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Academic literature on the topic 'Degenerate wave equation'

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Published: 11 March 2023

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Journal articles on the topic "Degenerate wave equation"

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Balkizov, G. A. "Boundary value problems with data on opposite characteristics for a second-order mixed-hyperbolic equation." REPORTS ADYGE (CIRCASSIAN) INTERNATIONAL ACADEMY OF SCIENCES 20, no. 3 (2020): 6–13. http://dx.doi.org/10.47928/1726-9946-2020-20-3-6-13.

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Within the framework of this work, solutions of boundary value problems with data on “opposite” (“parallel”) characteristics are found for one mixed-hyperbolic equation consisting of a wave operator in one part of the domain and a degenerate hyperbolic Gellerstedt operator in the other part. It is known that problems with data on opposite (parallel) characteristics for the wave equation in the characteristic quadrangle are posed incorrectly. However, as shown in this paper, the solution of similar problems for a mixed-hyperbolic equation consisting of a wave operator in one part of the domain and a degenerate hyperbolic Gellerstedt operator with an order of degeneracy in the other part of the domain, under certain conditions on the given functions, exists, is unique and is written explicitly.
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Stuhlmeier, R., and M. Stiassnie. "Evolution of statistically inhomogeneous degenerate water wave quartets." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2111 (December 11, 2017): 20170101. http://dx.doi.org/10.1098/rsta.2017.0101.

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A discretized equation for the evolution of random surface wave fields on deep water is derived from Zakharov's equation, allowing for a general treatment of the stability and long-time behaviour of broad-banded sea states. It is investigated for the simple case of degenerate four-wave interaction, and the instability of statistically homogeneous states to small inhomogeneous disturbances is demonstrated. Furthermore, the long-time evolution is studied for several cases and shown to lead to a complex spatio-temporal energy distribution. The possible impact of this evolution on the statistics of freak wave occurrence is explored. This article is part of the theme issue ‘Nonlinear water waves’.
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Angelopoulos, Yannis, Stefanos Aretakis, and Dejan Gajic. "A Non-degenerate Scattering Theory for the Wave Equation on Extremal Reissner–Nordström." Communications in Mathematical Physics 380, no. 1 (September 23, 2020): 323–408. http://dx.doi.org/10.1007/s00220-020-03857-3.

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Abstract It is known that sub-extremal black hole backgrounds do not admit a (bijective) non-degenerate scattering theory in the exterior region due to the fact that the redshift effect at the event horizon acts as an unstable blueshift mechanism in the backwards direction in time. In the extremal case, however, the redshift effect degenerates and hence yields a much milder blueshift effect when viewed in the backwards direction. In this paper, we construct a definitive (bijective) non-degenerate scattering theory for the wave equation on extremal Reissner–Nordström backgrounds. We make use of physical-space energy norms which are non-degenerate both at the event horizon and at null infinity. As an application of our theory we present a construction of a large class of smooth, exponentially decaying modes. We also derive scattering results in the black hole interior region.
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Rani, Neelam, and Manikant Yadav. "The Nonlinear Magnetosonic Waves in Magnetized Dense Plasma for Quantum Effects of Degenerate Electrons." 4, no. 4 (December 10, 2021): 180–88. http://dx.doi.org/10.26565/2312-4334-2021-4-24.

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The nonlinear magnetosonic solitons are investigated in magnetized dense plasma for quantum effects of degenerate electrons in this research work. After reviewing the basic introduction of quantum plasma, we described the nonlinear phenomenon of magnetosonic wave. The reductive perturbation technique is employed for low frequency nonlinear magnetosonic waves in magnetized quantum plasma. In this paper, we have derived the Korteweg-de Vries (KdV) equation of magnetosonic solitons in a magnetized quantum plasma with degenerate electrons having arbitrary electron temperature. It is observed that the propagation of magnetosonic solitons in a magnetized dense plasma with the quantum effects of degenerate electrons and Bohm diffraction. The quantum or degeneracy effects become relevant in plasmas when fermi temperature and thermodynamic temperatures of degenerate electrons have same order.
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Sánchez-Garduño, Faustino, and Judith Pérez-Velázquez. "Reactive-Diffusive-Advective Traveling Waves in a Family of Degenerate Nonlinear Equations." Scientific World Journal 2016 (2016): 1–21. http://dx.doi.org/10.1155/2016/5620839.

