in current and expected oil prices. Production interruptions due to a heavy winter, reduced inventories, and increased supply restrictions instituted by producing countries have also contributed to the increase. Meanwhile, market forecasts and recent Federal Reserve pronouncements suggest that the benchmark interest rate in the U.S. will remain stable for the next two years. Nevertheless, a significant increase in public spending in the country has fostered expectations for greater growth and inflation, as well as increased uncertainty over the moment in which a normalization of monetary policy might begin. This has been reflected in an increase in long-term interest rates. In this context, emerging market economies in the region, including Colombia, have registered increases in sovereign risk premiums and long-term domestic interest rates, and a depreciation of local currencies against the dollar. Recent outbreaks of COVID-19 in several of these economies; limits on vaccine supply and the slow pace of immunization campaigns in some countries; a significant increase in public debt; and tensions between the United States and China, among other factors, all add to a high level of uncertainty surrounding interest rate spreads, external financing conditions, and the future performance of risk premiums. The impact that this environment could have on the exchange rate and on domestic financing conditions represent risks to the macroeconomic and monetary policy forecasts. Domestic financial conditions continue to favor recovery in economic activity. The transmission of reductions to the policy interest rate on credit rates has been significant. The banking portfolio continues to recover amid circumstances that have affected both the supply and demand for loans, and in which some credit risks have materialized. Preferential and ordinary commercial interest rates have fallen to a similar degree as the benchmark interest rate. As is generally the case, this transmission has come at a slower pace for consumer credit rates, and has been further delayed in the case of mortgage rates. Commercial credit levels stabilized above pre-pandemic levels in March, following an increase resulting from significant liquidity requirements for businesses in the second quarter of 2020. The consumer credit portfolio continued to recover and has now surpassed February 2020 levels, though overall growth in the portfolio remains low. At the same time, portfolio projections and default indicators have increased, and credit establishment earnings have come down. Despite this, credit disbursements continue to recover and solvency indicators remain well above regulatory minimums. 1.2 Monetary policy decision In its meetings in March and April the BDBR left the benchmark interest rate unchanged at 1.75%.

Macroeconomic summary The Colombian economy sustained numerous shocks in the second quarter, pri¬marily related to costs and supply. The majority of these shocks were unantic¬ipated or proved more persistent than expected, interrupting the recovery in economic activity observed at the beginning of the year and pushing overall inflation above the target. Core inflation (excluding food and regulated items) increased but remained low, in line with the technical staff’s expectations. A third wave of the pandemic, which became more severe and prolonged than the previous outbreak, began in early April. This had both a high cost in terms of human life and a negative impact on Colombia's economic recovery. Between May and mid-June roadblocks and other disruptions to public order had a sig¬nificant negative effect on economic activity and inflation. The combination and magnitude of these two shocks likely led to a decline in gross domestic product (GDP) compared to the first quarter. Roadblocks also led to a significant in¬crease in food prices. The accumulated effects of global disruptions to certain value chains and increased international freight transportation prices, which since the end of 2020 have restricted supply and increased costs, also affected Colombia’s economy. The factors described above, which primarily affected the consumer price index (CPI) for goods and foods, explain to a significant degree the technical staff’s forecast errors and the increase in overall inflation above the 3% target. By contrast, increases in core inflation and in prices for regulated items were in line with the technical staff’s expectations, and can be explained largely by the elimination of various price relief measures put in place last year. An increase in perceived sovereign risk and the upward pressures that this im¬plies on international financing costs and the exchange rate were further con¬siderations. Despite significant negative shocks, economic growth in the first half of the year (9.1%) is now expected to be significantly higher than projected in the April re¬port (7.1%), a sign of a more dynamic economy that could recover more quickly than previously forecast. Diverse economic activity figures have indicated high¬er-than-expected growth since the end of 2020. This suggests that the negative effects on output from recurring waves of COVID-19 have grown weaker and less long-lasting with subsequent outbreaks. Nevertheless, the third wave of the coro¬navirus, and to an even greater degree the previously mentioned roadblocks and disruptions to public order, likely led to a decline in GDP in the second quar¬ter compared to the first. Despite this, data from the monthly economic tracking indicator (ISE) for April and