Articles de revues : « Hydrodynamic integrable systems »

1

Kupershmidt, Boris A. « Noncommutative Integrable Systems of Hydrodynamic Type ». Acta Applicandae Mathematicae 92, n^o 3 (21 septembre 2006) : 269–92. http://dx.doi.org/10.1007/s10440-006-9054-1.

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Fava, Michele, Sounak Biswas, Sarang Gopalakrishnan, Romain Vasseur et S. A. Parameswaran. « Hydrodynamic nonlinear response of interacting integrable systems ». Proceedings of the National Academy of Sciences 118, n^o 37 (7 septembre 2021) : e2106945118. http://dx.doi.org/10.1073/pnas.2106945118.

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We develop a formalism for computing the nonlinear response of interacting integrable systems. Our results are asymptotically exact in the hydrodynamic limit where perturbing fields vary sufficiently slowly in space and time. We show that spatially resolved nonlinear response distinguishes interacting integrable systems from noninteracting ones, exemplifying this for the Lieb–Liniger gas. We give a prescription for computing finite-temperature Drude weights of arbitrary order, which is in excellent agreement with numerical evaluation of the third-order response of the XXZ spin chain. We identify intrinsically nonperturbative regimes of the nonlinear response of integrable systems.

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Prykarpatskyy, Yarema. « On the Integrable Chaplygin Type Hydrodynamic Systems and Their Geometric Structure ». Symmetry 12, n^o 5 (1 mai 2020) : 697. http://dx.doi.org/10.3390/sym12050697.

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A class of spatially one-dimensional completely integrable Chaplygin hydrodynamic systems was studied within framework of Lie-algebraic approach. The Chaplygin hydrodynamic systems were considered as differential systems on the torus. It has been shown that the geometric structure of the systems under analysis has strong relationship with diffeomorphism group orbits on them. It has allowed to find a new infinite hierarchy of integrable Chaplygin like hydrodynamic systems.

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Mokhov, O. I. « Integrable bi-Hamiltonian systems of hydrodynamic type ». Russian Mathematical Surveys 57, n^o 1 (28 février 2002) : 153–54. http://dx.doi.org/10.1070/rm2002v057n01abeh000483.

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Ferapontov, E. V., et A. P. Fordy. « Separable Hamiltonians and integrable systems of hydrodynamic type ». Journal of Geometry and Physics 21, n^o 2 (janvier 1997) : 169–82. http://dx.doi.org/10.1016/s0393-0440(96)00013-7.

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Odesskii, A. V., et V. V. Sokolov. « Integrable (2+1)-dimensional systems of hydrodynamic type ». Theoretical and Mathematical Physics 163, n^o 2 (mai 2010) : 549–86. http://dx.doi.org/10.1007/s11232-010-0043-1.

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7

Kodama, Y., et B. G. Konopelchenko. « Confluence of hypergeometric functions and integrable hydrodynamic-type systems ». Theoretical and Mathematical Physics 188, n^o 3 (septembre 2016) : 1334–57. http://dx.doi.org/10.1134/s0040577916090051.

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8

Dubrovin, B. A. « Differential geometry of strongly integrable systems of hydrodynamic type ». Functional Analysis and Its Applications 24, n^o 4 (1991) : 280–85. http://dx.doi.org/10.1007/bf01077332.

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El, Gennady A. « Soliton gas in integrable dispersive hydrodynamics ». Journal of Statistical Mechanics : Theory and Experiment 2021, n^o 11 (1 novembre 2021) : 114001. http://dx.doi.org/10.1088/1742-5468/ac0f6d.

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Abstract We review the spectral theory of soliton gases in integrable dispersive hydrodynamic systems. We first present a phenomenological approach based on the consideration of phase shifts in pairwise soliton collisions and leading to the kinetic equation for a non-equilibrium soliton gas. Then, a more detailed theory is presented in which soliton gas dynamics are modelled by a thermodynamic type limit of modulated finite-gap spectral solutions of the Korteweg–de Vries and the focusing nonlinear Schrödinger (NLS) equations. For the focusing NLS equation the notions of soliton condensate and breather gas are introduced that are related to the phenomena of spontaneous modulational instability and the rogue wave formation. The integrability properties of the kinetic equation for soliton gas are discussed and some physically relevant solutions are presented and compared with direct numerical simulations of dispersive hydrodynamic systems.

