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Stability and Time-Step Constraints of Implicit-Explicit Runge-Kutta Methods for the Linearized Korteweg-de Vries Equation
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作者 Joseph Hunter Zheng Sun Yulong Xing 《Communications on Applied Mathematics and Computation》 EI 2024年第1期658-687,共30页
This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either... This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints. 展开更多
关键词 Linearized korteweg-de vries(kdv)equation Implicit-explicit(IMEX)Runge-Kutta(RK)method STABILITY Courant-Friedrichs-Lewy(CFL)condition Finite difference(FD)method Local discontinuous Galerkin(DG)method
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Analytical and Numerical Computations of Multi-Solitons in the Korteweg-de Vries (KdV) Equation
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作者 Hycienth O. Orapine Emem Ayankop-Andi Godwin J. Ibeh 《Applied Mathematics》 2020年第7期511-531,共21页
In this paper, an analytical and numerical computation of multi-solitons in Korteweg-de Vries (KdV) equation is presented. The KdV equation, which is classic of all model equations of nonlinear waves in the soliton ph... In this paper, an analytical and numerical computation of multi-solitons in Korteweg-de Vries (KdV) equation is presented. The KdV equation, which is classic of all model equations of nonlinear waves in the soliton phenomena, is described. In the analytical computation, the multi-solitons in KdV equation are computed symbolically using computer symbolic manipulator<span style="white-space:nowrap;">&#8212;</span>Wolfram Mathematica via Hirota method because of the lengthy algebraic computation in the method. For the numerical computation, Crank-Nicolson implicit scheme is used to obtain numerical algorithm for the KdV equation. The simulations of solitons in MATLAB as well as results concerning collision or interactions between solitons are presented. Comparing the analytical and numerical solutions, it is observed that the results are identically equal with little ripples in solitons after a collision in the numerical simulations;however there is no significant effect to cause a change in their properties. This supports the existence of solitons solutions and the theoretical assertion that solitons indeed collide with one another and come out without change of properties or identities. 展开更多
关键词 korteweg-de vries equation SOLITONS Hirota Method Crank-Nicolson Method
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Exact solutions of stochastic fractional Korteweg de–Vries equation with conformable derivatives 被引量:2
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作者 Hossam A.Ghany Abd-Allah Hyder M Zakarya 《Chinese Physics B》 SCIE EI CAS CSCD 2020年第3期62-69,共8页
We deal with the Wick-type stochastic fractional Korteweg de–Vries(KdV) equation with conformable derivatives.With the aid of the Exp-function method, white noise theory, and Hermite transform, we produce a novel set... We deal with the Wick-type stochastic fractional Korteweg de–Vries(KdV) equation with conformable derivatives.With the aid of the Exp-function method, white noise theory, and Hermite transform, we produce a novel set of exact soliton and periodic wave solutions to the fractional KdV equation with conformable derivatives. With the help of inverse Hermite transform, we get stochastic soliton and periodic wave solutions of the Wick-type stochastic fractional KdV equation with conformable derivatives. Eventually, by an application example, we show how the stochastic solutions can be given as Brownian motion functional solutions. 展开更多
关键词 korteweg de–vries(kdv)equation conformable DERIVATIVE stochastic BROWNIAN motion Expfunction method
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Artificial perturbation for solving the Korteweg-de Vries equation 被引量:1
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作者 KHELIL N. BENSALAH N. +1 位作者 SAIDI H. ZERARKA A. 《Journal of Zhejiang University-Science A(Applied Physics & Engineering)》 SCIE EI CAS CSCD 2006年第12期2079-2082,共4页
A perturbation method is introduced in the context of dynamical system for solving the nonlinear Korteweg-de Vries (KdV) equation. Best efficiency is obtained for few perturbative corrections. It is shown that, the qu... A perturbation method is introduced in the context of dynamical system for solving the nonlinear Korteweg-de Vries (KdV) equation. Best efficiency is obtained for few perturbative corrections. It is shown that, the question of convergence of this approach is completely guaranteed here, because a limited number of term included in the series can describe a sufficient exact solution. Comparisons with the solutions of the quintic spline, and finite difference are presented. 展开更多