May surpassed expectations, and new sector-level measures of economic activity suggest that the negative impact of the pandemic on output continues to moderate, amid reduced restrictions on mobility and im¬provements in the pace of vaccination programs. Freight transportation registers (June) and unregulated energy demand (July), among other indicators, suggest a significant recovery following the roadblocks in May. Given the above, annual GDP growth in the second quarter is expected to have been around 17.3% (previously 15.8%), explained in large part by a low basis of comparison. The technical staff revised its growth projection for 2021 upward from 6% to 7.5%. This forecast, which comes with an unusually high degree of uncertain¬ty, assumes no additional disruptions to public order and that any new waves of COVID-19 will not have significant additional negative effects on economic activity. Recovery in international demand, price levels for some of Colombia’s export com¬modities, and remittances from workers abroad have all performed better than projected in the previous report. This dynamic is expected to continue to drive recovery in the national income over the rest of the year. Continued ample international liquidity, an acceleration in vacci¬nation programs, and low interest rates can also be ex¬pected to favor economic activity. Improved performance in the second quarter, which led to an upward growth revision for all components of spending, is expected to continue, with the economy returning to 2019 production levels at the end of 2021, earlier than estimated in the April report. This forecast continues to account for the short-term effects on aggregate demand of a tax reform package along the lines of what is currently being pro-posed by the national government. Given the above, the central forecast scenario in this report projects growth in 2021 of 7.5% and in 2022 of 3.1% (Graph 1.1). In this scenar¬io, economic activity would nonetheless remain below potential. The noted improvement in these projections comes with a high degree of uncertainty. Annual inflation increased more than expected in June (3.63%) as a result of changes in food prices, while growth in core inflation (1.87%) was similar to projections.

Macroeconomic summary Annual inflation continued to rise in the first quarter (8.5%) and again outpaced both market expectations and the technical staff’s projections. Inflation in major consumer price index (CPI) baskets has accelerated year-to-date, rising in March at an annual rate above 3%. Food prices (25.4%) continued to contribute most to rising inflation, mainly affected by a deterioration in external supply and rising costs of agricultural inputs. Increases in transportation prices and in some utility rates (energy and gas) can explain the acceleration in regulated items prices (8.3%). For its part, the increase in inflation excluding food and regulated items (4.5%) would be the result of shocks in supply and external costs that have been more persistent than expected, the effects of indexation, accumulated inflationary pressures from the exchange rate, and a faster-than-anticipated tightening of excess productive capacity. Within the basket excluding food and regulated items, external inflationary pressures have meaningfully impacted on goods prices (6.4%), which have been accelerating since the last quarter of 2021. Annual growth in services prices (3.8%) above the target rate is due primarily to food away from home (14.1%), which was affected by significant increases in food and utilities prices and by a rise in the legal monthly minimum wage. Housing rentals and other services prices also increased, though at rates below 3%. Forecast and expected inflation have increased and remain above the target rate, partly due to external pressures (prices and costs) that have been more persistent than projected in the January report (Graphs 1.1 and 1.2). Russia’s invasion of Ukraine accentuated inflationary pressures, particularly on international prices for certain agricultural goods and inputs, energy, and oil. The current inflation projection assumes international food prices will increase through the middle of this year, then remain high and relatively stable for the remainder of 2022. Recovery in the perishable food supply is forecast to be less dynamic than previously anticipated due to high agricultural input prices. Oil prices should begin to recede starting in the second half of the year, but from higher levels than those presented in the previous report. Given the above, higher forecast inflation could accentuate indexation effects and increase inflation expectations. The reversion of a rebate on value-added tax (VAT) applied to cleaning and hygiene products, alongside the end of Colombia’s COVID-19 health emergency, could increase the prices of those goods. The elimination of excess productive capacity on the forecast horizon, with an output gap close to zero and somewhat higher than projected in January, is another factor to consider. As a consequence, annual inflation is expected to remain at high levels through June. Inflation should then decline, though at a slower pace than projected in the previous report. The adjustment process of the monetary policy rate wouldcontribute to pushing inflation and its expectations toward the target on the forecast horizon. Year-end inflation for 2022 is expected to be around 7.1%, declining to 4.8% in 2023. Economic activity again outperformed expectations. The technical staff’s growth forecast for 2022 