New Explicit Solitary Wave Solutions and Periodic Wave Solutions for the Generalized Coupled Hirota-Satsuma KdV System

In this paper, we study the generalized coupled Hirota Satsuma KdV system by using the new generalizedtransformation in homogeneous balance method. As a result, many explicit exact solutions, which contain new solitarywave solutions, periodic wave solutions, and the combined formal solitary wave solutions, and periodic wave solutions,are obtained.     

Variable Separation and Derivative-Dependent Functional Separable Solutions to Generalized KdV Equations

Using the generalized conditional symmetry approach, a complete list of canonicalforms for the Kortewegde-Vries type equations with which possessing derivative-dependent functional separable solutions (DDFSSs) is obtained.The exact DDFSSs of the resulting equations are explicitly exhibited.     

Nonclassical Symmetry Reductions for the Integrable Super KdV Equations

Using the direct method introduced by Clarkson and Kruskal (CK), we obtain similarity reductions of the integrable super Kd V equations. The group explanations of the results are also given.     

Deforming the Travelling Wave Solutions of the KdV Equation to Those of the Generalized Fifth Order KdV Equation

Using some limiting procedures, the solutions of the fifth order KdV equation u_t + (μu²+ υu_xx + αuu_xx + βu_x² + γu³ + δu_xxxx)_x = 0 would degenerate into the solutions of a simple equation, say KdV equation. In this letter, we analyze the possibility of the inverse procedure of the limiting process mentioned above for the travelling wave solutions. The results show that the procedure for deforming a travelling wave solution of the KdV equation to that of the generalized fifth order KdV equation can be accomplished by some pure algebraic tricks. Moreover, this inverse procedure is not unique in general.     

Solitary Wave and Non-traveling Wave Solutions to Two Nonlinear Evolution Equations

By applying the extended homogeneous balance method, we find some new explicit solutions to two nonlinear evolution equations, which include n-resonance plane solitary wave and non-traveling wave solutions.     

New Periodic Wave Solutions to Generalized Klein-Gordon and Benjamin Equations

By making use of extended mapping method and auxiliary equation for finding new periodic wave solu tions of nonlinear evolution equations in mathematical physics, we obtain some new periodic wave solutions for generalized Klein-Cordon equation and Benjamin equation, which cannot be found in previous work. This method also can be used to find new periodic wave solutions of other nonlinear evolution equations.     

Solitary Wave and Non-traveling Wave Solutions to Two Nonlinear Evolution Equations

By applying the extended homogeneous balance method, we find some new explicit solutions to two nonlinear evolution equations, which include n-resonance plane solitary wave and non-traveling wave solutions.     

The Periodic Wave Solutions for Two Nonlinear Evolution Equations

By using the F-expansion method proposed recently, the periodic wave solutions expressed by Jacobi elliptic functions for two nonlinear evolution equations are derived. In the limit cases, the solitary wave solutions and the other type of traveling wave solutions for the system are obtained.     

The Peridic Wave Solutions for Two Nonlinear Evolution Equations

By using the F-expansion method proposed recently, the periodic wave solutions expressed by Jacobielliptic functions for two nonlinear evolution equations are derived. In the limit cases, the solitary wave solutions andthe other type of traveling wave solutions for the system are obtained.     

Exact Solutions of Bogoyavlenskii Coupled KdV Equations

The special soliton solutions of Bogoyavlenskii coupled KdV equations are obtained by means of the standard Weiss-Tabor-Carnvale Painlev\'{e} truncation expansion and the nonstandard truncation of a modified Conte's invariant Painlev\'{e} expansion.     

Exact Solutions of Bogoyavlenskii Coupled KdV Equations

The special soliton solutions of Bogoyavlenskii coupled KdV equations are obtained by means of the standard Weiss-Tabor-Carnvale Painleve truncation expansion and the nonstandard truncation of a modified Conte‘s invariant Painleve expansion.     

Higher-Dimensional KdV Equations and Their Soliton Solutions

A (2+1)-dimensional KdV equation is obtained by use of Hirota method, which possesses N-soliton solution, specially its exact two-soliton solution is presented. By employing a proper algebraic transformation and the Riccati equation, a type of bell-shape soliton solutions are produced via regarding the variable in the Riccati equation as the independent variable. Finally, we extend the above (2+1)-dimensional KdV equation into (3+1)-dimensional equation, the two-soliton solutions are given.     

Soliton Solutions to Coupled Integrable Dispersionless Equations

In this paper, by introducing a new transformation, the bilinear form of the coupled integrable dispersionless （CID） equations is derived. It will be shown that this bilineax form is easier to perform the standard Hirota process. One-, two-, and three-soliton solutions are presented. Furthermore, the N-soliton solutions axe derived.     

Soliton Solutions to Coupled Integrable Dispersionless Equations

In this paper, by introducing a new transformation, the bilinear form of the coupled integrable dispersionless (CID) equations is derived. It will be shown that this bilinear form is easier to perform the standard Hirota process. One-, two-, and three-soliton solutions are presented. Furthermore, the N-soliton solutions are derived.     

Equivalence of Two Approaches to Integrable Hierarchies of KdV type

The equivalence between the approaches of Drinfeld-Sokolov and Feigin-Frenkel to the mKdV and KdV hierarchies is established. A new derivation of the mKdV equations in the zero curvature form is given. Connection with the Baker-Akhiezer function and the tau-function is also discussed. Received: Received: 1 July 1996 / Accepted: 21 October 1996     

On the Self-Similar Solutions of Generalized Hydrodynamics Equations and Nonlinear Wave Patterns

Abstract

Solutions of the system of dynamical equations of state and equations of the balance of mass and momentum are studied. The system possesses families of periodic, quasiperiodic and soliton-like invariant solutions. Self-similar solutions of this generalized hydrodynamic system are studied. Various complicated regimes, arising as a result with terms desribing relaxing and dissipative properties of the medium are described.     

Solitary Wave Solutions for the Coupled Ito System and a Generalized Hirota-Satsuma Coupled KdV System

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