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Journal articles on the topic 'Degenerate wave equation'

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Author: Grafiati

Published: 11 March 2023

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1

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

MANSOUR, M. B. A. "TRAVELLING WAVE SOLUTIONS FOR DOUBLY DEGENERATE REACTION–DIFFUSION EQUATIONS." ANZIAM Journal 52, no. 1 (July 2010): 101–9. http://dx.doi.org/10.1017/s144618111100054x.

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AbstractThis paper concerns a nonlinear doubly degenerate reaction–diffusion equation which appears in a bacterial growth model and is also of considerable mathematical interest. A travelling wave analysis for the equation is carried out. In particular, the qualitative behaviour of both sharp and smooth travelling wave solutions is analysed. This travelling wave behaviour is also verified by some numerical computations for a special case.

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12

ELIASSON, BENGT, and PADMA KANT SHUKLA. "Dispersion properties of electrostatic oscillations in quantum plasmas." Journal of Plasma Physics 76, no. 1 (October 27, 2009): 7–17. http://dx.doi.org/10.1017/s0022377809990316.

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AbstractWe present a derivation of the dispersion relation for electrostatic oscillations in a zero-temperature quantum plasma, in which degenerate electrons are governed by the Wigner equation, while non-degenerate ions follow the classical fluid equations. The Poisson equation determines the electrostatic wave potential. We consider parameters ranging from semiconductor plasmas to metallic plasmas and electron densities of compressed matter such as in laser compression schemes and dense astrophysical objects. Owing to the wave diffraction caused by overlapping electron wave function because of the Heisenberg uncertainty principle in dense plasmas, we have the possibility of Landau damping of the high-frequency electron plasma oscillations at large enough wavenumbers. The exact dispersion relations for the electron plasma oscillations are solved numerically and compared with the ones obtained by using approximate formulas for the electron susceptibility in the high- and low-frequency cases.

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13

Campos, Juan, Andrea Corli, and Luisa Malaguti. "Saturated Fronts in Crowds Dynamics." Advanced Nonlinear Studies 21, no. 2 (February 2, 2021): 303–26. http://dx.doi.org/10.1515/ans-2021-2118.

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Abstract We consider a degenerate scalar parabolic equation, in one spatial dimension, of flux-saturated type. The equation also contains a convective term. We study the existence and regularity of traveling-wave solutions; in particular we show that they can be discontinuous. Uniqueness is recovered by requiring an entropy condition, and entropic solutions turn out to be the vanishing-diffusion limits of traveling-wave solutions to the equation with an additional non-degenerate diffusion. Applications to crowds dynamics, which motivated the present research, are also provided.

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14

Kannai, Y., and S. Kiro. "The initial value problem for a degenerate wave equation." Proceedings of the American Mathematical Society 104, no. 1 (January 1, 1988): 125. http://dx.doi.org/10.1090/s0002-9939-1988-0958055-6.

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15

Li, Zongguang, and Rui Liu. "Bifurcations and Exact Solutions in a Nonlinear Wave Equation." International Journal of Bifurcation and Chaos 29, no. 07 (June 30, 2019): 1950098. http://dx.doi.org/10.1142/s0218127419500986.

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The dynamical model of a nonlinear wave is governed by a partial differential equation which is a special case of the [Formula: see text]-family equation. Its traveling system is a singular system with a singular straight line. On this line, there exist two degenerate nodes of the associated regular system. By using the method of dynamical systems and the theory of singular traveling wave systems, in this paper we show that, corresponding to global level curves, this wave equation has global periodic wave solutions and anti-solitary wave solutions. We obtain their exact representations. Specially, we discover some new phenomena. (i) Infinitely many periodic orbits of the traveling wave system pass through the singular straight line. (ii) Inside some homoclinic orbits of the traveling wave system there is not any singular point. (iii) There exist periodic wave bifurcation and double anti-solitary waves bifurcation.

