Some new exact solutions of an auxiliary ordinary differential equation are obtained, which were neglected by Sirendaoreji et al in their auxiliary equation method. By using this method and these new solutions the com...Some new exact solutions of an auxiliary ordinary differential equation are obtained, which were neglected by Sirendaoreji et al in their auxiliary equation method. By using this method and these new solutions the combined Korteweg-de Vries （KdV） and modified KdV （mKdV） equation and （2＋1）-dimensional Broer-Kaup-Kupershmidt system are investigated and abundant exact travelling wave solutions are obtained that include new solitary wave solutions and triangular periodic wave solutions.展开更多
In this paper the ( G'/G )-expansion method is used to find exact travelling wave solutions for a combined KdV and Schwarzian KdV equation. As a result, multiple travelling wave solutions with arbitrary parameters...In this paper the ( G'/G )-expansion method is used to find exact travelling wave solutions for a combined KdV and Schwarzian KdV equation. As a result, multiple travelling wave solutions with arbitrary parameters are obtained, which are expressed by hyperbolic functions, trigonometric functions and rational functions. When the parameters are taken as special values, the solitary waves are derived from the travelling waves. The (G'/G)-expansion method presents a wider applicability for handling nonlinear wave equations.展开更多
This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries （KdV） equation and a coupled modified Korteweg-de Vries （mKdV）...This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries （KdV） equation and a coupled modified Korteweg-de Vries （mKdV） equation. This method provides a sequence Of functions which converges to the exact solution of the problem and is based on the use of Lagrange multiplier for identification of optimal values of parameters in a functional. Some examples are given to demonstrate the reliability and convenience of the method and comparisons are made with the exact solutions.展开更多
The authors develop a direct approach to the soliton perturbation based on the separation of variables. With the aid of approach, the first-order effects of perturbation on a KdV-MKdV soliton are derived, both the slo...The authors develop a direct approach to the soliton perturbation based on the separation of variables. With the aid of approach, the first-order effects of perturbation on a KdV-MKdV soliton are derived, both the slow time-dependence of the soliton parameters and the first-order correction are obtained.展开更多
In this paper, we consider small perturbations of the KdV-mKdV equation u_t =-u_(xxx) + 6 uu_x + 6 u^2 u_x on the real line with periodic boundary conditions. It is shown that the above equation admits a Cantor family...In this paper, we consider small perturbations of the KdV-mKdV equation u_t =-u_(xxx) + 6 uu_x + 6 u^2 u_x on the real line with periodic boundary conditions. It is shown that the above equation admits a Cantor family of small amplitude quasi-periodic solutions under such perturbations. The proof is based on an abstract infinite dimensional KAM theorem.展开更多
Higher-order Korteweg-de Vries （KdV）-modified KdV （mKdV） equations with a higher-degree of nonlinear terms are derived from a simple incompressible non-hydrostatic Boussinesq equation set in atmosphere and are use...Higher-order Korteweg-de Vries （KdV）-modified KdV （mKdV） equations with a higher-degree of nonlinear terms are derived from a simple incompressible non-hydrostatic Boussinesq equation set in atmosphere and are used to investigate gravity waves in atmosphere. By taking advantage of the auxiliary nonlinear ordinary differential equation, periodic wave and solitary wave solutions of the fifth-order KdV-mKdV models with higher-degree nonlinear terms are obtained under some constraint conditions. The analysis shows that the propagation and the periodic structures of gravity waves depend on the properties of the slope of line of constant phase and atmospheric stability. The Jacobi elliptic function wave and solitary wave solutions with slowly varying amplitude are transformed into triangular waves with the abruptly varying amplitude and breaking gravity waves under the effect of atmospheric instability.展开更多
基金Project supported by the National Natural Science Foundation of China （Grant No 10472029）.
文摘Some new exact solutions of an auxiliary ordinary differential equation are obtained, which were neglected by Sirendaoreji et al in their auxiliary equation method. By using this method and these new solutions the combined Korteweg-de Vries （KdV） and modified KdV （mKdV） equation and （2＋1）-dimensional Broer-Kaup-Kupershmidt system are investigated and abundant exact travelling wave solutions are obtained that include new solitary wave solutions and triangular periodic wave solutions.
基金Supported by the Natural Science Foundation of Education Department of Henan Province（2011Bl10013） Supported by the Youth Science Foundation of Henan University of Science and Tech- nology（2008QN026）
文摘In this paper the ( G'/G )-expansion method is used to find exact travelling wave solutions for a combined KdV and Schwarzian KdV equation. As a result, multiple travelling wave solutions with arbitrary parameters are obtained, which are expressed by hyperbolic functions, trigonometric functions and rational functions. When the parameters are taken as special values, the solitary waves are derived from the travelling waves. The (G'/G)-expansion method presents a wider applicability for handling nonlinear wave equations.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.10771019 and 10826107)
文摘This paper applies the variational iteration method to obtain approximate analytic solutions of a generalized Hirota-Satsuma coupled Korteweg-de Vries （KdV） equation and a coupled modified Korteweg-de Vries （mKdV） equation. This method provides a sequence Of functions which converges to the exact solution of the problem and is based on the use of Lagrange multiplier for identification of optimal values of parameters in a functional. Some examples are given to demonstrate the reliability and convenience of the method and comparisons are made with the exact solutions.
基金the National Science Foundation of China(19775013)
文摘The authors develop a direct approach to the soliton perturbation based on the separation of variables. With the aid of approach, the first-order effects of perturbation on a KdV-MKdV soliton are derived, both the slow time-dependence of the soliton parameters and the first-order correction are obtained.
基金Supported by NSFC(11601036,11401041)Science and Technology Foundation of Shandong Province(J16LI52)
文摘In this paper, we consider small perturbations of the KdV-mKdV equation u_t =-u_(xxx) + 6 uu_x + 6 u^2 u_x on the real line with periodic boundary conditions. It is shown that the above equation admits a Cantor family of small amplitude quasi-periodic solutions under such perturbations. The proof is based on an abstract infinite dimensional KAM theorem.
基金Project supported by the National Natural Science Foundation of China (Grant No 40775069)
文摘Higher-order Korteweg-de Vries （KdV）-modified KdV （mKdV） equations with a higher-degree of nonlinear terms are derived from a simple incompressible non-hydrostatic Boussinesq equation set in atmosphere and are used to investigate gravity waves in atmosphere. By taking advantage of the auxiliary nonlinear ordinary differential equation, periodic wave and solitary wave solutions of the fifth-order KdV-mKdV models with higher-degree nonlinear terms are obtained under some constraint conditions. The analysis shows that the propagation and the periodic structures of gravity waves depend on the properties of the slope of line of constant phase and atmospheric stability. The Jacobi elliptic function wave and solitary wave solutions with slowly varying amplitude are transformed into triangular waves with the abruptly varying amplitude and breaking gravity waves under the effect of atmospheric instability.
