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Existence and Stability of Standing Waves for the Nonlinear Schrödinger Equation with Combined Nonlinearities and a Partial Harmonic Potential
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作者 Wei Wang 《Journal of Applied Mathematics and Physics》 2024年第5期1606-1615,共10页
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
关键词 nonlinear Schrödinger equation Orbital Stability Standing waves
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On Two Types of Stability of Solutions to a Pair of Damped Coupled Nonlinear Evolution Equations
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作者 Mark Jones 《Advances in Pure Mathematics》 2024年第5期354-366,共13页
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
关键词 nonlinear Schrödinger equation STABILITY Capillary-Gravity waves
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Breather and its interaction with rogue wave of the coupled modified nonlinear Schrodinger equation
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作者 王明 徐涛 +1 位作者 何国亮 田雨 《Chinese Physics B》 SCIE EI CAS CSCD 2023年第5期350-356,共7页
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
关键词 coupled modified nonlinear Schr?dinger equation Darboux transformation BREATHER rouge wave
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Selective Impact of Dispersion and Nonlinearity on Waves and Solitary Wave in a Strongly Nonlinear and Flattened Waveguide
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作者 Christian Regis Ngouo Tchinda Marcelle Nina Zambo Abou’ou Jean Roger Bogning 《Open Journal of Applied Sciences》 2024年第7期1730-1753,共24页
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
关键词 Flattened waveguide Solitary wave Characteristic Coefficient Probabilities Propagation nonlinear DISPERSIVE Partial Differential equation
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An Extended Riccati Equation Method to Find New Solitary Wave Solutions of the Burgers-Fisher Equation
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作者 Yixinni Liu Hongyan Pan 《Open Journal of Applied Sciences》 2023年第8期1418-1432,共15页
In this paper, our objective is to explore novel solitary wave solutions of the Burgers-Fisher equation, which characterizes the interplay between diffusion and reaction phenomena. Understanding this equation is cruci... In this paper, our objective is to explore novel solitary wave solutions of the Burgers-Fisher equation, which characterizes the interplay between diffusion and reaction phenomena. Understanding this equation is crucial for addressing challenges in fluid, chemical kinetics and population dynamics. We tackle this task by employing the Riccati equation and employing various function transformations to solve the Burgers-Fisher equation. By adopting different coefficients in the Riccati equation, we obtain a wide range of exact solutions, many of which have not been previously documented. These abundant solitary wave solutions serve as valuable tools for comprehending the Burgers-Fisher equation and contribute to expanding our knowledge in this field. 展开更多
关键词 Solitary wave SOLITON Burger-Fisher equation Riccati equation nonlinear Evolution equation
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New explicit multi-symplectic scheme for nonlinear wave equation 被引量:4
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作者 李昊辰 孙建强 秦孟兆 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2014年第3期369-380,共12页
Based on splitting multi-symplectic structures, a new multi-symplectic scheme is proposed and applied to a nonlinear wave equation. The explicit multi-symplectic scheme of the nonlinear wave equation is obtained, and ... Based on splitting multi-symplectic structures, a new multi-symplectic scheme is proposed and applied to a nonlinear wave equation. The explicit multi-symplectic scheme of the nonlinear wave equation is obtained, and the corresponding multi-symplectic conservation property is proved. The backward error analysis shows that the explicit multi-symplectic scheme has good accuracy. The sine-Gordon equation and the Klein-Gordon equation are simulated by an explicit multi-symplectic scheme. The numerical results show that the new explicit multi-symplectic scheme can well simulate the solitary wave behaviors of the nonlinear wave equation and approximately preserve the relative energy error of the equation. 展开更多
关键词 nonlinear wave equation multi-symplectic method backward error analysis
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TRAVELLING WAVE SOLUTIONS OF NONLINEAR EVOLUTION EQUATIONS BY USING SYMBOLIC COMPUTATION 被引量:4
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作者 Fan Engui E mail:faneg@fudan.edu.cnInstituteofMath.,FudanUniv.,Shanghai200433 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2001年第2期149-155,共7页
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
关键词 nonlinear evolution equation travelling wave solution symbolic computation.