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16

Holzegel, Gustav, Jonathan Luk, Jacques Smulevici, and Claude Warnick. "Asymptotic Properties of Linear Field Equations in Anti-de Sitter Space." Communications in Mathematical Physics 374, no. 2 (November 4, 2019): 1125–78. http://dx.doi.org/10.1007/s00220-019-03601-6.

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Abstract We study the global dynamics of the wave equation, Maxwell’s equation and the linearized Bianchi equations on a fixed anti-de Sitter (AdS) background. Provided dissipative boundary conditions are imposed on the dynamical fields we prove uniform boundedness of the natural energy as well as both degenerate (near the AdS boundary) and non-degenerate integrated decay estimates. Remarkably, the non-degenerate estimates “lose a derivative”. We relate this loss to a trapping phenomenon near the AdS boundary, which itself originates from the properties of (approximately) gliding rays near the boundary. Using the Gaussian beam approximation we prove that non-degenerate energy decay without loss of derivatives does not hold. As a consequence of the non-degenerate integrated decay estimates, we also obtain pointwise-in-time decay estimates for the energy. Our paper provides the key estimates for a proof of the non-linear stability of the anti-de Sitter spacetime under dissipative boundary conditions. Finally, we contrast our results with the case of reflecting boundary conditions.

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17

Wu, Jiu Hui, Lin Zhang, and Ke jiang Zhou. "A novel kind of equations linking the quantum dynamics and the classical wave motions based on the catastrophe theory." Europhysics Letters 136, no. 4 (November 1, 2021): 40004. http://dx.doi.org/10.1209/0295-5075/ac2b5b.

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Abstract Considering that the catastrophe theory could describe quantitatively any phase transition process, we adopt the folding and cusp catastrophe types as the potential functions in the Schrödinger equation to attempt to link the quantum dynamics and the classical wave motions. Thus, through the dimensionless analysis a novel kind of partial differential equations is derived out. When the scaling parameter of the novel equation is equal to the Planck's constant, this equation becomes a detailed time-independent Schrödinger equation, from which Bohr correspondence principle can be found. On the other hand, when the scaling parameter tends to zero, this equation could degenerate to the classical Helmholtz equation. Therefore, this novel kind of equations could describe quantitatively the variation process of the wave functions from the macroscopic level to the quantum size.

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18

Schleicher, Jörg, and Jessé C. Costa. "A separable strong-anisotropy approximation for pure qP-wave propagation in transversely isotropic media." GEOPHYSICS 81, no. 6 (November 2016): C337—C354. http://dx.doi.org/10.1190/geo2016-0138.1.

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The wave equation can be tailored to describe wave propagation in vertical-symmetry axis transversely isotropic (VTI) media. The qP- and qS-wave eikonal equations derived from the VTI wave equation indicate that in the pseudoacoustic approximation, their dispersion relations degenerate into a single one. Therefore, when using this dispersion relation for wave simulation, for instance, by means of finite-difference approximations, both events are generated. To avoid the occurrence of the pseudo-S-wave, the qP-wave dispersion relation alone needs to be approximated. This can be done with or without the pseudoacoustic approximation. A Padé expansion of the exact qP-wave dispersion relation leads to a very good approximation. Our implementation of a separable version of this equation in the mixed space-wavenumber domain permits it to be compared with a low-rank solution of the exact qP-wave dispersion relation. Our numerical experiments showed that this approximation can provide highly accurate wavefields, even in strongly anisotropic inhomogeneous media.

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19

Wang, Qinlong, Yu'e Xiong, Wentao Huang, and Valery Romanovski. "Isolated periodic wave trains in a generalized Burgers–Huxley equation." Electronic Journal of Qualitative Theory of Differential Equations, no. 4 (2022): 1–16. http://dx.doi.org/10.14232/ejqtde.2022.1.4.