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10

De Nardis, Jacopo, Benjamin Doyon, Marko Medenjak et Miłosz Panfil. « Correlation functions and transport coefficients in generalised hydrodynamics ». Journal of Statistical Mechanics : Theory and Experiment 2022, n^o 1 (1 janvier 2022) : 014002. http://dx.doi.org/10.1088/1742-5468/ac3658.

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Abstract We review the recent advances on exact results for dynamical correlation functions at large scales and related transport coefficients in interacting integrable models. We discuss Drude weights, conductivity and diffusion constants, as well as linear and nonlinear response on top of equilibrium and non-equilibrium states. We consider the problems from the complementary perspectives of the general hydrodynamic theory of many-body systems, including hydrodynamic projections, and form-factor expansions in integrable models, and show how they provide a comprehensive and consistent set of exact methods to extract large scale behaviours. Finally, we overview various applications in integrable spin chains and field theories.

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Bonnemain, Thibault, Benjamin Doyon et Gennady El. « Generalized hydrodynamics of the KdV soliton gas ». Journal of Physics A : Mathematical and Theoretical 55, n^o 37 (19 août 2022) : 374004. http://dx.doi.org/10.1088/1751-8121/ac8253.

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Abstract We establish the explicit correspondence between the theory of soliton gases in classical integrable dispersive hydrodynamics, and generalized hydrodynamics (GHD), the hydrodynamic theory for many-body quantum and classical integrable systems. This is done by constructing the GHD description of the soliton gas for the Korteweg–de Vries equation. We further predict the exact form of the free energy density and flux, and of the static correlation matrices of conserved charges and currents, for the soliton gas. For this purpose, we identify the solitons’ statistics with that of classical particles, and confirm the resulting GHD static correlation matrices by numerical simulations of the soliton gas. Finally, we express conjectured dynamical correlation functions for the soliton gas by simply borrowing the GHD results. In principle, other conjectures are also immediately available, such as diffusion and large-deviation functions for fluctuations of soliton transport.

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Kokotov, A., et D. Korotkin. « A new hierarchy of integrable systems associated to Hurwitz spaces ». Philosophical Transactions of the Royal Society A : Mathematical, Physical and Engineering Sciences 366, n^o 1867 (22 juin 2007) : 1055–88. http://dx.doi.org/10.1098/rsta.2007.2061.

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In this paper, we introduce a new class of integrable systems, naturally associated to Hurwitz spaces (spaces of meromorphic functions over Riemann surfaces). The critical values of the meromorphic functions play the role of ‘times’. Our systems give a natural generalization of the Ernst equation; in genus zero, they realize the scheme of deformation of integrable systems proposed by Burtsev, Mikhailov and Zakharov. We show that any solution of these systems in rank 1 defines a flat diagonal metric (Darboux–Egoroff metric) together with a class of corresponding systems of hydrodynamic type and their solutions.

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Artemovych, Orest D., Denis Blackmore et Anatolij K. Prykarpatski. « Poisson brackets, Novikov-Leibniz structures and integrable Riemann hydrodynamic systems ». Journal of Nonlinear Mathematical Physics 24, n^o 1 (22 décembre 2016) : 41–72. http://dx.doi.org/10.1080/14029251.2016.1274114.

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Manno, Gianni, et Maxim V. Pavlov. « Hydrodynamic-type systems describing 2-dimensional polynomially integrable geodesic flows ». Journal of Geometry and Physics 113 (mars 2017) : 197–205. http://dx.doi.org/10.1016/j.geomphys.2016.10.023.

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Bogdanov, L. V., et B. G. Konopelchenko. « Symmetry constraints for dispersionless integrable equations and systems of hydrodynamic type ». Physics Letters A 330, n^o 6 (octobre 2004) : 448–59. http://dx.doi.org/10.1016/j.physleta.2004.08.024.

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Błaszak, Maciej, et Wen-Xiu Ma. « Separable Hamiltonian equations on Riemann manifolds and related integrable hydrodynamic systems ». Journal of Geometry and Physics 47, n^o 1 (juillet 2003) : 21–42. http://dx.doi.org/10.1016/s0393-0440(02)00173-0.