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This paper deals with the analysis of existence of traveling wave solutions (TWS) for a diffusion-degenerate (atD(0)=0) and advection-degenerate (ath′(0)=0) reaction-diffusion-advection (RDA) equation. Diffusion is a strictly increasing function and the reaction term generalizes the kinetic part of the Fisher-KPP equation. We consider different forms of the convection termh(u):(1) h′(u)is constantk,(2) h′(u)=kuwithk>0, and(3)it is a quite general form which guarantees the degeneracy in the advective term. In Case 1, we prove that the task can be reduced to that for the corresponding equation, wherek=0, and then previous results reported from the authors can be extended. For the other two cases, we use both analytical and numerical tools. The analysis we carried out is based on the restatement of searching TWS for the full RDA equation into a two-dimensional dynamical problem. This consists of searching for the conditions on the parameter values for which there exist heteroclinic trajectories of the ordinary differential equations (ODE) system in the traveling wave coordinates. Throughout the paper we obtain the dynamics by using tools coming from qualitative theory of ODE.
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Xu, Tianyuan, Shanming Ji, Ming Mei, and Jingxue Yin. "Critical sharp front for doubly nonlinear degenerate diffusion equations with time delay." Nonlinearity 35, no. 7 (June 16, 2022): 3358–84. http://dx.doi.org/10.1088/1361-6544/ac72e8.

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Abstract This paper is concerned with the critical sharp travelling wave for doubly nonlinear diffusion equation with time delay, where the doubly nonlinear degenerate diffusion is defined by ( ( u m ) x p − 2 ( u m ) x ) x with m > 0 and p > 1. The doubly nonlinear diffusion equation is proved to admit a unique sharp type travelling wave for the degenerate case m(p − 1) > 1, the so-called slow-diffusion case. This sharp travelling wave associated with the minimal wave speed c*(m, p, r) is monotonically increasing, where the minimal wave speed satisfies c*(m, p, r) < c*(m, p, 0) for any time delay r > 0. The sharp front is C 1-smooth for 1 p − 1 < m < p p − 1 , and piecewise smooth for m ⩾ p p − 1 . Our results indicate that time delay slows down the minimal travelling wave speed for the doubly nonlinear degenerate diffusion equations. The approach adopted for proof is the phase transform method combining the variational method. The main technical issue for the proof is to overcome the obstacle caused by the doubly nonlinear degenerate diffusion.
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DING HONG-YU, WU BAI-SHI, YIN DA-JUN, and LI JIN-QUAN. "RATE EQUATION THEORY OF DEGENERATE FOUR WAVE MIXING." Acta Physica Sinica 37, no. 3 (1988): 408. http://dx.doi.org/10.7498/aps.37.408.

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MEDVEDEV, G. S., K. ONO, and P. J. HOLMES. "Travelling wave solutions of the degenerate KolmogorovPetrovskiPiskunov equation." European Journal of Applied Mathematics 14, no. 3 (June 2003): 343–67. http://dx.doi.org/10.1017/s0956792503005102.

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Yuan, Xu. "Construction of excited multi-solitons for the 5D energy-critical wave equation." Journal of Hyperbolic Differential Equations 18, no. 02 (June 2021): 397–434. http://dx.doi.org/10.1142/s0219891621500120.

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For the 5D energy-critical wave equation, we construct excited [Formula: see text]-solitons with collinear speeds, i.e. solutions [Formula: see text] of the equation such that [Formula: see text] where for [Formula: see text], [Formula: see text] is the Lorentz transform of a non-degenerate and sufficiently decaying excited state, each with different but collinear speeds. The existence proof follows the ideas of Martel–Merle [Construction of multi-solitons for the energy-critical wave equation in dimension 5, Arch. Ration. Mech. Anal. 222(3) (2016) 1113–1160] and Côte–Martel [Multi-travelling waves for the nonlinear Klein–Gordon equation, Trans. Amer. Math. Soc. 370(10) (2018) 7461–7487] developed for the energy-critical wave and nonlinear Klein–Gordon equations. In particular, we rely on an energy method and on a general coercivity property for the linearized operator.
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Chouaou, Fatiha, Chahira Aichi, and Abbes Benaissa. "Decay estimates for a degenerate wave equation with a dynamic fractional feedback acting on the degenerate boundary." Filomat 35, no. 10 (2021): 3219–39. http://dx.doi.org/10.2298/fil2110219c.