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We study the isolated periodic wave trains in a class of modified generalized Burgers–Huxley equation. The planar systems with a degenerate equilibrium arising after the traveling transformation are investigated. By finding certain positive definite Lyapunov functions in the neighborhood of the degenerate singular points and the Hopf bifurcation points, the number of possible limit cycles in the corresponding planar systems is determined. The existence of isolated periodic wave trains in the equation is established, which is universal for any positive integer n in this model. Within the process, one interesting example is obtained, namely a series of limit cycles bifurcating from a semi-hyperbolic singular point with one zero eigenvalue and one non-zero eigenvalue for its Jacobi matrix.

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20

Gürses, Metin, and Aslı Pekcan. "Traveling wave solutions of degenerate coupled Korteweg-de Vries equation." Journal of Mathematical Physics 55, no. 9 (September 2014): 091501. http://dx.doi.org/10.1063/1.4893636.

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21

Fu, Ying, and Changzheng Qu. "Well-posedness and wave breaking of the degenerate Novikov equation." Journal of Differential Equations 263, no. 8 (October 2017): 4634–57. http://dx.doi.org/10.1016/j.jde.2017.05.027.

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22

Zhu, Wen-Jing, Ai-Yong Chen, and Qi-Huai Liu. "Periodic Wave, Solitary Wave and Compacton Solutions of a Nonlinear Wave Equation with Degenerate Dispersion." Communications in Theoretical Physics 63, no. 1 (January 2015): 57–62. http://dx.doi.org/10.1088/0253-6102/63/1/10.

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23

HO, CHOON-LIN, and YUTAKA HOSOTANI. "ANYON EQUATION ON A TORUS." International Journal of Modern Physics A 07, no. 23 (September 20, 1992): 5797–831. http://dx.doi.org/10.1142/s0217751x92002647.

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Starting from the quantum field theory of nonrelativistic matter on a torus interacting with Chern-Simons gauge fields, we derive the Schrödinger equation for an anyon system. The nonintegrable phases of the Wilson line integrals on a torus play an essential role. In addition to generating degenerate vacua, they enter in the definition of a many-body Schrödinger wave function in quantum mechanics, which can be defined as a regular function of the coordinates of anyons. It obeys a non-Abelian representation of the braid group algebra, being related to Einarsson’s wave function by a singular gauge transformation.

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24

Aassila, Mohammed. "Global existence of solutions to degenerate wave equations with dissipative terms." Bulletin of the Australian Mathematical Society 60, no. 1 (August 1999): 1–10. http://dx.doi.org/10.1017/s000497270003327x.

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25

Deka, Manoj Kr, and Apul N. Dev. "Solitary Wave with Quantisation of Electron’s Orbit in a Magnetised Plasma in the Presence of Heavy Negative Ions." Zeitschrift für Naturforschung A 75, no. 3 (March 26, 2020): 211–23. http://dx.doi.org/10.1515/zna-2019-0296.

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AbstractThe propagation characteristics of solitary wave in a degenerate plasma in the presence of Landau-quantised magnetic field and heavy negative ion are studied. The nature of solitary wave in such plasma under the influence of magnetic quantisation and the concentration of both electrons and negative ions, as well as in the presence of degenerate temperature, are studied with the help of a time-independent analytical scheme of the solution of Zakharov–Kuznetsov equation. The electron density, as well as the magnetic quantisation parameter, has an outstanding effect on the features of solitary wave proliferation in such plasma. Interestingly, for any fixed electron density, the magnetic quantisation parameter has an equal control on the maximum height and dispersive properties of the solitary wave. Toward higher temperatures and higher magnetic fields, the width of the solitary wave decreases. For a lower magnetic field, the maximum amplitude of the solitary wave decreases rapidly at higher values of degenerate temperature and negative ion concentration; however, at a lower value of degenerate temperature, the maximum amplitude increases with increasing negative ion concentration.

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26

JOSEPH, ANCEMMA, and K. PORSEZIAN. "PERIODIC WAVE SOLUTIONS TO MODIFIED NONLINEAR SCHRÖDINGER EQUATION PERTAINING TO NEGATIVE INDEX MATERIALS." Journal of Nonlinear Optical Physics & Materials 19, no. 01 (March 2010): 177–87. http://dx.doi.org/10.1142/s0218863510005005.

