Application of the generalized Kudryashov method to the Eckhaus equation

In this paper, the generalized Kudryashov method is presented to seek exact solutions of the Eckhaus equation. From these solutions, we can derive solitary wave solutions as a special case. The proposed method is direct, effective and convenient and can be applied to many nonlinear evolution equations in mathematical physics

Integrable nonlinear evolution equations.

by Zheng Yu-kun.Thesis (Ph.D.)--Chinese University of Hong Kong, 1991.Includes bibliographical references.Preface --- p.1Chapter Chapter 1. --- Gauge Transformation and the Higher Order Korteweg-de Vries Equations --- p.6Chapter 1. --- Higher order KdV equations --- p.6Chapter 2. --- η2-dependent higher order mKdV equations --- p.9Chapter 3. --- η2-dependent Miura transformation and Backlund transformation --- p.13Chapter 4. --- Gauge transformation of the wave function --- p.15Chapter 5. --- Backlund transformation for the η2 -dependent higher order mKdV equation --- p.24Chapter 6. --- Applications --- p.25Chapter 7. --- References --- p.30Chapter Chapter 2. --- Solutions of a Nonisospectral and Variable Coefficient Korteweg-de Vries Equation --- p.31Chapter 1. --- Introduction --- p.31Chapter 2. --- Nonisospectral variable coefficient KdV-type equations --- p.32Chapter 3. --- Invariance of LP under the Crum transformation --- p.34Chapter 4. --- Backlund transformation for the h-t-KdV equation --- p.35Chapter 5. --- Solutions --- p.39Chapter 6. --- References --- p.43Chapter Chapter 3. --- Nonisospectral Variable Coefficient Higher Order Korteweg-de Vries Equations --- p.45Chapter 1. --- Introduction --- p.45Chapter 2. --- Nonisospectral t-ho-KdV equations --- p.47Chapter 3. --- Nonisospectral η2 dependent variable coefficient higher order modified KdV equation --- p.50Chapter 4. --- Backlund transformation and gauge transformation --- p.57Chapter 5. --- Example. Solutions of second order ni-t-KdV equation and its corresponding ni-t-η2-mKdV equation --- p.61Chapter 6. --- References --- p.66Chapter Chapter 4. --- Gauge and Backlund Transformations for the Variable Coefficient Higher-Order Modified Korteweg-de Vries Equation --- p.67Chapter 1. --- Introduction --- p.67Chapter 2. --- The t-ho-mKdV equation --- p.68Chapter 3. --- Some results about the t-ho-KdV equation --- p.74Chapter 4. --- A Backlund transformation for the t-ho-mKdV equation --- p.76Chapter 5. --- Gauge transformat ion and the Backlund transformation --- p.78Chapter 6. --- References --- p.85Chapter Chapter 5. --- Gauge and Backlund Transformat ions for the Generalized Sine-Gordon Equation and Its η Dependent Modified Equation --- p.86Chapter 1. --- Introduction --- p.86Chapter 2. --- Generalized Sine-Gordon equation --- p.87Chapter 3. --- Backlund transformation for the GSGE --- p.92Chapter 4. --- Gauge transformations for AKNS systems --- p.98Chapter 5. --- η dependent modified GSGE and its Backlund transformation --- p.102Chapter 6. --- Summary and example --- p.105Chapter 7. --- References --- p.110Chapter Chapter 6. --- Backlund Transformation for the Nonisospectral and Variable Coefficient Nonlinear Schrodinger Equation --- p.111Chapter 1. --- Introduction --- p.111Chapter 2. --- A generalized NLSE --- p.112Chapter 3. --- Γ Riccati equation system --- p.114Chapter 4. --- Invariance of the Γ-system --- p.116Chapter 5. --- Lax pair corresponding to the GNLSE --- p.119Chapter 6. --- BT´ةs for the Γ evolution equation and the GNLSE --- p.121Chapter 7. --- References --- p.126Chapter Chapter 7. --- Backlund Transformations for the Caudrey-Dodd-Gibbon-Sawada-Kotera Equation and Its λ-Modified Equation --- p.127Chapter 1. --- Introduction --- p.127Chapter 2. --- The CDGSKE and the λ-mCDGSKE --- p.128Chapter 3. --- The general solution for the scattering problem of the CDGSKE --- p.130Chapter 4. --- The BT for the λ-mCDGSKE --- p.135Chapter 5. --- The BT for the CDGSKE --- p.136Chapter 6. --- References --- p.139Summary --- p.14

Multiple Soliton Solutions for a New Generalization of the Associated Camassa-Holm Equation by Exp-Function Method

The Exp-function method is generalized to construct N-soliton solutions of a new generalization of the associated Camassa-Holm equation. As a result, one-soliton, two-soliton, and three-soliton solutions are obtained, from which the uniform formulae of N-soliton solutions are derived. It is shown that the Exp-function method may provide us with a straightforward, effective, and alternative mathematical tool for generating N-soliton solutions of nonlinear evolution equations in mathematical physics

Comment on “Application of (G′/G)-expansion method to travelling-wave solutions of three nonlinear evolution equation" [Comput Fluids 2010;39;1957-63]

In a recent paper [Abazari R. Application of (G′ G )-expansion method to travelling wave solutions of three nonlinear evolution equation. Computers & Fluids 2010;39:1957–1963], the (G′/G)-expansion method was used to find travelling-wave solutions to three nonlinear evolution equations that arise in the mathematical modelling of fluids. The author claimed that the method delivers more general forms of solution than other methods. In this note we point out that not only is this claim false but that the delivered solutions are cumbersome and misleading. The extended tanh-function expansion method, for example, is not only entirely equivalent to the (G′/G)-expansion method but is more efficient and user-friendly, and delivers solutions in a compact and elegant form

New multi-soliton solutions for generalized Burgers-Huxley equation

The double exp-function method is used to obtain a two-soliton solution of the generalized Burgers-Huxley equation. The wave has two different velocities and two different frequencies