The stability of a set of spatially constant plane wave solutions to a pair of damped coupled nonlinear Schrödinger evolution equations is considered. The equations could model physical phenomena arising in fluid...The stability of a set of spatially constant plane wave solutions to a pair of damped coupled nonlinear Schrödinger evolution equations is considered. The equations could model physical phenomena arising in fluid dynamics, fibre optics or electron plasmas. The main result is that any small perturbation to the solution remains small for all time. Here small is interpreted as being both in the supremum sense and the square integrable sense.展开更多
In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L<sup>2</sup>-supercriti...In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L<sup>2</sup>-supercritical case, we obtain the existence and stability of standing waves. Our results are complements to the results of Carles and Il’yasov’s artical, where orbital stability of standing waves have been studied for the 2D Schrödinger equation with combined nonlinearities and harmonic potential.展开更多
A numerical wave model based on the modified four-order nonlinear Schoedinger (NKS) equation in deep water is developed to simulate freak waves. A standard split-step, pseudo-spectral method is used to solve NLS equ...A numerical wave model based on the modified four-order nonlinear Schoedinger (NKS) equation in deep water is developed to simulate freak waves. A standard split-step, pseudo-spectral method is used to solve NLS equation. The validation of the model is firstly verified, and then the simulation of freak waves is perforated by changing sideband condi- tions. Results show that freak waves entirely consistent with the definition in the evolution of wave trains are obtained. The possible occurrence mechanism of freak waves is discussed and the relevant characteristics are also analyzed.展开更多
The waveguide which is at the center of our concerns in this work is a strongly flattened waveguide, that is to say characterized by a strong dispersion and in addition is strongly nonlinear. As this type of waveguide...The waveguide which is at the center of our concerns in this work is a strongly flattened waveguide, that is to say characterized by a strong dispersion and in addition is strongly nonlinear. As this type of waveguide contains multiple dispersion coefficients according to the degrees of spatial variation within it, our work in this article is to see how these dispersions and nonlinearities each influence the wave or the signal that can propagate in the waveguide. Since the partial differential equation which governs the dynamics of propagation in such transmission medium presents several dispersion and nonlinear coefficients, we check how they contribute to the choices of the solutions that we want them to verify this nonlinear partial differential equation. This effectively requires an adequate choice of the form of solution to be constructed. Thus, this article is based on three main pillars, namely: first of all, making a good choice of the solution function to be constructed, secondly, determining the exact solutions and, if necessary, remodeling the main equation such that it is possible;then check the impact of the dispersion and nonlinear coefficients on the solutions. Finally, the reliability of the solutions obtained is tested by a study of the propagation. Another very important aspect is the use of notions of probability to select the predominant solutions.展开更多
We investigate the coupled modified nonlinear Schr?dinger equation.Breather solutions are constructed through the traditional Darboux transformation with nonzero plane-wave solutions.To obtain the higher-order localiz...We investigate the coupled modified nonlinear Schr?dinger equation.Breather solutions are constructed through the traditional Darboux transformation with nonzero plane-wave solutions.To obtain the higher-order localized wave solution,the N-fold generalized Darboux transformation is given.Under the condition that the characteristic equation admits a double-root,we present the expression of the first-order interactional solution.Then we graphically analyze the dynamics of the breather and rogue wave.Due to the simultaneous existence of nonlinear and self-steepening terms in the equation,different profiles in two components for the breathers are presented.展开更多
A Riccati equation involving a parameter and symbolic computation are used to uniformly construct the different forms of travelling wave solutions for nonlinear evolution equations.It is shown that the sign of the pa...A Riccati equation involving a parameter and symbolic computation are used to uniformly construct the different forms of travelling wave solutions for nonlinear evolution equations.It is shown that the sign of the parameter can be applied in judging the existence of various forms of travelling wave solutions.An efficiency of this method is demonstrated on some equations,which include Burgers Huxley equation,Caudrey Dodd Gibbon Kawada equation,generalized Benjamin Bona Mahony equation and generalized Fisher equation.展开更多