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In this paper, we intend to determine periodic wave solutions for the modified nonlinear Schrödinger equation pertaining to negative index materials. We have treated the propagation equation possessing higher order linear and nonlinear dispersion terms with Jacobian elliptic function expansion method and arrived at the Jacobian elliptic periodic wave solutions. When the module of the Jacobian elliptic function m → 1, these solutions degenerate to the solitary wave solutions of the governing system.

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27

Wei, Changhua. "Classical solutions to the relativistic Euler equations for a linearly degenerate equation of state." Journal of Hyperbolic Differential Equations 14, no. 03 (September 2017): 535–63. http://dx.doi.org/10.1142/s0219891617500187.

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We are concerned with the global existence and blowup of the classical solutions to the Cauchy problem of one-dimensional isentropic relativistic Euler equations (Chaplygin gas, pressureless perfect fluid and stiff matter) with linearly degenerate characteristics. We at first derive the exact representation formula for all the fluids by the property of linearly degenerate. Then for the Chaplygin gas and the pressureless perfect fluid, we give a classification of the initial data that leads to the global existence and the blowup of the classical solution, respectively. We construct, especially, a class of initial data that contributes to the formation of “cusp-type” singularity and study the evolution of the solution after blowup by introducing a weak solution called delta shock wave. At last, for the stiff matter, we show that this system is indeed a linear system and prove the global existence of the classical solution to this fluid.

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28

RAHMAN, ATA-UR, S. ALI, A. MUSHTAQ, and A. QAMAR. "Nonlinear ion acoustic excitations in relativistic degenerate, astrophysical electron–positron–ion plasmas." Journal of Plasma Physics 79, no. 5 (May 28, 2013): 817–23. http://dx.doi.org/10.1017/s0022377813000524.

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AbstractThe dynamics and propagation of ion acoustic (IA) waves are considered in an unmagnetized collisionless plasma, whose constituents are the relativistically degenerate electrons and positrons as well as the inertial cold ions. At a first step, a linear dispersion relation for IA waves is derived and analysed numerically. For nonlinear analysis, the reductive perturbation technique is used to derive a Korteweg–deVries equation, which admits a localized wave solution in the presence of relativistic degenerate electrons and positrons. It is shown that only compressive IA solitary waves can propagate, whose amplitude, width and phase velocity are significantly modified due to the positron concentration. The latter also strongly influences all the relativistic plasma parameters. Our present analysis is aimed to understand collective interactions in dense astrophysical objects, e.g. white dwarfs, where the lighter species electrons and positrons are taken as relativistically degenerate.

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29

Borsch, Vladimir V. "ON INITIAL BOUNDARY VALUE PROBLEMS FOR THE DEGENERATE 1D WAVE EQUATION." Journal of Optimization, Differential Equations and their Applications 27, no. 2 (December 1, 2019): 23. http://dx.doi.org/10.15421/141906.

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30

Paudel, L., and J. Iaia. "Traveling wave solutions of the porous medium equation with degenerate interfaces." Nonlinear Analysis: Theory, Methods & Applications 81 (April 2013): 110–29. http://dx.doi.org/10.1016/j.na.2012.10.016.

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Kim, Sangil, Jong-Yeoul Park, and Yong Han Kang. "Stochastic quasilinear viscoelastic wave equation with degenerate damping and source terms." Computers & Mathematics with Applications 75, no. 11 (June 2018): 3987–94. http://dx.doi.org/10.1016/j.camwa.2018.03.008.

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32

Cavalcanti, Marcelo M., Valeria N. Domingos Cavalcanti, Marcio A. Jorge Silva, and Claudete M. Webler. "Exponential stability for the wave equation with degenerate nonlocal weak damping." Israel Journal of Mathematics 219, no. 1 (April 2017): 189–213. http://dx.doi.org/10.1007/s11856-017-1478-y.