关键词 PERTURBATION Taylor series Quintic spline korteweg-de vries kdv equation
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Travelling Solitary Wave Solutions to Higher Order Korteweg-de Vries Equation 被引量:3
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作者 Chunhuan Xiang Honglei Wang 《Open Journal of Applied Sciences》 2019年第5期354-360,共7页
The travelling solitary wave solutions to the higher order Korteweg-de Vries equation are obtained by using tanh-polynomial method. The method is effective and concise, which is also applied to various partial differe... The travelling solitary wave solutions to the higher order Korteweg-de Vries equation are obtained by using tanh-polynomial method. The method is effective and concise, which is also applied to various partial differential equations to obtain traveling wave solutions. The numerical simulation of the solutions is given for completeness. Numerical results show that the tanh-polynomial method works quite well. 展开更多
关键词 Higher Order korteweg-DE vries equation TRAVELLING WAVE Solutions SOLITARY WAVE
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Discrete Singular Convolution Method for Numerical Solutions of Fifth Order Korteweg-De Vries Equations 被引量:2
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作者 Edson Pindza Eben Maré 《Journal of Applied Mathematics and Physics》 2013年第7期5-15,共11页
A new computational method for solving the fifth order Korteweg-de Vries (fKdV) equation is proposed. The nonlinear partial differential equation is discretized in space using the discrete singular convolution (DSC) s... A new computational method for solving the fifth order Korteweg-de Vries (fKdV) equation is proposed. The nonlinear partial differential equation is discretized in space using the discrete singular convolution (DSC) scheme and an exponential time integration scheme combined with the best rational approximations based on the Carathéodory-Fejér procedure for time discretization. We check several numerical results of our approach against available analytical solutions. In addition, we computed the conservation laws of the fKdV equation. We find that the DSC approach is a very accurate, efficient and reliable method for solving nonlinear partial differential equations. 展开更多
关键词 FIFTH Order korteweg-DE vries equations Discrete Singular Convolution Exponential Time Discretization METHOD Soliton Solutions Conservation LAWS
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Martingale Solution to Stochastic Extended Korteweg-de Vries Equation 被引量:1
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作者 Anna Karczewska Maciej Szczeciński 《Advances in Pure Mathematics》 2018年第12期863-878,共16页
The deterministic extended Korteweg-de Vries equation plays an essential role in the description of the creation and propagation of nonlinear waves in many fields. We study a stochastic extended Korteweg-de Vries equa... The deterministic extended Korteweg-de Vries equation plays an essential role in the description of the creation and propagation of nonlinear waves in many fields. We study a stochastic extended Korteweg-de Vries equation driven by a multiplicative noise in the form of a cylindrical Wiener process. We prove the existence of a martingale solution to the equation studied for all physically relevant initial conditions. The proof of the solution is based on two approximations of the problem considered and the compactness method. 展开更多
关键词 EXTENDED korteweg-DE vries equation MARTINGALE SOLUTION STOCHASTIC Fluid Dynamics
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The extended symmetry approach for studying the general Korteweg-de Vries-type equation 被引量:1
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作者 李志芳 阮航宇 《Chinese Physics B》 SCIE EI CAS CSCD 2010年第4期3-10,共8页
The extended symmetry approach is used to study the general Korteweg-de Vries-type (KdV-type) equation. Several variable-coefficient equations are obtained. The solutions of these resulting equations can be construc... The extended symmetry approach is used to study the general Korteweg-de Vries-type (KdV-type) equation. Several variable-coefficient equations are obtained. The solutions of these resulting equations can be constructed by the solutions of original models if their solutions are well known, such as the standard constant coefficient KdV equation and the standard compound KdV--Burgers equation, and so on. Then any one of these variable-coefficient equations can be considered as an original model to obtain new variable-coefficient equations whose solutions can also be known by means of transformation relations between solutions of the resulting new variable-coefficient equations and the original equation. 展开更多