has been revised upward from 4.3% to 5% (Graph 1.3). Output increased more than expected in annual terms in the fourth quarter of 2021 (10.7%), driven by domestic demand that came primarily because of private consumption above pre-pandemic levels. Investment also registered a significant recovery without returning to 2019 levels and with mixed performance by component. The trade deficit increased, with significant growth in imports similar to that for exports. The economic tracking indicator (ISE) for January and February suggested that firstquarter output would be higher than previously expected and that the positive demand shock observed at the end of 2021 could be fading slower than anticipated. Imports in consumer goods, retail sales figures, real restaurant and hotel income, and credit card purchases suggest that household spending continues to be dynamic, with levels similar to those registered at the end of 2021. Project launch and housing starts figures and capital goods import data suggest that investment also continues to recover but would remain below pre-pandemic levels. Consumption growth is expected to decelerate over the year from high levels reached over the last two quarters. This would come amid tighter domestic and external financial conditions, the exhaustion of suppressed demand, and a deterioration of available household income due to increased inflation. Investment is expected to continue to recover, while the trade deficit should tighten alongside high oil and other export commodity prices. Given all of the above, first-quarter economic growth is now expected to be 7.2% (previously 5.2%) and 5.0% for 2022 as a whole (previously 4.3%). Output growth would continue to moderate in 2023 (2.9%, previously 3.1%), converging similar to long-term rates. The technical staff’s revised projections suggest that the output gap would remain at levels close to zero on the forecast horizon but be tighter than forecast in January (Graph 1.4). These estimates continue to be affected by significant uncertainty associated with geopolitical tensions, external financial conditions, Colombia’s electoral cycle, and the COVID-19 pandemic. External demand is now projected to grow at a slower pace than previously expected amid increased global inflationary pressures, high oil prices, and tighter international financial conditions than forecast in January. The Russian invasion of Ukraine and its inflationary effects on prices for oil and certain agricultural goods and inputs accentuated existing global inflationary pressures originating in supply restrictions and increased international costs. A decline in the supply of Russian oil, low inventory levels, and continued production limits on behalf of the Organization of Petroleum Exporting Countries and its allies (OPEC+) can explain increased projected oil prices for 2022 (USD 100.8/barrel, previously USD 75.3) and 2023 (USD 86.8/barrel, previously USD 71.2). The forecast trajectory for the U.S. Federal Reserve (Fed) interest rate has increased for this and next year to reflect higher real and expected inflation and positive performance in the labormarket and economic activity. The normalization of monetary policy in various developed and emerging market economies, more persistent supply and cost shocks, and outbreaks of COVID-19 in some Asian countries contributed to a reduction in the average growth outlook for Colombia’s trade partners for 2022 (2.8%, previously 3.3%) and 2023 (2.4%, previously 2.6%). In this context, the projected path for Colombia’s risk premium increased, partly due to increased geopolitical global tensions, less expansionary monetary policy in the United States, an increase in perceived risk for emerging markets, and domestic factors such as accumulated macroeconomic imbalances and political uncertainty. Given all the above, external financial conditions are tighter than projected in January report. External forecasts and their impact on Colombia’s macroeconomic scenario continue to be affected by considerable uncertainty, given the unpredictability of both the conflict between Russia and Ukraine and the pandemic. The current macroeconomic scenario, characterized by high real inflation levels, forecast and expected inflation above 3%, and an output gap close to zero, suggests an increased risk of inflation expectations becoming unanchored. This scenario offers very limited space for expansionary monetary policy. Domestic demand has been more dynamic than projected in the January report and excess productive capacity would have tightened more quickly than anticipated. Headline and core inflation rose above expectations, reflecting more persistent and important external shocks on supply and costs. The Russian invasion of Ukraine accentuated supply restrictions and pressures on international costs. This partly explains the increase in the inflation forecast trajectory to levels above the target in the next two years. Inflation expectations increased again and are above 3%. All of this increased the risk of inflation expectations becoming unanchored and could generate indexation effects that move inflation still further from the target rate. This macroeconomic context also implies reduced space for expansionary monetary policy. 1.2 Monetary policy decision Banco de la República’s board of directors (BDBR) continues to adjust its monetary policy. In its meetings both in March and April of 2022, it decided by majority to increase the monetary policy rate by 100 basis points, bringing it to 6.0% (Graph 1.5).