This paper aims to propose an improved numerical model for wave breaking in the nearshore region based on the fully nonlinear form of Boussinesq equations. The model uses the κ equation turbulence scheme to determine...This paper aims to propose an improved numerical model for wave breaking in the nearshore region based on the fully nonlinear form of Boussinesq equations. The model uses the κ equation turbulence scheme to determine the eddy viscosity in the Boussinesq equations. To calculate the turbulence production term in the equation, a new formula is derived based on the concept of surface roller. By use of this formula, the turbulence production in the one-equation turbulence scheme is directly related to the difference between the water particle velocity and the wave celerity. The model is verified by Hansen and Svendsen's experimental data (1979) in terms of wave height and setup and setdown. The comparison between the model and experimental results of wave height and setup and setdown shows satisfactory agreement. The modeled turbulence energy decreases as waves attenuate in the surf zone. The modeled production term peaks at the breaking point and decreases as waves propagate shoreward. It is also suggested that both convection and diffusion play their important roles in the transport of turbulence energy immediately after wave breaking. When waves approach to the shoreline, the production and dissipation of turbulence energy are almost balanced. By use of the slot technique for the simulation of the movable shoreline boundary, wave runup in the swash zone is well simulated by the present model.展开更多
In this paper, a new auxiliary equation method is presented of constructing more new non-travelling wave solutions of nonlinear differential equations in mathematical physics, which is direct and more powerful than pr...In this paper, a new auxiliary equation method is presented of constructing more new non-travelling wave solutions of nonlinear differential equations in mathematical physics, which is direct and more powerful than projective Riccati equation method. In order to illustrate the validity and the advantages of the method, (2+1)-dimensional asymmetric Nizhnik-Novikov-Vesselov equation is employed and many new double periodic non-travelling wave solutions are obtained. This algorithm can also be applied to other nonlinear differential equations.展开更多
The elliptic equation is taken as a transformation and applied to solve nonlinear wave equations. It is shown that this method is more powerful to give more kinds of solutions, such as rational solutions, solitary wav...The elliptic equation is taken as a transformation and applied to solve nonlinear wave equations. It is shown that this method is more powerful to give more kinds of solutions, such as rational solutions, solitary wave solutions,periodic wave solutions and so on, so it can be taken as a generalized method.展开更多
The split-step pseudo-spectral method is a useful method for solving nonlinear wave equations. However, it is not widely used because of the limitation of the periodic boundary condition. In this paper, the method is ...The split-step pseudo-spectral method is a useful method for solving nonlinear wave equations. However, it is not widely used because of the limitation of the periodic boundary condition. In this paper, the method is modified at its second step by avoiding transforming the wave height function into a frequency domain function. Thus, the periodic boundary condition is not required, and the new method is easy to implement. In order to validate its performance, the proposed method was used to solve the nonlinear parabolic mild-slope equation and the spatial modified nonlinear Schrodinger (MNLS) equation, which were used to model the wave propagation under different bathymetric conditions. Good agreement between the numerical and experimental results shows that the present method is effective and efficient in solving nonlinear wave eouations.展开更多
From the nonlinear sine-Gordon equation, new transformations are obtained in this paper, which are applied to propose a new approach to construct exact periodic solutions to nonlinear wave equations. It is shown that ...From the nonlinear sine-Gordon equation, new transformations are obtained in this paper, which are applied to propose a new approach to construct exact periodic solutions to nonlinear wave equations. It is shown that more new periodic solutions can be obtained by this new approach, and more shock wave solutions or solitary wave solutions can be got under their limit conditions.展开更多
This article discusses spherical pulse like solutions of the system of semilinear wave equations with the pulses focusing at a point and emerging outgoing in three space variables. In small initial data case, it shows...This article discusses spherical pulse like solutions of the system of semilinear wave equations with the pulses focusing at a point and emerging outgoing in three space variables. In small initial data case, it shows that the nonlinearities have a strong effect at the focal point. Scattering operator is introduced to describe the caustic crossing. With the aid of the L^∞ norms, it analyzes the relative errors in approximate solutions.展开更多
Based on the high order nonlinear and dispersive wave equation with a dissipative term, a numerical model for nonlinear waves is developed, It is suitable to calculate wave propagation in water areas with an arbitrari...Based on the high order nonlinear and dispersive wave equation with a dissipative term, a numerical model for nonlinear waves is developed, It is suitable to calculate wave propagation in water areas with an arbitrarily varying bottom slope and a relative depth h/L(0)less than or equal to1. By the application of the completely implicit stagger grid and central difference algorithm, discrete governing equations are obtained. Although the central difference algorithm of second-order accuracy both in time and space domains is used to yield the difference equations, the order of truncation error in the difference equation is the same as that of the third-order derivatives of the Boussinesq equation. In this paper, the correction to the first-order derivative is made, and the accuracy of the difference equation is improved. The verifications of accuracy show that the results of the numerical model are in good agreement with those of analytical Solutions and physical models.展开更多