关键词 extended symmetry approach general korteweg-de vries-type kdv-type) equation variable-coefficient equation
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Second order conformal multi-symplectic method for the damped Korteweg–de Vries equation
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作者 Feng Guo 《Chinese Physics B》 SCIE EI CAS CSCD 2019年第5期20-26,共7页
A conformal multi-symplectic method has been proposed for the damped Korteweg–de Vries(DKdV) equation, which is based on the conformal multi-symplectic structure. By using the Strang-splitting method and the Preissma... A conformal multi-symplectic method has been proposed for the damped Korteweg–de Vries(DKdV) equation, which is based on the conformal multi-symplectic structure. By using the Strang-splitting method and the Preissmann box scheme,we obtain a conformal multi-symplectic scheme for multi-symplectic partial differential equations(PDEs) with added dissipation. Applying it to the DKdV equation, we construct a conformal multi-symplectic algorithm for it, which is of second order accuracy in time. Numerical experiments demonstrate that the proposed method not only preserves the dissipation rate of mass exactly with periodic boundary conditions, but also has excellent long-time numerical behavior. 展开更多
关键词 CONFORMAL MULTI-SYMPLECTIC METHOD DAMPED korteweg–de vries (kdv) equation DISSIPATION preservation
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An improved element-free Galerkin method for solving the generalized fifth-order Korteweg-de Vries equation
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作者 冯昭 王晓东 欧阳洁 《Chinese Physics B》 SCIE EI CAS CSCD 2013年第7期320-327,共8页
In this paper, an improved element-free Galerkin (IEFG) method is proposed to solve the generalized fifth-order Korteweg-de Vries (gfKdV) equation. When the traditional element-free Galerkin (EFG) method is used... In this paper, an improved element-free Galerkin (IEFG) method is proposed to solve the generalized fifth-order Korteweg-de Vries (gfKdV) equation. When the traditional element-free Galerkin (EFG) method is used to solve such an equation, unstable or even wrong numerical solutions may be obtained due to the violation of the consistency conditions of the moving least-squares (MLS) shape functions. To solve this problem, the EFG method is improved by employing the improved moving least-squares (IMLS) approximation based on the shifted polynomial basis functions. The effectiveness of the IEFG method for the gfKdV equation is investigated by using some numerical examples. Meanwhile, the motion of single solitary wave and the interaction of two solitons are simulated using the IEFG method. 展开更多
关键词 element-free Galerkin method shifted polynomial basis generalized fifth-order korteweg–de vries equation solitary wave
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Exact explicit solitary wave and periodic wave solutions and their dynamical behaviors for the Schamel–Korteweg–de Vries equation
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作者 Bin He Qing Meng 《Chinese Physics B》 SCIE EI CAS CSCD 2021年第6期62-76,共15页
The Schamel–Korteweg–de Vries equation is investigated by the approach of dynamics.The existences of solitary wave including ω-shape solitary wave and periodic wave are proved via investigating the dynamical behavi... The Schamel–Korteweg–de Vries equation is investigated by the approach of dynamics.The existences of solitary wave including ω-shape solitary wave and periodic wave are proved via investigating the dynamical behaviors with phase space analyses.The sufficient conditions to guarantee the existences of the above solutions in different regions of the parametric space are given.All possible exact explicit parametric representations of the waves are also presented.Along with the details of the analyses,the analytical results are numerically simulated lastly. 展开更多
关键词 Schamel–korteweg–de vries equation dynamical behavior solitary wave solution periodic wave solution
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A second-order convergent and linearized difference schemefor the initial-boundary value problemof the Korteweg-de Vries equation
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作者 Wang Xuping Sun Zhizhong 《Journal of Southeast University(English Edition)》 EI CAS 2022年第2期203-212,共10页