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33

Nakao, Mitsuhiro. "Energy decay for the wave equation with a degenerate dissipative term." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 100, no. 1-2 (1985): 19–27. http://dx.doi.org/10.1017/s0308210500013603.

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SynopsisDecay estimates for the energy are derived for the initial boundary value problem of the wave equation with a degenerate dissipative term:where Ω is a bounded domain in Rn, a(×) is a nonnegative function such that a1 ∊Lp(Ω) for some p > 0 and f is a function tending to 0 rapidly as t → ∞

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34

Han, Zhong-Jie, Gengsheng Wang, and Jing Wang. "Explicit decay rate for a degenerate hyperbolic-parabolic coupled system." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 116. http://dx.doi.org/10.1051/cocv/2020040.

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This paper studies the stability of a 1-dim system which comprises a wave equation and a degenerate heat equation in two connected bounded intervals. The coupling between these two different components occurs at the interface with certain transmission conditions. We find an explicit polynomial decay rate for solutions of this system. This rate depends on the degree of the degeneration for the diffusion coefficient near the interface. Besides, the well-posedness of this degenerate coupled system is proved by the semigroup theory.

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ZHUANG, BINXIAN, YUANJIANG XIANG, XIAOYU DAI, and SHUANGCHUN WEN. "BOUNDED TRAVELING WAVE SOLUTIONS TO THE SHORT PULSE EQUATION." Journal of Nonlinear Optical Physics & Materials 21, no. 04 (December 2012): 1250049. http://dx.doi.org/10.1142/s021886351250049x.

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In this paper, we try to find bounded traveling wave solutions to short pulse equation (SPE) under different combinations of the coefficients of SPE. Results show that the global bounded traveling wave solutions to SPE are possible only in the focusing nonlinearity and impossible in the defocusing nonlinearity. The periodic loop or inverted loop, and the periodic hump or inverted hump global bounded traveling solutions are obtained in the focusing nonlinearity for three cases separately. The former can degenerate into loop or inverted loop solitary solutions.

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Rizvi, S. T. R., Aly R. Seadawy, S. Ahmed, M. Younis, and K. Ali. "Lump, rogue wave, multi-waves and Homoclinic breather solutions for (2+1)-Modified Veronese Web equation." International Journal of Modern Physics B 35, no. 04 (January 28, 2021): 2150055. http://dx.doi.org/10.1142/s0217979221500557.

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This work addresses the four main inducements: Lump, rogue wave, Homoclinic breather and multi-wave solutions for (2+1)-Modified Veronese Web (MVW) equation via Hirota bilinear approach and the ansatz technique. This model is a linearly degenerate integrable nonlinear partial differential equation (NLPDE) and can also be used to admit a differential covering with nonremoval physical parameters. By assuming the function [Formula: see text] in the Hirota bilinear form of the presented model as the general quadratic function, trigonometric function and exponential function form, also with appropriate set of parameters, we have prevented the lump, rogue wave, breather and multi-wave solutions successfully. A precise compatible wave transformation is utilized to obtain multi-wave solutions of governing model. Also, the motion track of the lump, Rogue wave and multi-waves is also explained both physically and theoretically. These new results contain some special arbitrary constants that can be useful to spell out diversity in qualitative features of wave phenomena.

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SHUKLA, P. K., L. STENFLO, and R. BINGHAM. "Instability of plasma waves caused by incoherent photons in dense plasmas." Journal of Plasma Physics 76, no. 6 (August 17, 2010): 845–51. http://dx.doi.org/10.1017/s0022377810000437.