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Pavlov, Maxim V. « Algebro-Geometric Approach in the Theory of Integrable Hydrodynamic Type Systems ». Communications in Mathematical Physics 272, n^o 2 (3 avril 2007) : 469–505. http://dx.doi.org/10.1007/s00220-007-0235-1.

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Ferapontov, E. V. « Isoparametric hypersurfaces in spheres, integrable nondiagonalizable systems of hydrodynamic type, and N-wave systems ». Differential Geometry and its Applications 5, n^o 4 (décembre 1995) : 335–69. http://dx.doi.org/10.1016/0926-2245(95)00022-4.

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Konopelchenko, B. G., et W. K. Schief. « Integrable discretization of hodograph-type systems, hyperelliptic integrals and Whitham equations ». Proceedings of the Royal Society A : Mathematical, Physical and Engineering Sciences 470, n^o 2172 (8 décembre 2014) : 20140514. http://dx.doi.org/10.1098/rspa.2014.0514.

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Based on the well-established theory of discrete conjugate nets in discrete differential geometry, we propose and examine discrete analogues of important objects and notions in the theory of semi-Hamiltonian systems of hydrodynamic type. In particular, we present discrete counterparts of (generalized) hodograph equations, hyperelliptic integrals and associated cycles, characteristic speeds of Whitham-type and (implicitly) the corresponding Whitham equations. By construction, the intimate relationship with integrable system theory is maintained in the discrete setting.

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20

Ferapontov, E. V., et A. Moro. « Dispersive deformations of hydrodynamic reductions of (2 + 1)D dispersionless integrable systems ». Journal of Physics A : Mathematical and Theoretical 42, n^o 3 (9 décembre 2008) : 035211. http://dx.doi.org/10.1088/1751-8113/42/3/035211.

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21

Doyon, Benjamin, et Jason Myers. « Fluctuations in Ballistic Transport from Euler Hydrodynamics ». Annales Henri Poincaré 21, n^o 1 (15 novembre 2019) : 255–302. http://dx.doi.org/10.1007/s00023-019-00860-w.

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AbstractWe propose a general formalism, within large-deviation theory, giving access to the exact statistics of fluctuations of ballistically transported conserved quantities in homogeneous, stationary states. The formalism is expected to apply to any system with an Euler hydrodynamic description, classical or quantum, integrable or not, in or out of equilibrium. We express the exact scaled cumulant generating function (or full counting statistics) for any (quasi-)local conserved quantity in terms of the flux Jacobian. We show that the “extended fluctuation relations” of Bernard and Doyon follow from the linearity of the hydrodynamic equations, forming a marker of “freeness” much like the absence of hydrodynamic diffusion does. We show how an extension of the formalism gives exact exponential behaviours of spatio-temporal two-point functions of twist fields, with applications to order-parameter dynamical correlations in arbitrary homogeneous, stationary state. We explain in what situations the large-deviation principle at the basis of the results fail, and discuss how this connects with nonlinear fluctuating hydrodynamics. Applying the formalism to conformal hydrodynamics, we evaluate the exact cumulants of energy transport in quantum critical systems of arbitrary dimension at low but nonzero temperatures, observing a phase transition for Lorentz boosts at the sound velocity.

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Ferapontov, E. V., et K. R. Khusnutdinova. « The characterization of two-component (2+1)-dimensional integrable systems of hydrodynamic type ». Journal of Physics A : Mathematical and General 37, n^o 8 (11 février 2004) : 2949–63. http://dx.doi.org/10.1088/0305-4470/37/8/007.

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Umurzhakhova, Zh B., M. D. Koshanova, Zh Pashen et K. R. Yesmakhanova. « QUASICLASSICAL LIMIT OF THE SCHRÖDINGER-MAXWELL- BLOCH EQUATIONS ». PHYSICO-MATHEMATICAL SERIES 2, n^o 336 (15 avril 2021) : 179–84. http://dx.doi.org/10.32014/2021.2518-1726.39.