To numerically solve the initial-boundary value problem of the Korteweg-de Vries equation,an equivalent coupled system of nonlinear equations is obtained by the method of reduction of order.Then,a difference scheme is... To numerically solve the initial-boundary value problem of the Korteweg-de Vries equation,an equivalent coupled system of nonlinear equations is obtained by the method of reduction of order.Then,a difference scheme is constructed for the system.The new variable introduced can be separated from the difference scheme to obtain another difference scheme containing only the original variable.The energy method is applied to the theoretical analysis of the difference scheme.Results show that the difference scheme is uniquely solvable and satisfies the energy conservation law corresponding to the original problem.Moreover,the difference scheme converges when the step ratio satisfies a constraint condition,and the temporal and spatial convergence orders are both two.Numerical examples verify the convergence order and the invariant of the difference scheme.Furthermore,the step ratio constraint is unnecessary for the convergence of the difference scheme.Compared with a known two-level nonlinear difference scheme,the proposed difference scheme has more advantages in numerical calculation. 展开更多
关键词 korteweg-de vries(kdv)equation linearized difference scheme conservation convergence
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Deformed two-dimensional rogue waves in the (2+1)-dimensional Korteweg–de Vries equation
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作者 Yulei Cao Peng-Yan Hu +1 位作者 Yi Cheng Jingsong He 《Chinese Physics B》 SCIE EI CAS CSCD 2021年第3期205-214,共10页
Within the(2+1)-dimensional Korteweg–de Vries equation framework,new bilinear B¨acklund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation.By introducing an a... Within the(2+1)-dimensional Korteweg–de Vries equation framework,new bilinear B¨acklund transformation and Lax pair are presented based on the binary Bell polynomials and gauge transformation.By introducing an arbitrary functionφ(y),a family of deformed soliton and deformed breather solutions are presented with the improved Hirota’s bilinear method.By choosing the appropriate parameters,their interesting dynamic behaviors are shown in three-dimensional plots.Furthermore,novel rational solutions are generated by taking the limit of the obtained solitons.Additionally,twodimensional(2D)rogue waves(localized in both space and time)on the soliton plane are presented,we refer to them as deformed 2D rogue waves.The obtained deformed 2D rogue waves can be viewed as a 2D analog of the Peregrine soliton on soliton plane,and its evolution process is analyzed in detail.The deformed 2D rogue wave solutions are constructed successfully,which are closely related to the arbitrary functionφ(y).This new idea is also applicable to other nonlinear systems. 展开更多
关键词 two-dimensional(2D)korteweg-de vries(kdv)equation Bilinear method Backlund transformation Lax pair deformed 2D rogue wave
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The Pfaffian Technique: A (2 + 1)-Dimensional Korteweg de Vries Equation
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作者 Lixiao Zhai Junxiao Zhao 《Journal of Applied Mathematics and Physics》 2016年第10期1930-1935,共6页
The (2 + 1)-dimensional Korteweg de Vries (KdV) equation, which was first derived by Boiti et al., has been studied by various distinct methods. It is known that this (2 + 1)-dimensional KdV equation has rich solution... The (2 + 1)-dimensional Korteweg de Vries (KdV) equation, which was first derived by Boiti et al., has been studied by various distinct methods. It is known that this (2 + 1)-dimensional KdV equation has rich solutions, such as multi-soliton solutions and dromion solutions. In the present article, a unified representation of its N-soliton solution is given by means of pfaffian. We’ll show that this (2 + 1)-dimensional KdV equation is nothing but the Plücker identity when its &#964-function is given by pfaffian. 展开更多
关键词 The (2 + 1)-Dimensional korteweg de vries equation Hirota Bilinear Method PFAFFIAN Plücker Identity
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Existence for a Higher Order Coupled System of Korteweg-de Vries Equations
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作者 Min Liu 《Applied Mathematics》 2021年第4期298-310,共13页