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An Improved Nearshore Wave Breaking Model Based on the Fully Nonlinear Boussinesq Equations 被引量:2
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作者 李绍武 李春颖 +1 位作者 时钟 谷汉斌 《China Ocean Engineering》 SCIE EI 2005年第1期61-71,共11页
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
关键词 wave breaking surface roller κ equation Boussinesq equations fully nonlinear
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Analytical solutions and rogue waves in (3+1)-dimensional nonlinear SchrSdinger equation 被引量:2
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《Chinese Physics B》 SCIE EI CAS CSCD 2012年第3期138-144,共7页
Analytical solutions in terms of rational-like functions are presented for a (3+1)-dimensional nonlinear Schrodinger equation with time-varying coefficients and a harmonica potential using the similarity transforma... Analytical solutions in terms of rational-like functions are presented for a (3+1)-dimensional nonlinear Schrodinger equation with time-varying coefficients and a harmonica potential using the similarity transformation and a direct ansatz. Several free functions of time t are involved to generate abundant wave structures. Three types of elementary functions are chosen to exhibit the corresponding nonlinear rogue wave propagations. 展开更多
关键词 nonlinear SchrSdinger equation similarity transformation rational-like solution rogue wave
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A new variable coefficient algebraic method and non-travelling wave solutions of nonlinear equations 被引量:2
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作者 陆斌 张鸿庆 《Chinese Physics B》 SCIE EI CAS CSCD 2008年第11期3974-3984,共11页
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
关键词 nonlinear partial differential equations non-travelling wave solutions asymmetric Nizhnik-Novikov- Vesselov equation
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Solving Nonlinear Wave Equations by Elliptic Equation 被引量:12
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作者 FUZun-Tao LIUShi-Da LIUShi-Kuo 《Communications in Theoretical Physics》 SCIE CAS CSCD 2003年第5期531-536,共6页
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
关键词 非线性波动方程 椭圆方程 雅各比椭圆函数 求解方法 微分方程 JACOBI椭圆函数
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Numerical solutions for two nonlinear wave equations 被引量:2
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作者 Yi-feng ZHANG Rui-jie LI 《Water Science and Engineering》 EI CAS 2012年第4期410-418,共9页
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
关键词 nonlinear water wave equation parabolic mild-slope equation spatial MNLSequation numerical method
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Travelling solitary wave solutions for the generalized Burgers-Huxley equation with nonlinear terms of any order 被引量:2
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作者 邓习军 燕子宗 韩立波 《Chinese Physics B》 SCIE EI CAS CSCD 2009年第8期3169-3173,共5页
In this paper, the travelling wave solutions for the generalized Burgers-Huxley equation with nonlinear terms of any order are studied. By using the first integral method, which is based on the divisor theorem, some e... In this paper, the travelling wave solutions for the generalized Burgers-Huxley equation with nonlinear terms of any order are studied. By using the first integral method, which is based on the divisor theorem, some exact explicit travelling solitary wave solutions for the above equation are obtained. As a result, some minor errors and some known results in the previousl literature are clarified and improved. 展开更多
关键词 travelling wave solutions first integral method generalized Burgers-Huxley equation with nonlinear terms of any order
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A New Approach to Solve Nonlinear Wave Equations 被引量:15
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作者 FUZun-Tao LIUShi-Kuo 《Communications in Theoretical Physics》 SCIE CAS CSCD 2003年第1期27-30,共4页
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
关键词 非线性波方程 JACOBI椭圆函数 周期波解
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SPHERICAL NONLINEAR PULSES FOR THE SOLUTIONS OF NONLINEAR WAVE EQUATIONSⅡ, NONLINEAR CAUSTIC 被引量:1
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作者 袁明生 《Acta Mathematica Scientia》 SCIE CSCD 2007年第2期381-394,共14页