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In this paper, we consider a one-dimensional weakly degenerate wave equation with a dynamic nonlocal boundary feedback of fractional type acting at a degenerate point. First We show well-posedness by using the semigroup theory. Next, we show that our system is not uniformly stable by spectral analysis. Hence, we look for a polynomial decay rate for a smooth initial data by using a result due Borichev and Tomilov which reduces the problem of estimating the rate of energy decay to finding a growth bound for the resolvent of the generator associated with the semigroup. This analysis proves that the degeneracy affect the energy decay rates.
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Dissertations / Theses on the topic "Degenerate wave equation"

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Civin, Damon. "Stability of charged rotating black holes for linear scalar perturbations." Thesis, University of Cambridge, 2015. https://www.repository.cam.ac.uk/handle/1810/247397.

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In this thesis, the stability of the family of subextremal Kerr-Newman space- times is studied in the case of linear scalar perturbations. That is, nondegenerate energy bounds (NEB) and integrated local energy decay (ILED) results are proved for solutions of the wave equation on the domain of outer communications. The main obstacles to the proof of these results are superradiance, trapping and their interaction. These difficulties are surmounted by localising solutions of the wave equation in phase space and applying the vector field method. Miraculously, as in the Kerr case, superradiance and trapping occur in disjoint regions of phase space and can be dealt with individually. Trapping is a high frequency obstruction to the proof whereas superradiance occurs at both high and low frequencies. The construction of energy currents for superradiant frequencies gives rise to an unfavourable boundary term. In the high frequency regime, this boundary term is controlled by exploiting the presence of a large parameter. For low superradiant frequencies, no such parameter is available. This difficulty is overcome by proving quantitative versions of mode stability type results. The mode stability result on the real axis is then applied to prove integrated local energy decay for solutions of the wave equation restricted to a bounded frequency regime. The (ILED) statement is necessarily degenerate due to the trapping effect. This implies that a nondegenerate (ILED) statement must lose differentiability. If one uses an (ILED) result that loses differentiability to prove (NEB), this loss is passed onto the (NEB) statement as well. Here, the geometry of the subextremal Kerr-Newman background is exploited to obtain the (NEB) statement directly from the degenerate (ILED) with no loss of differentiability.
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Sánchez, Garduño Faustino. "Travelling waves in one-dimensional degenerate non-linear reaction-diffussion equations." Thesis, University of Oxford, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.334929.

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Karimou, Gazibo Mohamed. "Etudes mathématiques et numériques des problèmes paraboliques avec des conditions aux limites." Phd thesis, Université de Franche-Comté, 2013. http://tel.archives-ouvertes.fr/tel-00950759.

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Cette thèse est centrée autour de l'étude théorique et de l'analyse numérique des équations paraboliques non linéaires avec divers conditions aux limites. La première partie est consacrée aux équations paraboliques dégénérées mêlant des phénomènes non-linéaires de diffusion et de transport. Nous définissons des notions de solutions entropiques adaptées pour chacune des conditions aux limites (flux nul, Robin, Dirichlet). La difficulté principale dans l'étude de ces problèmes est due au manque de régularité du flux pariétal pour traiter les termes de bords. Ceci pose un problème pour la preuve d'unicité. Pour y remédier, nous tirons profit du fait que ces résultats de régularités sur le bord sont plus faciles à obtenir pour le problème stationnaire et particulièrement en dimension un d'espace. Ainsi par la méthode de comparaison "fort-faible" nous arrivons à déduire l'unicité avec le choix d'une fonction test non symétrique et en utilisant la théorie des semi-groupes non linéaires. L'existence de solution se démontre en deux étapes, combinant la méthode de régularisation parabolique et les approximations de Galerkin. Nous développons ensuite une approche directe en construisant des solutions approchées par un schéma de volumes finis implicite en temps. Dans les deux cas, on combine les estimations dans les espaces fonctionnels bien choisis avec des arguments de compacité faible ou forte et diverses astuces permettant de passer à la limite dans des termes non linéaires. Notamment, nous introduisons une nouvelle notion de solution appelée solution processus intégrale dont l'objectif, dans le cadre de notre étude, est de pallier à la difficulté de prouver la convergence vers une solution entropique d'un schéma volumes finis pour le problème de flux nul au bord. La deuxième partie de cette thèse traite d'un problème à frontière libre décrivant la propagation d'un front de combustion et l'évolution de la température dans un milieu hétérogène. Il s'agit d'un système d'équations couplées constitué de l'équation de la chaleur bidimensionnelle et d'une équation de type Hamilton-Jacobi. L'objectif de cette partie est de construire un schéma numérique pour ce problème en combinant des discrétisations du type éléments finis avec les différences finies. Ceci nous permet notamment de vérifier la convergence de la solution numérique vers une solution onde pour un temps long. Dans un premier temps, nous nous intéressons à l'étude d'un problème unidimensionnel. Très vite, nous nous heurtons à un problème de stabilité du schéma. Cela est dû au problème de prise en compte de la condition de Neumann au bord. Par une technique de changement d'inconnue et d'approximation nous remédions à ce problème. Ensuite, nous adaptons cette technique pour la résolution du problème bidimensionnel. A l'aide d'un changement de variables, nous obtenons un domaine fixe facile pour la discrétisation. La monotonie du schéma obtenu est prouvée sous une hypothèse supplémentaire de propagation monotone qui exige que la frontière libre se déplace dans les directions d'un cône prescrit à l'avance.
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Hsu, Wei-wen, and 許維文. "The Behavior of Solutions for Some Degenerate Quasilinear Wave Equations." Thesis, 2007. http://ndltd.ncl.edu.tw/handle/06358046749418249041.