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The study of integrable equations is one of the most important aspects of modern mathematical and theoretical physics. Currently, there are a large number of nonlinear integrable equations that have a physical application. The concept of nonlinear integrable equations is closely related to solitons. An object being in a nonlinear medium that maintains its shape at moving, as well as when interacting with its own kind, is called a soliton or a solitary wave. In many physical processes, nonlinearity is closely related to the concept of dispersion. Soliton solutions have dispersionless properties. Connection with the fact that the nonlinear component of the equation compensates for the dispersion term. In addition to integrable nonlinear differential equations, there is also an important class of integrable partial differential equations (PDEs), so-called the integrable equations of hydrodynamic type or dispersionless (quasiclassical) equations [1-13]. Nonlinear dispersionless equations arise as a dispersionless (quasiclassical) limit of known integrable equations. In recent years, the study of dispersionless systems has become of great importance, since they arise as a result of the analysis of various problems, such as physics, mathematics, and applied mathematics, from the theory of quantum fields and strings to the theory of conformal mappings on the complex plane. Well-known classical methods of the theory of intrinsic systems are used to study dispersionless equations. In this paper, we present the quasicalassical limit of the system of (1+1)-dimensional Schrödinger-Maxwell- Bloch (NLS-MB) equations. The system of the NLS-MB equations is one of the classic examples of the theory of nonlinear integrable equations. The NLS-MB equations describe the propagation of optical solitons in fibers with resonance and doped with erbium. And we will also show the integrability of the quasiclassical limit of the NLS-MB using the obtained Lax representation.

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Durnin, Joseph, Andrea De Luca, Jacopo De Nardis et Benjamin Doyon. « Diffusive hydrodynamics of inhomogenous Hamiltonians ». Journal of Physics A : Mathematical and Theoretical 54, n^o 49 (24 novembre 2021) : 494001. http://dx.doi.org/10.1088/1751-8121/ac2c57.

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Abstract We derive a large-scale hydrodynamic equation, including diffusive and dissipative effects, for systems with generic static position-dependent driving forces coupling to local conserved quantities. We show that this equation predicts entropy increase and thermal states as the only stationary states. The equation applies to any hydrodynamic system with any number of local, parity and time-symmetric conserved quantities, in arbitrary dimension. It is fully expressed in terms of elements of an extended Onsager matrix. In integrable systems, this matrix admits an expansion in the density of excitations. We evaluate exactly its two-particle–hole contribution, which dominates at low density, in terms of the scattering phase and dispersion of the quasiparticles, giving a lower bound for the extended Onsager matrix and entropy production. We conclude with a molecular dynamics simulation, demonstrating thermalisation over diffusive time scales in the Toda interacting particle model with an inhomogeneous energy field.

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Ferapontov, E. V., A. V. Odesskii et N. M. Stoilov. « Classification of integrable two-component Hamiltonian systems of hydrodynamic type in 2 + 1 dimensions ». Journal of Mathematical Physics 52, n^o 7 (juillet 2011) : 073505. http://dx.doi.org/10.1063/1.3602081.

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Makridin, Z. V. « An Effective Algorithm for Finding Multidimensional Conservation Laws for Integrable Systems of Hydrodynamic Type ». Theoretical and Mathematical Physics 194, n^o 2 (février 2018) : 274–83. http://dx.doi.org/10.1134/s0040577918020071.

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Spohn, Herbert. « Hydrodynamic equations for the Ablowitz–Ladik discretization of the nonlinear Schrödinger equation ». Journal of Mathematical Physics 63, n^o 3 (1 mars 2022) : 033305. http://dx.doi.org/10.1063/5.0075670.

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Ablowitz and Ladik discovered a discretization that preserves the integrability of the nonlinear Schrödinger equation in one dimension. We compute the generalized free energy of this model and determine the generalized Gibbs ensemble averaged fields and their currents. They are linked to the mean-field circular unitary matrix ensemble. The resulting hydrodynamic equations follow the pattern already known from other integrable many-body systems. The discretized modified Korteweg–de-Vries equation is also studied, which turns out to be related to the beta Jacobi log gas.

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Alba, Vincenzo, Bruno Bertini, Maurizio Fagotti, Lorenzo Piroli et Paola Ruggiero. « Generalized-hydrodynamic approach to inhomogeneous quenches : correlations, entanglement and quantum effects ». Journal of Statistical Mechanics : Theory and Experiment 2021, n^o 11 (1 novembre 2021) : 114004. http://dx.doi.org/10.1088/1742-5468/ac257d.