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
关键词 nonlinear wave equations spherical pulses caustic crossing nonlinearity
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A more general form of lump solution,lumpoff,and instanton/rogue wave solutions of a reduced (3+1)-dimensional nonlinear evolution equation 被引量:2
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作者 郑攀峰 贾曼 《Chinese Physics B》 SCIE EI CAS CSCD 2018年第12期147-156,共10页
In this manuscript,a reduced(3+1)-dimensional nonlinear evolution equation is studied.We first construct the bilinear formalism of the equation by using the binary Bell polynomials theory,then explore a lump solution ... In this manuscript,a reduced(3+1)-dimensional nonlinear evolution equation is studied.We first construct the bilinear formalism of the equation by using the binary Bell polynomials theory,then explore a lump solution to the special case for z=x.Furthermore,a more general form of lump solution of the equation is found which possesses seven arbitrary parameters and four constraint conditions.By cutting the lump by the induced soliton(s),lumpoff and instanton/rogue wave solutions are also constructed by the more general form of lump solution. 展开更多
关键词 a reduced(3 + 1)-dimensional nonlinear evolution equation more general form of lump solution soliton induced by lump lumpoff and instanton/rogue wave solutions
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Numerical Calculation for Nonlinear Waves in Water of Arbitrarily Varying Depth with Boussinesq Equations 被引量:1
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作者 朱良生 洪广文 《China Ocean Engineering》 SCIE EI 2001年第3期355-369,共15页
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
关键词 nonlinear wave Boussinesq equation arbitrarily varying depth numerical calculation
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General Energy Decay of Solutions for A Wave Equation with Nonlocal Nonlinear Damping and Source Terms 被引量:3
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作者 ZHANG Hong-wei LI Dong-hao HU Qing-ying 《Chinese Quarterly Journal of Mathematics》 2020年第3期302-310,共9页
We consider a wave equation with nonlocal nonlinear damping and source terms.We prove a general energy decay property for solutions by constructing a stable set and using the multiplier technique.The main difficult is... We consider a wave equation with nonlocal nonlinear damping and source terms.We prove a general energy decay property for solutions by constructing a stable set and using the multiplier technique.The main difficult is how to handle with the nonlocal nonlinear damping term.Our result extends and improves the result in the literature such as the work by Jorge Silva and Narciso(Evolution Equation and Control Theory,2017(6):437-470)and Narciso(Evolution Equations and Control Theory,2020,9(2):487-508). 展开更多
关键词 wave equation Initial boundary value problem nonlinear nonlocal damping Energy decay
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Initial value problem for a class of fourth-order nonlinear wave equations 被引量:1
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作者 陈国旺 侯长顺 Shi-qiang DAI 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2009年第3期391-401,共11页
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
关键词 fourth-order nonlinear wave equation initial value problem global solution blow up of solution
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Solution of ODE u″+p(u)(u′)2+q(u)=0 and Applications to Classifications of All Single Travelling Wave Solutions to Some Nonlinear Mathematical Physics Equations 被引量:8
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作者 LIU Cheng-Shi 《Communications in Theoretical Physics》 SCIE CAS CSCD 2008年第2期291-296,共6页
在旅行波浪转变下面,一些非线性的部分微分方程例如 Camassa 河边肥沃的低地方程,高顺序的 KdV 方程,等等,被归结为 u&#8243; 表示的一首 integrable 颂诗+ p (u)(u')<SUP>2</SUP>+ q (u)= 0 答案能被给其将军... 在旅行波浪转变下面,一些非线性的部分微分方程例如 Camassa 河边肥沃的低地方程,高顺序的 KdV 方程,等等,被归结为 u&#8243; 表示的一首 integrable 颂诗+ p (u)(u')<SUP>2</SUP>+ q (u)= 0 答案能被给其将军。而且,为多项式联合完全的辨别系统,这些方程的所有单个旅行波浪答案的分类被获得。方程 u&#8243;0 包括的 + p (u)(u')<SUP>2</SUP>+ q (u)= 是的方程(u')<SUP>2</SUP>= f (u) 一种特殊情况,建议方法能也因此被用于很多非线性的方程。这些完全的结果不能被任何间接方法获得。 展开更多
关键词 数学物理方法 非线性数学物理方程 孤波解 对称群 非线性偏微分方程
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