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碩士
國立中央大學
數學研究所
95
In this paper we consider the Cauchy problem of some degenerate quasilinear wave equations. We first study the behavior of solutions to the linear degenerate wave equation. We obtain the -stability of solutions for the linear case just by the d''Almbert formula. To the nonlinear degenerate case, the Lax method and Glimm method in hyperbolic systems of conservation laws are used to construct the approximate solution of Cauchy problem in the first time step. As we demonstrate in this paper, the total variation of approximate solution may go to infinity due to the degeneracy of equation. We will do the case study for the behavior of solutions for some particular case of degenerate quasilinear wave equations.
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Book chapters on the topic "Degenerate wave equation"

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Garrett, Steven L. "Three-Dimensional Enclosures." In Understanding Acoustics, 621–72. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-44787-8_13.

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Abstract In this chapter, solutions to the wave equation that satisfies the boundary conditions within three-dimensional enclosures of different shapes are derived. This treatment is very similar to the two-dimensional solutions for waves on a membrane of Chap. 10.1007/978-3-030-44787-8_6. Many of the concepts introduced in Sect. 10.1007/978-3-030-44787-8_6#Sec1 for rectangular membranes and Sect. 10.1007/978-3-030-44787-8_6#Sec5 for circular membranes are repeated here with only slight modifications. These concepts include separation of variables, normal modes, modal degeneracy, and density of modes, as well as adiabatic invariance and the splitting of degenerate modes by perturbations. Throughout this chapter, familiarity with the results of Chap. 10.1007/978-3-030-44787-8_6 will be assumed. The similarities between the standing-wave solutions within enclosures of different shapes are stressed. At high enough frequencies, where the individual modes overlap, statistical energy analysis will be introduced to describe the diffuse (reverberant) sound field.
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Li, Zhiyan, Linghui Liu, Jingguo Qu, and Yuhuan Cui. "Superconvergence Analysis of Anisotropic Finite Element Method for a Kind of Nonlinear Degenerate Wave Equation." In Communications in Computer and Information Science, 341–47. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-16339-5_45.

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Airapetyan, Ruben, and Ingo Witt. "Propagation of Smoothness for Edge-degenerate Wave Equations." In Hyperbolic Problems: Theory, Numerics, Applications, 11–18. Basel: Birkhäuser Basel, 2001. http://dx.doi.org/10.1007/978-3-0348-8370-2_2.

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Sánchez-Garduño, Faustino, and Philip K. Maini. "Wave Patterns in One-Dimensional Nonlinear Degenerate Diffusion Equations." In Experimental and Theoretical Advances in Biological Pattern Formation, 83–86. Boston, MA: Springer US, 1993. http://dx.doi.org/10.1007/978-1-4615-2433-5_10.

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Danilov, V. G., V. P. Maslov, and K. A. Volosov. "Wave Asymptotic Solutions of Degenerate Semilinear Parabolic and Hyperbolic Equations." In Mathematical Modelling of Heat and Mass Transfer Processes, 127–200. Dordrecht: Springer Netherlands, 1995. http://dx.doi.org/10.1007/978-94-011-0409-8_5.

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Okamoto, Hisashi. "Applications of degenerate bifurcation equations to the taylor problem and the water wave problem." In The Navier-Stokes Equations Theory and Numerical Methods, 117–27. Berlin, Heidelberg: Springer Berlin Heidelberg, 1990. http://dx.doi.org/10.1007/bfb0086062.