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Abstract We give a pedagogical introduction to the generalized hydrodynamic approach to inhomogeneous quenches in integrable many-body quantum systems. We review recent applications of the theory, focusing in particular on two classes of problems: bipartitioning protocols and trap quenches, which represent two prototypical examples of broken translational symmetry in either the system initial state or post-quench Hamiltonian. We report on exact results that have been obtained for generic time-dependent correlation functions and entanglement evolution, and discuss in detail the range of applicability of the theory. Finally, we present some open questions and suggest perspectives on possible future directions.

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Prykarpatski, Anatolij K. « Quantum Current Algebra in Action : Linearization, Integrability of Classical and Factorization of Quantum Nonlinear Dynamical Systems ». Universe 8, n^o 5 (20 mai 2022) : 288. http://dx.doi.org/10.3390/universe8050288.

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This review is devoted to the universal algebraic and geometric properties of the non-relativistic quantum current algebra symmetry and to their representations subject to applications in describing geometrical and analytical properties of quantum and classical integrable Hamiltonian systems of theoretical and mathematical physics. The Fock space, the non-relativistic quantum current algebra symmetry and its cyclic representations on separable Hilbert spaces are reviewed and described in detail. The unitary current algebra family of operators and generating functional equations are described. A generating functional method to constructing irreducible current algebra representations is reviewed, and the ergodicity of the corresponding representation Hilbert space measure is mentioned. The algebraic properties of the so called coherent states are also reviewed, generated by cyclic representations of the Heisenberg algebra on Hilbert spaces. Unbelievable and impressive applications of coherent states to the theory of nonlinear dynamical systems on Hilbert spaces are described, along with their linearization and integrability. Moreover, we present a further development of these results within the modern Lie-algebraic approach to nonlinear dynamical systems on Poissonian functional manifolds, which proved to be both unexpected and important for the classification of integrable Hamiltonian flows on Hilbert spaces. The quantum current Lie algebra symmetry properties and their functional representations, interpreted as a universal algebraic structure of symmetries of completely integrable nonlinear dynamical systems of theoretical and mathematical physics on functional manifolds, are analyzed in detail. Based on the current algebra symmetry structure and their functional representations, an effective integrability criterion is formulated for a wide class of completely integrable Hamiltonian systems on functional manifolds. The related algebraic structure of the Poissonian operators and an effective algorithm of their analytical construction are described. The current algebra representations in separable Hilbert spaces and the factorized structure of quantum integrable many-particle Hamiltonian systems are reviewed. The related current algebra-based Hamiltonian reconstruction of the many-particle oscillatory and Calogero–Moser–Sutherland quantum models are reviewed and discussed in detail. The related quasi-classical quantum current algebra density representations and the collective variable approach in equilibrium statistical physics are reviewed. In addition, the classical Wigner type current algebra representation and its application to non-equilibrium classical statistical mechanics are described, and the construction of the Lie–Poisson structure on the phase space of the infinite hierarchy of distribution functions is presented. The related Boltzmann–Bogolubov type kinetic equation for the generating functional of many-particle distribution functions is constructed, and the invariant reduction scheme, compatible with imposed correlation functions constraints, is suggested and analyzed in detail. We also review current algebra functional representations and their geometric structure subject to the analytical description of quasi-stationary hydrodynamic flows and their magneto-hydrodynamic generalizations. A unified geometric description of the ideal idiabatic liquid dynamics is presented, and its Hamiltonian structure is analyzed. A special chapter of the review is devoted to recent results on the description of modified current Lie algebra symmetries on torus and their Lie-algebraic structures, related to integrable so-called heavenly type spatially many-dimensional dynamical systems on functional manifolds.

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Ferapontov, E. V. « Dupin hypersurfaces and integrable hamiltonian systems of hydrodynamic type, which do not possess Riemann invariants ». Differential Geometry and its Applications 5, n^o 2 (juin 1995) : 121–52. http://dx.doi.org/10.1016/0926-2245(95)00011-r.

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31

Procopio, Giuseppe, et Massimiliano Giona. « Stochastic Modeling of Particle Transport in Confined Geometries : Problems and Peculiarities ». Fluids 7, n^o 3 (12 mars 2022) : 105. http://dx.doi.org/10.3390/fluids7030105.