In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractio...In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractional order Sobolev space using the contraction mapping principle and the extension theorem. The sufficient conditions for the blow up of the solution to the initial value problem are given.展开更多
The Painleve integrability and exact solutions to a coupled nonlinear Schrodinger (CNLS) equation applied in atmospheric dynamics are discussed. Some parametric restrictions of the CNLS equation are given to pass th...The Painleve integrability and exact solutions to a coupled nonlinear Schrodinger (CNLS) equation applied in atmospheric dynamics are discussed. Some parametric restrictions of the CNLS equation are given to pass the Painleve test. Twenty periodic cnoidal wave solutions are obtained by applying the rational expansions of fundamental Jacobi elliptic functions. The exact solutions to the CNLS equation are used to explain the generation and propagation of atmospheric gravity waves.展开更多
We study the nonlinear stability of viscous shock waves for the Cauchy problem of one-dimensional nonisentropic compressible Navier–Stokes equations for a viscous and heat conducting ideal polytropic gas. The viscous...We study the nonlinear stability of viscous shock waves for the Cauchy problem of one-dimensional nonisentropic compressible Navier–Stokes equations for a viscous and heat conducting ideal polytropic gas. The viscous shock waves are shown to be time asymptotically stable under large initial perturbation with no restriction on the range of the adiabatic exponent provided that the strengths of the viscous shock waves are assumed to be sufficiently small.The proofs are based on the nonlinear energy estimates and the crucial step is to obtain the positive lower and upper bounds of the density and the temperature which are uniformly in time and space.展开更多
A class of strongly damped nonlinear wave equations are studied by using the technique of the operator decomposition,and the existence of the global compact attractor in space D(A)×V is obtained.
Generalized functional separation of variables to nonlinear evolution equations is studied in terms of the extended group foliation method, which is based on the Lie point symmetry method. The approach is applied to n...Generalized functional separation of variables to nonlinear evolution equations is studied in terms of the extended group foliation method, which is based on the Lie point symmetry method. The approach is applied to nonlinear wave equations with variable speed and external force. A complete classification for the wave equation which admits functional separable solutions is presented. Some known results can be recovered by this approach.展开更多
In this article, we consider quasilinear <span style="white-space:nowrap;">Schrödinger</span> equations of the form <img src="Edit_4d91f4a8-f399-4895-9edd-b0d77ec07654.bmp" ...In this article, we consider quasilinear <span style="white-space:nowrap;">Schrödinger</span> equations of the form <img src="Edit_4d91f4a8-f399-4895-9edd-b0d77ec07654.bmp" alt="" /> Such equations have been derived as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics. Unlike all known results in the literature, the nonlinearity is allowed to be indefinite. It is very interesting from physical and mathematical viewpoint. By mountain pass theorem and some special techniques, we prove the existence of solutions for the quasilinear <span style="white-space:nowrap;">Schrödinger</span> equations with indefinite nonlinearity. This indefinite problem had never been considered so far. So our main results can be regarded as complementary work in the literature.展开更多
文摘The stability of a set of spatially constant plane wave solutions to a pair of damped coupled nonlinear Schrödinger evolution equations is considered. The equations could model physical phenomena arising in fluid dynamics, fibre optics or electron plasmas. The main result is that any small perturbation to the solution remains small for all time. Here small is interpreted as being both in the supremum sense and the square integrable sense.
文摘In this paper, we study the existence of standing waves for the nonlinear Schrödinger equation with combined power-type nonlinearities and a partial harmonic potential. In the L<sup>2</sup>-supercritical case, we obtain the existence and stability of standing waves. Our results are complements to the results of Carles and Il’yasov’s artical, where orbital stability of standing waves have been studied for the 2D Schrödinger equation with combined nonlinearities and harmonic potential.
基金This paperis part of the research project (104061)supported by the keytechnology program,the Ministry of Educa-tion of China
文摘A numerical wave model based on the modified four-order nonlinear Schoedinger (NKS) equation in deep water is developed to simulate freak waves. A standard split-step, pseudo-spectral method is used to solve NLS equation. The validation of the model is firstly verified, and then the simulation of freak waves is perforated by changing sideband condi- tions. Results show that freak waves entirely consistent with the definition in the evolution of wave trains are obtained. The possible occurrence mechanism of freak waves is discussed and the relevant characteristics are also analyzed.