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AbstractWe consider the nonlinear instability of modified Langmuir and ion–sound waves caused by partially coherent photons in dense quantum plasmas. In our model, the dynamics of the photons is governed by a wave kinetic equation. The evolution equations for the Langmuir and ion–sound waves are deduced from the quantum hydrodynamic equations accounting for the incoherent photon pressure, the quantum statistical electron pressure, and the quantum Bohm force acting on the degenerate electrons. The governing equations are Fourier analyzed to obtain nonlinear dispersion relations. The latter are analyzed to predict instability of the modified Langmuir and ion–sound waves in the presence of partially coherent photons. Possible applications of our investigation to the next generation of intense laser–solid dense plasma experiments and compact dense astrophysical bodies are mentioned.

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38

Ratliff, Daniel J., and Thomas J. Bridges. "Multiphase wavetrains, singular wave interactions and the emergence of the Korteweg–de Vries equation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 472, no. 2196 (December 2016): 20160456. http://dx.doi.org/10.1098/rspa.2016.0456.

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Multiphase wavetrains are multiperiodic travelling waves with a set of distinct wavenumbers and distinct frequencies. In conservative systems, such families are associated with the conservation of wave action or other conservation law. At generic points (where the Jacobian of the wave action flux is non-degenerate), modulation of the wavetrain leads to the dispersionless multiphase conservation of wave action. The main result of this paper is that modulation of the multiphase wavetrain, when the Jacobian of the wave action flux vector is singular, morphs the vector-valued conservation law into the scalar Korteweg–de Vries (KdV) equation. The coefficients in the emergent KdV equation have a geometrical interpretation in terms of projection of the vector components of the conservation law. The theory herein is restricted to two phases to simplify presentation, with extensions to any finite dimension discussed in the concluding remarks. Two applications of the theory are presented: a coupled nonlinear Schrödinger equation and two-layer shallow-water hydrodynamics with a free surface. Both have two-phase solutions where criticality and the properties of the emergent KdV equation can be determined analytically.

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Akter, Jhorna, and Abdullah Al Mamun. "Nucleus-Acoustic Solitary Waves in Warm Degenerate Magneto-Rotating Quantum Plasmas." Fluids 7, no. 9 (September 16, 2022): 305. http://dx.doi.org/10.3390/fluids7090305.

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A warm degenerate magneto-rotating quantum plasma (WDMRQP) model consisting of a static heavy nucleus, inertial non-degenerate light nucleus, and warm non-relativistic or ultra-relativistic electrons has been considered to observe the generation of nucleus-acoustic (NA) solitary waves (NASWs). A Korteweg–de-Vries-type equation is derived by using the reductive perturbation method to describe the characteristics of the NASWs. It has been observed that the temperature of warm degenerate species, rotational speed of the plasma system, and the presence of heavy nucleus species modify the basic features (height and width) of NASWs in the WDMRQP system and support the existence of positive NA wave potential only. The applications of the present investigation have been briefly discussed.

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Anikin, Anatoly, Sergey Dobrokhotov, and Vladimir Nazaikinskii. "Asymptotic Solutions of the Wave Equation with Degenerate Velocity and with Right-Hand Side Localized in Space and Time." Zurnal matematiceskoj fiziki, analiza, geometrii 14, no. 4 (December 25, 2018): 393–405. http://dx.doi.org/10.15407/mag14.04.393.

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Yan, Weifang, and Rui Liu. "Existence and Asymptotic Behavior of Traveling Wave Fronts for a Time-Delayed Degenerate Diffusion Equation." Abstract and Applied Analysis 2013 (2013): 1–20. http://dx.doi.org/10.1155/2013/578345.

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This paper is concerned with traveling wave fronts for a degenerate diffusion equation with time delay. We first establish the necessary and sufficient conditions to the existence of monotone increasing and decreasing traveling wave fronts, respectively. Moreover, special attention is paid to the asymptotic behavior of traveling wave fronts connecting two uniform steady states. Some previous results are extended.

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42

Lazzaroni, Giuliano, and Lorenzo Nardini. "On the 1d wave equation in time-dependent domains and the problem of debond initiation." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 80. http://dx.doi.org/10.1051/cocv/2019006.