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The equivalence between parabolic transport equations for solute concentrations and stochastic dynamics for solute particle motion represents one of the most fertile correspondences in statistical physics originating from the work by Einstein on Brownian motion. In this article, we analyze the problems and the peculiarities of the stochastic equations of motion in microfluidic confined systems. The presence of solid boundaries leads to tensorial hydrodynamic coefficients (hydrodynamic resistance matrix) that depend also on the particle position. Singularity issues, originating from the non-integrable divergence of the entries of the resistance matrix near a solid no-slip boundary, determine some mass-transport paradoxes whenever surface phenomena, such as surface chemical reactions at the walls, are considered. These problems can be overcome by considering the occurrence of non vanishing slippage. Added-mass effects and the influence of fluid inertia in confined geometries are also briefly addressed.

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BUROVSKIY, P. A., E. V. FERAPONTOV et S. P. TSAREV. « SECOND-ORDER QUASILINEAR PDEs AND CONFORMAL STRUCTURES IN PROJECTIVE SPACE ». International Journal of Mathematics 21, n^o 06 (juin 2010) : 799–841. http://dx.doi.org/10.1142/s0129167x10006215.

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We investigate second-order quasilinear equations of the form fijuxixj = 0, where u is a function of n independent variables x1, …, xn, and the coefficients fij depend on the first-order derivatives p1 = ux1, …, pn = uxn only. We demonstrate that the natural equivalence group of the problem is isomorphic to SL(n + 1, R), which acts by projective transformations on the space Pn with coordinates p1, …, pn. The coefficient matrix fij defines on Pn a conformal structure fij(p)dpidpj. The necessary and sufficient conditions for the integrability of such equations by the method of hydrodynamic reductions are derived, implying that the moduli space of integrable equations is 20-dimensional. Any equation satisfying the integrability conditions is necessarily conservative, and possesses a dispersionless Lax pair. The integrability conditions imply that the conformal structure fij(p) dpidpj is conformally flat, and possesses infinitely many three-conjugate null coordinate systems parametrized by three arbitrary functions of one variable. Integrable equations provide examples of such conformal structures parametrized by elementary functions, elliptic functions and modular forms.

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Zakharov, Vladimir E. « Description of the $n$ -orthogonal curvilinear coordinate systems and Hamiltonian integrable systems of hydrodynamic type, I : Integration of the Lamé equations ». Duke Mathematical Journal 94, n^o 1 (juillet 1998) : 103–39. http://dx.doi.org/10.1215/s0012-7094-98-09406-6.

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Błaszak, Maciej, et Artur Sergyeyev. « A coordinate-free construction of conservation laws and reciprocal transformations for a class of integrable hydrodynamic-type systems ». Reports on Mathematical Physics 64, n^o 1-2 (août 2009) : 341–54. http://dx.doi.org/10.1016/s0034-4877(09)90038-6.

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Ferapontov, E. V., et O. I. Mokhov. « The associativity equations in the two-dimensional topological field theory as integrable Hamiltonian nondiagonalizable systems of hydrodynamic type ». Functional Analysis and Its Applications 30, n^o 3 (juillet 1996) : 195–203. http://dx.doi.org/10.1007/bf02509506.

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Ferapontov, E. V. « On the matrix Hopf equation and integrable Hamiltonian systems of hydrodynamic type, which do not possess Riemann invariants ». Physics Letters A 179, n^o 6 (août 1993) : 391–97. http://dx.doi.org/10.1016/0375-9601(93)90096-i.

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Moore, Joel E. « A perspective on quantum integrability in many-body-localized and Yang–Baxter systems ». Philosophical Transactions of the Royal Society A : Mathematical, Physical and Engineering Sciences 375, n^o 2108 (30 octobre 2017) : 20160429. http://dx.doi.org/10.1098/rsta.2016.0429.