文摘The waveguide which is at the center of our concerns in this work is a strongly flattened waveguide, that is to say characterized by a strong dispersion and in addition is strongly nonlinear. As this type of waveguide contains multiple dispersion coefficients according to the degrees of spatial variation within it, our work in this article is to see how these dispersions and nonlinearities each influence the wave or the signal that can propagate in the waveguide. Since the partial differential equation which governs the dynamics of propagation in such transmission medium presents several dispersion and nonlinear coefficients, we check how they contribute to the choices of the solutions that we want them to verify this nonlinear partial differential equation. This effectively requires an adequate choice of the form of solution to be constructed. Thus, this article is based on three main pillars, namely: first of all, making a good choice of the solution function to be constructed, secondly, determining the exact solutions and, if necessary, remodeling the main equation such that it is possible;then check the impact of the dispersion and nonlinear coefficients on the solutions. Finally, the reliability of the solutions obtained is tested by a study of the propagation. Another very important aspect is the use of notions of probability to select the predominant solutions.
基金the National Natural Science Foundation of China(Grant Nos.11871232 and 12201578)Natural Science Foundation of Henan Province,China(Grant Nos.222300420377 and 212300410417)。
文摘We investigate the coupled modified nonlinear Schr?dinger equation.Breather solutions are constructed through the traditional Darboux transformation with nonzero plane-wave solutions.To obtain the higher-order localized wave solution,the N-fold generalized Darboux transformation is given.Under the condition that the characteristic equation admits a double-root,we present the expression of the first-order interactional solution.Then we graphically analyze the dynamics of the breather and rogue wave.Due to the simultaneous existence of nonlinear and self-steepening terms in the equation,different profiles in two components for the breathers are presented.
基金Supported by the Postdoctoral Science Foundation of ChinaChinese Basic Research Plan"MathematicsMechanization and A Platform
文摘A Riccati equation involving a parameter and symbolic computation are used to uniformly construct the different forms of travelling wave solutions for nonlinear evolution equations.It is shown that the sign of the parameter can be applied in judging the existence of various forms of travelling wave solutions.An efficiency of this method is demonstrated on some equations,which include Burgers Huxley equation,Caudrey Dodd Gibbon Kawada equation,generalized Benjamin Bona Mahony equation and generalized Fisher equation.
基金This study was supported by the National Natural Science Foundation of China (Grant No.50479047) and partly by the National Science Fund for Distinguished Young Scholars of China (Estuarine and Coastal Science, Grant No.40225014)
文摘This paper aims to propose an improved numerical model for wave breaking in the nearshore region based on the fully nonlinear form of Boussinesq equations. The model uses the κ equation turbulence scheme to determine the eddy viscosity in the Boussinesq equations. To calculate the turbulence production term in the equation, a new formula is derived based on the concept of surface roller. By use of this formula, the turbulence production in the one-equation turbulence scheme is directly related to the difference between the water particle velocity and the wave celerity. The model is verified by Hansen and Svendsen's experimental data (1979) in terms of wave height and setup and setdown. The comparison between the model and experimental results of wave height and setup and setdown shows satisfactory agreement. The modeled turbulence energy decreases as waves attenuate in the surf zone. The modeled production term peaks at the breaking point and decreases as waves propagate shoreward. It is also suggested that both convection and diffusion play their important roles in the transport of turbulence energy immediately after wave breaking. When waves approach to the shoreline, the production and dissipation of turbulence energy are almost balanced. By use of the slot technique for the simulation of the movable shoreline boundary, wave runup in the swash zone is well simulated by the present model.
基金Project supported by the State Key Program for Basic Research of China (Grant No 2004CB318000)
文摘In this paper, a new auxiliary equation method is presented of constructing more new non-travelling wave solutions of nonlinear differential equations in mathematical physics, which is direct and more powerful than projective Riccati equation method. In order to illustrate the validity and the advantages of the method, (2+1)-dimensional asymmetric Nizhnik-Novikov-Vesselov equation is employed and many new double periodic non-travelling wave solutions are obtained. This algorithm can also be applied to other nonlinear differential equations.
文摘The elliptic equation is taken as a transformation and applied to solve nonlinear wave equations. It is shown that this method is more powerful to give more kinds of solutions, such as rational solutions, solitary wave solutions,periodic wave solutions and so on, so it can be taken as a generalized method.
基金supported by the Central Public-Interest Scientific Institution Basal Research Fund of China(Grant No.TKS100108)
文摘The split-step pseudo-spectral method is a useful method for solving nonlinear wave equations. However, it is not widely used because of the limitation of the periodic boundary condition. In this paper, the method is modified at its second step by avoiding transforming the wave height function into a frequency domain function. Thus, the periodic boundary condition is not required, and the new method is easy to implement. In order to validate its performance, the proposed method was used to solve the nonlinear parabolic mild-slope equation and the spatial modified nonlinear Schrodinger (MNLS) equation, which were used to model the wave propagation under different bathymetric conditions. Good agreement between the numerical and experimental results shows that the present method is effective and efficient in solving nonlinear wave eouations.