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Betancourt, Fernando, Raimund Bürger, and Kenneth H. Karlsen. "Well-posedness and Travelling Wave Analysis for a Strongly Degenerate Parabolic Aggregation Equation." In Series in Contemporary Applied Mathematics, 312–19. Co-Published with Higher Education Press, 2012. http://dx.doi.org/10.1142/9789814417099_0027.

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"NONLINEAR WAVE EQUATIONS WITH DEGENERATE DAMPING AND SOURCE TERMS." In Control and Boundary Analysis, 73–82. CRC Press, 2005. http://dx.doi.org/10.1201/9781420027426-9.

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Rammaha, Mohammad, Irena Lasiecka, and Viorel Barbu. "Nonlinear Wave Equations With Degenerate Damping and Source Terms." In Control and Boundary Analysis, 53–62. CRC Press, 2005. http://dx.doi.org/10.1201/9781420027426.ch4.

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Nakao, Mitsuhiro. "Energy Decay for Nonlinear Wave Equations with Degenerate Dissipative Terms." In Studies in Mathematics and Its Applications, 583–96. Elsevier, 1986. http://dx.doi.org/10.1016/s0168-2024(08)70147-4.

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Conference papers on the topic "Degenerate wave equation"

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Nikolov, Aleksey, and Nedyu Popivanov. "Singular solutions to Protter's problem for (3+1)-D degenerate wave equation." In APPLICATIONS OF MATHEMATICS IN ENGINEERING AND ECONOMICS (AMEE '12): Proceedings of the 38th International Conference Applications of Mathematics in Engineering and Economics. AIP, 2012. http://dx.doi.org/10.1063/1.4766790.

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Nikolov, Aleksey, and Nedyu Popivanov. "Asymptotic expansion of singular solutions to Protter problem for (2+1)-D degenerate wave equation." In 39TH INTERNATIONAL CONFERENCE APPLICATIONS OF MATHEMATICS IN ENGINEERING AND ECONOMICS AMEE13. AIP, 2013. http://dx.doi.org/10.1063/1.4854763.

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Kazakov, A. L., P. A. Kuznetsov, and L. F. Spevak. "A heat wave problem for a degenerate nonlinear parabolic equation with a specified source function." In MECHANICS, RESOURCE AND DIAGNOSTICS OF MATERIALS AND STRUCTURES (MRDMS-2018): Proceedings of the 12th International Conference on Mechanics, Resource and Diagnostics of Materials and Structures. Author(s), 2018. http://dx.doi.org/10.1063/1.5084385.

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Novosadov, B. K. "On degenerate bispinor states of the Dirac wave equation system for an electron in an electrostatic field." In SPIE Proceedings, edited by Yuri I. Ozhigov. SPIE, 2008. http://dx.doi.org/10.1117/12.801909.

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Wang, S. L., and H. F. Pan. "Photorefractive four wave mixing and photorefractive circuits." In Nonlinear Optics. Washington, D.C.: Optica Publishing Group, 1992. http://dx.doi.org/10.1364/nlo.1992.md17.

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Degenerate four wave mixing in photorefractive media(PFWM) has been intensively studied theoretically. A complete solution of the coupled wave equation is steel not available. This makes some problems such as the uncertainty of the solution, lack of understanding of phase dependence and energy flux, and unable to treat those of multi coupled interaction regions. This paper is to give a general solution so that the PFWM can be described as a phase sensitive eight pin element arid apply it to various PFWM circuits by composing the elements for different uses.
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Simpson, T. B., and J. M. Liu. "Four-Wave Mixing and Optical Modulation in Laser Diodes." In Nonlinear Optics. Washington, D.C.: Optica Publishing Group, 1992. http://dx.doi.org/10.1364/nlo.1992.we4.

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Nearly-degenerate four-wave mixing in laser oscillators has attracted much recent attention.[1-4] In laser diodes, the interaction has usually been described by models emphasizing the spatial dependence of the nonlinear coupling.[2] Alternatively, injection locking of laser diodes is most often described by models emphasizing the optical resonance of the laser cavity and the time dependence of the interaction.[5] Both situations, however, represent nearly identical experimental configurations and little effort has been made to connect the two pictures. Recent four-wave mixing experiments in an argon-ion laser have been modeled using the lumped-circuit rate-equation formalism[3] and an earlier rate-equation analysis was shown to be consistent with laser diode four-wave mixing data.[4]
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Norwood, D. P., H. E. Swoboda, M. D. Dawson, T. F. Boggess, A. L. Smirl, and T. C. Hasenberg. "Femtosecond degenerate four-wave mixing in multiple quantum wells at 300 K." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1990. http://dx.doi.org/10.1364/oam.1990.thee2.