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Two of the most active areas in quantum many-particle dynamics involve systems with an unusually large number of conservation laws. Many-body-localized systems generalize ideas of Anderson localization by disorder to interacting systems. While localization still exists with interactions and inhibits thermalization, the interactions between conserved quantities lead to some dramatic differences from the Anderson case. Quantum integrable models such as the XXZ spin chain or Bose gas with delta-function interactions also have infinite sets of conservation laws, again leading to modifications of conventional thermalization. A practical way to treat the hydrodynamic evolution from local equilibrium to global equilibrium in such models is discussed. This paper expands upon a presentation at a discussion meeting of the Royal Society on 7 February 2017. The work described was carried out with a number of collaborators, including Jens Bardarson, Vir Bulchandani, Roni Ilan, Christoph Karrasch, Siddharth Parameswaran, Frank Pollmann and Romain Vasseur. This article is part of the themed issue ‘Breakdown of ergodicity in quantum systems: from solids to synthetic matter’.

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Ferapontov, E. V. « Several conjectures and results in the theory of integrable Hamiltonian systems of hydrodynamic type, which do not possess Riemann invariants ». Theoretical and Mathematical Physics 99, n^o 2 (mai 1994) : 567–70. http://dx.doi.org/10.1007/bf01016140.

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Kambe, Tsutomu. « Geometrical Aspects in Hydrodynamics and Integrable Systems ». Theoretical and Computational Fluid Dynamics 10, n^o 1-4 (1 janvier 1998) : 249–61. http://dx.doi.org/10.1007/s001620050062.

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MARSHAKOV, A. « EXACT SOLUTIONS TO QUANTUM FIELD THEORIES AND INTEGRABLE EQUATIONS ». Modern Physics Letters A 11, n^o 14 (10 mai 1996) : 1169–83. http://dx.doi.org/10.1142/s021773239600120x.

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The exact solutions to quantum string and gauge field theories are discussed and their formulation in the framework of integrable systems is presented. In particular we consider in detail several examples of the appearance of solutions to the first-order integrable equations of hydrodynamical type and stress that all known examples can be treated as partial solutions to the same problem in the theory of integrable systems.

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Cubero, Axel Cortés, Takato Yoshimura et Herbert Spohn. « Form factors and generalized hydrodynamics for integrable systems ». Journal of Statistical Mechanics : Theory and Experiment 2021, n^o 11 (1 novembre 2021) : 114002. http://dx.doi.org/10.1088/1742-5468/ac2eda.

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Bulchandani, Vir B. « On classical integrability of the hydrodynamics of quantum integrable systems ». Journal of Physics A : Mathematical and Theoretical 50, n^o 43 (3 octobre 2017) : 435203. http://dx.doi.org/10.1088/1751-8121/aa8c62.

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Bastianello, Alvise, Bruno Bertini, Benjamin Doyon et Romain Vasseur. « Introduction to the Special Issue on Emergent Hydrodynamics in Integrable Many-Body Systems ». Journal of Statistical Mechanics : Theory and Experiment 2022, n^o 1 (1 janvier 2022) : 014001. http://dx.doi.org/10.1088/1742-5468/ac3e6a.

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Prykarpatsky, Yarema A., Ilona Urbaniak, Radosław A. Kycia et Anatolij K. Prykarpatski. « Dark Type Dynamical Systems : The Integrability Algorithm and Applications ». Algorithms 15, n^o 8 (28 juillet 2022) : 266. http://dx.doi.org/10.3390/a15080266.

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Based on a devised gradient-holonomic integrability testing algorithm, we analyze a class of dark type nonlinear dynamical systems on spatially one-dimensional functional manifolds possessing hidden symmetry properties and allowing their linearization on the associated cotangent spaces. We described main spectral properties of nonlinear Lax type integrable dynamical systems on periodic functional manifolds particular within the classical Floquet theory, as well as we presented the determining functional relationships between the conserved quantities and related geometric Poisson and recursion structures on functional manifolds. For evolution flows on functional manifolds, parametrically depending on additional functional variables, naturally related with the classical Bellman-Pontriagin optimal control problem theory, we studied a wide class of nonlinear dynamical systems of dark type on spatially one-dimensional functional manifolds, which are both of diffusion and dispersion classes and can have interesting applications in modern physics, optics, mechanics, hydrodynamics and biology sciences. We prove that all of these dynamical systems possess rich hidden symmetry properties, are Lax type linearizable and possess finite or infinite hierarchies of suitably ordered conserved quantities.

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De Nardis, Jacopo, Denis Bernard et Benjamin Doyon. « Hydrodynamic Diffusion in Integrable Systems ». Physical Review Letters 121, n^o 16 (17 octobre 2018). http://dx.doi.org/10.1103/physrevlett.121.160603.