Consider the following system of coupled Korteweg-de Vries equations, <img src="Edit_81ea1215-e696-403f-9d6c-1449e107359f.bmp" alt="" /><span style="white-space:nowrap;">where... Consider the following system of coupled Korteweg-de Vries equations, <img src="Edit_81ea1215-e696-403f-9d6c-1449e107359f.bmp" alt="" /><span style="white-space:nowrap;">where<em> u</em>, <em>v </em><span style="white-space:nowrap;">&#8838;</span> <em>W</em><sup>2,2</sup>, 2≤<em>N</em>≤7 and <em>λ</em><sub><em>i</em></sub>,<em>β</em> > 0, <em>β</em> </span>denotes a real coupling parameter. Firstly, we prove the existence of the solutions of a coupled system of Korteweg-de Vries equations using variation approach and minimization techniques on Nehari manifold. Then, we show the multiplicity of the equations by a bifurcation theory which is rare for studying higher order equations. 展开更多
关键词 System of korteweg-de vries equations Normalized Vector Solitary Waves Variation Approach
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Double Elzaki Transform Decomposition Method for Solving Third Order Korteweg-De-Vries Equations
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作者 Moh A. Hassan Tarig M. Elzaki 《Journal of Applied Mathematics and Physics》 2021年第1期21-30,共10页
In this study, we used Double Elzaki Transform (DET) coupled with Adomian polynomial to produce a new method to solve Third Order Korteweg-De Vries Equations (KdV) equations. We will provide the necessary explanation ... In this study, we used Double Elzaki Transform (DET) coupled with Adomian polynomial to produce a new method to solve Third Order Korteweg-De Vries Equations (KdV) equations. We will provide the necessary explanation for this method with addition some examples to demonstrate the effectiveness of this method. 展开更多
关键词 Double Elzaki Transform Adomian Polynomial korteweg-De vries equations
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The Dispersion Relation of Internal Wave Extended-Korteweg-de Vries Equation in a Two-Layer Fluid
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作者 Pingping Feng Xianghua Meng 《Journal of Applied Mathematics and Physics》 2021年第5期1056-1064,共9页
To understand the characteristics of ocean internal waves better, we study the dispersion relation of extended-Korteweg-de Vries (EKdV) equation with quadratic and cubic nonlinear terms in a two-layer fluid by using t... To understand the characteristics of ocean internal waves better, we study the dispersion relation of extended-Korteweg-de Vries (EKdV) equation with quadratic and cubic nonlinear terms in a two-layer fluid by using the Poincaré-Lighthill-Kuo (PLK) method which is one of the perturbation methods. Starting from the partial differential equation, the PLK method can be used to solve the dispersion relation of the equation. In this paper, we use PLK method to solve the equation and derive the dispersion relation of EKdV equation which is related to wave number and amplitude. Based on the dispersion relation obtained in this paper, the expressions of group velocity and phase velocity of the equation are obtained. Under the actual hydrological data, the influence of hydrological parameters on the dispersion relation for descending internal wave is discussed. It is hope that the obtained results will be helpful to the study of energy transfer and other internal wave parameters in the future. 展开更多
关键词 Ocean Internal Waves Dispersion Relation Extended-korteweg-de vries equation Poincaré-Lighthill-Kuo Method
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从Newton定律到广义Hamiltonian系统(Ⅲ)——关于Korteweg-de Vries(KdV)类型的非线性发展方程的一个估计
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作者 张海清 黄迅成 《扬州职业大学学报》 2007年第1期26-31,共6页
Newton定律是描述物体运动的基本定律,Hamiltonian方程则为运动的基本规律提供了另外一种表达。由Hamiltonian方程发展而来的Hamiltonian可积系统是现代孤立子理论的重要组成部分。文中证明了一个关于Korteweg-de Vries(KdV)类型的非线... Newton定律是描述物体运动的基本定律,Hamiltonian方程则为运动的基本规律提供了另外一种表达。由Hamiltonian方程发展而来的Hamiltonian可积系统是现代孤立子理论的重要组成部分。文中证明了一个关于Korteweg-de Vries(KdV)类型的非线性发展方程的在加权Sobolev空间中的估计式。这一估计式对证明一类一般的非线性扩散型发展方程的不变性质是非常有用的。 展开更多
关键词 Hamiltonian方程 korteweg—de vries方程 非线性发展方程
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基于Korteweg-de Vries方程解析解的海洋内波模拟研究 被引量:2
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作者 李娟 顾行发 +4 位作者 余涛 孙源 郭丁 徐京萍 董文 《海洋通报》 CAS CSCD 北大核心 2011年第1期23-28,共6页
主要是利用非线性发展方程丰富的解析解,基于含有散射项和微扰项的Korteweg-de Vries方程对台湾东北部东海海域内波的传播特性进行研究,并着重分析海洋内波在SAR图像上的信号特征,进而讨论耗散项和微扰项对海洋内波所引起的表层流速变... 主要是利用非线性发展方程丰富的解析解,基于含有散射项和微扰项的Korteweg-de Vries方程对台湾东北部东海海域内波的传播特性进行研究,并着重分析海洋内波在SAR图像上的信号特征,进而讨论耗散项和微扰项对海洋内波所引起的表层流速变化的影响。 展开更多
关键词 海洋内波 korteweg-DE vries方程 解析解
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一类复合Burgers-Korteweg-de Vries方程的行波解和稳定性分析 被引量:3
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作者 苗宝军 李鹏 申建伟 《四川师范大学学报(自然科学版)》 CAS CSCD 北大核心 2009年第5期602-605,共4页
利用微分方程定性理论的相关方法和定理,对一类复合Burgers-Korteweg-de Vries方程进行研究,获得了方程行波解的存在性和唯一性,并给出了方程行波解的表达式,与此同时,研究了行波解的动力学行为和它们不同解的分岔.
关键词 定性理论 行波解 复合Burgers-korteweg-de vries方程 稳定性分析
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