文摘From the nonlinear sine-Gordon equation, new transformations are obtained in this paper, which are applied to propose a new approach to construct exact periodic solutions to nonlinear wave equations. It is shown that more new periodic solutions can be obtained by this new approach, and more shock wave solutions or solitary wave solutions can be got under their limit conditions.
基金The study is supported by National Natural Science Foundation of China (10131050)the Educational Ministry of Chinathe Shanghai Science and Technology Committee foundation (03QMH1407)
文摘This article discusses spherical pulse like solutions of the system of semilinear wave equations with the pulses focusing at a point and emerging outgoing in three space variables. In small initial data case, it shows that the nonlinearities have a strong effect at the focal point. Scattering operator is introduced to describe the caustic crossing. With the aid of the L^∞ norms, it analyzes the relative errors in approximate solutions.
基金This research was financially supported by China National Key Basic Research Project "Circulation Principal and Mathematic Model" (Grant No. 1999043810) Guangdong Science and Technology Innovation Project: "Disaster Diagnoses of Sea Walls" (99B07102G)
文摘Based on the high order nonlinear and dispersive wave equation with a dissipative term, a numerical model for nonlinear waves is developed, It is suitable to calculate wave propagation in water areas with an arbitrarily varying bottom slope and a relative depth h/L(0)less than or equal to1. By the application of the completely implicit stagger grid and central difference algorithm, discrete governing equations are obtained. Although the central difference algorithm of second-order accuracy both in time and space domains is used to yield the difference equations, the order of truncation error in the difference equation is the same as that of the third-order derivatives of the Boussinesq equation. In this paper, the correction to the first-order derivative is made, and the accuracy of the difference equation is improved. The verifications of accuracy show that the results of the numerical model are in good agreement with those of analytical Solutions and physical models.
基金supported by the National Natural Science Foundation of China (No. 10671182)
文摘In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractional order Sobolev space using the contraction mapping principle and the extension theorem. The sufficient conditions for the blow up of the solution to the initial value problem are given.
基金Project supported by the National Natural Science Foundation of China (Nos. 10735030and 40775069)the Natural Science Foundation of Guangdong Province of China(No. 10452840301004616)the Scientific Research Foundation for the Doctors of University of Electronic Science and Technology of China Zhongshan Institute (No. 408YKQ09)
文摘The Painleve integrability and exact solutions to a coupled nonlinear Schrodinger (CNLS) equation applied in atmospheric dynamics are discussed. Some parametric restrictions of the CNLS equation are given to pass the Painleve test. Twenty periodic cnoidal wave solutions are obtained by applying the rational expansions of fundamental Jacobi elliptic functions. The exact solutions to the CNLS equation are used to explain the generation and propagation of atmospheric gravity waves.
文摘We study the nonlinear stability of viscous shock waves for the Cauchy problem of one-dimensional nonisentropic compressible Navier–Stokes equations for a viscous and heat conducting ideal polytropic gas. The viscous shock waves are shown to be time asymptotically stable under large initial perturbation with no restriction on the range of the adiabatic exponent provided that the strengths of the viscous shock waves are assumed to be sufficiently small.The proofs are based on the nonlinear energy estimates and the crucial step is to obtain the positive lower and upper bounds of the density and the temperature which are uniformly in time and space.
文摘A class of strongly damped nonlinear wave equations are studied by using the technique of the operator decomposition,and the existence of the global compact attractor in space D(A)×V is obtained.
文摘Generalized functional separation of variables to nonlinear evolution equations is studied in terms of the extended group foliation method, which is based on the Lie point symmetry method. The approach is applied to nonlinear wave equations with variable speed and external force. A complete classification for the wave equation which admits functional separable solutions is presented. Some known results can be recovered by this approach.
文摘In this article, we consider quasilinear <span style="white-space:nowrap;">Schrödinger</span> equations of the form <img src="Edit_4d91f4a8-f399-4895-9edd-b0d77ec07654.bmp" alt="" /> Such equations have been derived as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics. Unlike all known results in the literature, the nonlinearity is allowed to be indefinite. It is very interesting from physical and mathematical viewpoint. By mountain pass theorem and some special techniques, we prove the existence of solutions for the quasilinear <span style="white-space:nowrap;">Schrödinger</span> equations with indefinite nonlinearity. This indefinite problem had never been considered so far. So our main results can be regarded as complementary work in the literature.