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We have investigated the decay of optically-induced, short-period (≅120 nm) gratings in GaAs/AlGaAs multiple-quantum-well structures with a barrier thickness LB of 1.4-15 nm. The crystals were excited with 100 fs pulses at wavelengths either resonant with the n = 1 exciton or into the bands. The decay of the diffraction efficiency of the gratings shows two components: a fast initial contribution that is independent of lb and has a decay time of 200-300 fs and a slower exponential contribution that depends on lb and has decay times between 1.5-90 ps. The latter component is determined by perpendicular transport by means of over-barrier hopping and phonon-assisted tunneling among adjacent wells and is consistent with a simple-rate-equation model based on these transport mechanisms and the recombination of the excited carriers. The fast initial component is observed only for resonant excitation and may be connected with the exciton ionization and/or reorientation. To investigate this feature, we performed two-beam self-diffraction experiments with cross-polarized pulses. Here, we observed signals in the background-free direction consistent with self-diffraction from an orientational grating, whereas we saw a fast increase of the transmission in the direction of the transmitted beam
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Haus, Joseph W., M. Scalora, L. Wang, and C. M. Bowden. "Longitudinal and transverse inhomogeneities in a nonlinear oscillator." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1987. http://dx.doi.org/10.1364/oam.1987.tuj9.

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We consider a nonlinear medium whose polarization P is described by a nonlinear oscillator equation.1,2 The steady-state fields are analytically solved for plane waves in a slowly varying envelope approximation. We find the conditions for the appearance of a boundary in the medium separating a high-polarization branch of stationary solutions from a low-polarization branch. We show how this boundary can be probed by the reflected intensity and by degenerate four-wave mixing spectroscopy, and results are given for these cases. We also use a numerical procedure to solve the steady-state equations which include one transverse dimension. There is an interesting change in the on-axis behavior related to the self-focusing effects for Fresnel numbers of order unity.
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Simpson, T. B., and J. M. Liu. "Saturation of four-wave mixing signals in laser diodes." In OSA Annual Meeting. Washington, D.C.: Optica Publishing Group, 1992. http://dx.doi.org/10.1364/oam.1992.tuz21.

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Under very weak, nearly degenerate optical injection, laser diodes undergo amplitude and phase modulations that can be modeled by using the lumped-circuit rate equation approach. Population pulsations of the free carriers induce four-wave mixing between the injected optical field and the oscillating laser diode field. By measuring the amplitude modulation signal, regeneratively amplified input signal, and four-wave mixing signal, key diode parameters such as the carrier lifetime, relaxation resonance, and anti-guiding factor can be calculated. As the injection field is increased above this small signal regime the four-wave mixing signal saturates and broadens. Over certain detuning ranges, where the broadening is most pronounced, the nonlinear interaction can induce a longitudinal mode hop. Depending on the frequency difference between the injected optical field and the near-resonant laser mode, the laser diode will oscillate on one of the two longitudinal modes. Associated with this bistable operation is the appearance of a resonantly enhanced highly nondegenerate four-wave mixing signal on a third longitudinal mode.
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Jana, Anirban, and Arvind Raman. "Aeroelastic Flutter of a Rotating Disk in an Unbounded Compressible Inviscid Fluid." In ASME 2003 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2003. http://dx.doi.org/10.1115/detc2003/vib-48466.

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The aeroelastic flutter of an unbaffled flexible disk rotating in an unbounded fluid is investigated, by modeling the disk-fluid system as a rotating Kirchhoff plate coupled to irrotational flow of a compressible inviscid fluid. The fluid motions are governed by the wave equation of linear acoustics. Fluid-structure coupling is achieved between the disk and the fluid by means of the fluid loading on the disk and the velocity matching boundary conditions on the disk surface. A perturbed eigenvalue formulation is used to compute systematically the coupled system eigenvalues. A series solution is presented for the dual integral equations, arising from the mixed boundary value problem governing the fluid motions. It is found that two distinct aerodynamic effects occur — radiation damping into the surrounding fluid and added fluid inertia effect. Provided the disk has zero material damping, the radiation damping causes the flutter speed to coincide with the critical speed. This flutter instability is a degenerate bifurcation with eigenvalues crossing into the right half plane through the origin with zero speed. The added fluid inertia effect modifies the frequencies of the traveling waves but does not affect the critical speed.
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