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Doyon, Benjamin. « Lecture notes on Generalised Hydrodynamics ». SciPost Physics Lecture Notes, 28 août 2020. http://dx.doi.org/10.21468/scipostphyslectnotes.18.

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These are lecture notes for a series of lectures given at the Les Houches Summer School on Integrability in Atomic and Condensed Matter Physics, 30 July to 24 August 2018. The same series of lectures has also been given at the Tokyo Institute of Technology, October 2019. I overview in a pedagogical fashion the main aspects of the theory of generalised hydrodynamics, a hydrodynamic theory for quantum and classical many-body integrable systems. Only very basic knowledge of hydrodynamics and integrable systems is assumed.

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Perfetto, Gabriele, et Benjamin Doyon. « Euler-scale dynamical fluctuations in non-equilibrium interacting integrable systems ». SciPost Physics 10, n^o 5 (27 mai 2021). http://dx.doi.org/10.21468/scipostphys.10.5.116.

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We derive an exact formula for the scaled cumulant generating function of the time-integrated current associated to an arbitrary ballistically transported conserved charge. Our results rely on the Euler-scale description of interacting, many-body, integrable models out of equilibrium given by the generalized hydrodynamics, and on the large deviation theory. Crucially, our findings extend previous studies by accounting for inhomogeneous and dynamical initial states in interacting systems. We present exact expressions for the first three cumulants of the time-integrated current. Considering the non-interacting limit of our general expression for the scaled cumulant generating function, we further show that for the partitioning protocol initial state our result coincides with previous results of the literature. Given the universality of the generalized hydrodynamics, the expression obtained for the scaled cumulant generating function is applicable to any interacting integrable model obeying the hydrodynamic equations, both classical and quantum.

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VU, Dinh-Long, et Takato Yoshimura. « Equations of state in generalized hydrodynamics ». SciPost Physics 6, n^o 2 (15 février 2019). http://dx.doi.org/10.21468/scipostphys.6.2.023.

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We, for the first time, report a first-principle proof of the equations of state used in the hydrodynamic theory for integrable systems, termed generalized hydrodynamics (GHD). The proof makes full use of the graph theoretic approach to Thermodynamic Bethe ansatz (TBA) that was proposed recently. This approach is purely combinatorial and relies only on common structures shared among Bethe solvable models, suggesting universal applicability of the method. To illustrate the idea of the proof, we focus on relativistic integrable quantum field theories with diagonal scatterings and without bound states such as strings.

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Ferapontov, E. V., et M. V. Pavlov. « Kinetic Equation for Soliton Gas : Integrable Reductions ». Journal of Nonlinear Science 32, n^o 2 (4 mars 2022). http://dx.doi.org/10.1007/s00332-022-09782-0.

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AbstractMacroscopic dynamics of soliton gases can be analytically described by the thermodynamic limit of the Whitham equations, yielding an integro-differential kinetic equation for the density of states. Under a delta-functional ansatz, the kinetic equation for soliton gas reduces to a non-diagonalisable system of hydrodynamic type whose matrix consists of several $$2\times 2$$ 2 × 2 Jordan blocks. Here we demonstrate the integrability of this system by showing that it possesses a hierarchy of commuting hydrodynamic flows and can be solved by an extension of the generalised hodograph method. Our approach is a generalisation of Tsarev’s theory of diagonalisable systems of hydrodynamic type to quasilinear systems with non-trivial Jordan block structure.

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Fagotti, Maurizio. « Locally quasi-stationary states in noninteracting spin chains ». SciPost Physics 8, n^o 3 (27 mars 2020). http://dx.doi.org/10.21468/scipostphys.8.3.048.

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Locally quasi-stationary states (LQSS) were introduced as inhomogeneous generalisations of stationary states in integrable systems. Roughly speaking, LQSSs look like stationary states, but only locally. Despite their key role in hydrodynamic descriptions, an unambiguous definition of LQSSs was not given. By solving the dynamics in inhomogeneous noninteracting spin chains, we identify the set of LQSSs as a subspace that is invariant under time evolution, and we explicitly construct the latter in a generalised XY model. As a by-product, we exhibit an exact generalised hydrodynamic theory (including ``quantum corrections'').

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