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Motivated by a debonding model for a thin film peeled from a substrate, we analyse the one-dimensional wave equation, in a time-dependent domain which is degenerate at the initial time. In the first part of the paper we prove existence for the wave equation when the evolution of the domain is given; in the second part of the paper, the evolution of the domain is unknown and is governed by an energy criterion coupled with the wave equation. Our existence result for such coupled problem is a contribution to the study of crack initiation in dynamic fracture.

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43

Zennir, Khaled, and Svetlin G. Georgiev. "New results on Blow-up of solutions for Emden-Fowler type degenerate wave equation with memory." Boletim da Sociedade Paranaense de Matemática 39, no. 2 (January 1, 2021): 163–79. http://dx.doi.org/10.5269/bspm.40397.

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In this article we consider a new class of a Emden-Fowler type semilinear degenerate wave equation with memory. The main contributions here is to show that the memory lets the global solutions of the degenerate problem still non-exist without any conditions on the nature of growth of the relaxation function. This is to extend the paper in \cite{L11} for the dissipative case.

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44

Shah, S. A. "Study Of Three Dimensional Propagation Of Waves In Hollow Poroelastic Circular Cylinders." International Journal of Applied Mechanics and Engineering 20, no. 3 (August 1, 2015): 565–87. http://dx.doi.org/10.1515/ijame-2015-0037.

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Abstract Employing Biot’s theory of wave propagation in liquid saturated porous media, waves propagating in a hollow poroelastic circular cylinder of infinite extent are investigated. General frequency equations for propagation of waves are obtained each for a pervious and an impervious surface. Degenerate cases of the general frequency equations of pervious and impervious surfaces, when the longitudinal wavenumber k and angular wavenumber n are zero, are considered. When k=0, the plane-strain vibrations and longitudinal shear vibrations are uncoupled and when k≠0 these are coupled. It is seen that the frequency equation of longitudinal shear vibrations is independent of the nature of the surface. When the angular (or circumferential) wavenumber is zero, i.e., n=0, axially symmetric vibrations and torsional vibrations are uncoupled. For n≠0 these vibrations are coupled. The frequency equation of torsional vibrations is independent of the nature of the surface. By ignoring liquid effects, the results of a purely elastic solid are obtained as a special case.

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45

Hossen, M. R., L. Nahar, and A. A. Mamun. "Planar and Nonplanar Solitary Waves in a Four-Component Relativistic Degenerate Dense Plasma." Journal of Astrophysics 2014 (October 23, 2014): 1–8. http://dx.doi.org/10.1155/2014/653065.

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The nonlinear propagation of electrostatic perturbation modes in an unmagnetized, collisionless, relativistic, degenerate plasma (containing both nonrelativistic and ultrarelativistic degenerate electrons, nonrelativistic degenerate ions, and arbitrarily charged static heavy ions) has been investigated theoretically. The Korteweg-de Vries (K-dV) equation has been derived by employing the reductive perturbation method. Their solitary wave solution is obtained and numerically analyzed in case of both planar and nonplanar (cylindrical and spherical) geometry. It has been observed that the ion-acoustic (IA) and modified ion-acoustic (mIA) solitary waves have been significantly changed due to the effects of degenerate plasma pressure and number densities of the arbitrarily charged heavy ions. It has been also found that properties of planar K-dV solitons are quite different from those of nonplanar K-dV solitons. There are numerous variations in case of mIA solitary waves due to the polarity of heavy ions. The basic features and the underlying physics of IA and mIA solitary waves, which are relevant to some astrophysical compact objects, are briefly discussed.

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46

Benaissa, Abbes, and Chahira Aichi. "Energy decay for a degeneratewave equation under fractional derivative controls." Filomat 32, no. 17 (2018): 6045–72. http://dx.doi.org/10.2298/fil1817045b.

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In this article, we consider a one-dimensional degenerate wave equation with a boundary control condition of fractional derivative type. We show that the problem is not uniformly stable by a spectrum method and we study the polynomial stability using the semigroup theory of linear operators.

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47

Ma, Hongcai, Xiaoyu Chen, and Aiping Deng. "Novel Exact Solution for the Bidirectional Sixth-Order Sawada–Kotera Equation." Universe 9, no. 1 (January 15, 2023): 55. http://dx.doi.org/10.3390/universe9010055.

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In this paper, we take the bidirectional sixth-order Sawada–Kotera equation as an instance and use a new limit approach to generate a multiple-pole solution and the degenerate of the breather wave from the N-order soliton solution. We show not only the substitution method, but also the specific mathematical expression of the double-pole, triple-pole, and the degenerate breather solution after the substitution. Meanwhile, we give the dynamic images and trajectories of the different multiple-pole solution. Moreover, we also acquire the interaction between two double-pole solutions and different nonlinear superposition solutions.

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Huang, Yehui, Hongqing Jing, Min Li, Zhenjun Ye, and Yuqin Yao. "On Solutions of an Extended Nonlocal Nonlinear Schrödinger Equation in Plasmas." Mathematics 8, no. 7 (July 5, 2020): 1099. http://dx.doi.org/10.3390/math8071099.

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The parity-time symmetric nonlocal nonlinear Schrödinger equation with self-consistent sources (PTNNLSESCS) is used to describe the interaction between an high-frequency electrostatic wave and an ion-acoustic wave in plasmas. In this paper, the soliton solutions, rational soliton solutions and rogue wave solutions are derived for the PTNNLSESCS via the generalized Darboux transformation. We find that the soliton solutions can exhibit the elastic interactions of different type of solutions such as antidark-antidark, dark-antidark, and dark-dark soliton pairs on a continuous wave background. Also, we discuss the degenerate case in which only one antidark or dark soliton remains. The rogue wave solution is derived in some specially chosen situations.

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MOSLEM, W. M., U. M. ABDELSALAM, R. SABRY, E. F. EL-SHAMY, and S. K. EL-LABANY. "Three-dimensional cylindrical Kadomtsev–Petviashvili equation in a dusty electronegative plasma." Journal of Plasma Physics 76, no. 3-4 (January 29, 2010): 453–66. http://dx.doi.org/10.1017/s0022377809990808.

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AbstractThe hydrodynamic equations of positive and negative ions, Boltzmann electron density distribution and Poisson equation with stationary dust particles are used along with the reductive perturbation method to derive a three-dimensional cylindrical Kadomtsev–Petviashvili equation. The generalized expansion method, used to obtain a new class of solutions, admits a train of well-separated bell-shaped periodic pulses. At certain condition, these periodic pulses degenerate to solitary wave solutions. The effects of the physical parameters on the solitary pulses are examined. Finally, the present results should elucidate the properties of ion-acoustic solitary pulses in multi-component plasmas, particularly in Earth's ionosphere.

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GUSTAFSON, STEPHEN, KENJI NAKANISHI, and TAI-PENG TSAI. "SCATTERING THEORY FOR THE GROSS–PITAEVSKII EQUATION IN THREE DIMENSIONS." Communications in Contemporary Mathematics 11, no. 04 (August 2009): 657–707. http://dx.doi.org/10.1142/s0219199709003491.

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We study the global behavior of small solutions of the Gross–Pitaevskii equation in three dimensions. We prove that disturbances from the constant equilibrium with small, localized energy, disperse for large time, according to the linearized equation. Translated to the defocusing nonlinear Schrödinger equation, this implies asymptotic stability of all plane wave solutions for such disturbances. We also prove that every linearized solution with finite energy has a nonlinear solution which is asymptotic to it. The key ingredients are: (1) some quadratic transforms of the solutions, which effectively linearize the nonlinear energy space, (2) a bilinear Fourier multiplier estimate, which allows irregular denominators due to a degenerate non-resonance property of the quadratic interactions, and (3) geometric investigation of the degeneracy in the Fourier space to minimize its influence.

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