Breather-type soliton and two-soliton solutions for modified Korteweg-de Vries equation

In this Letter, different kinds of solutions including breather-type soliton and two-soliton solutions, are obtained for the modified Korteweg-de Vries (M-KdV) equation by using bilinear form, the extended homoclinic test approach and dependent variable transformations. Moreover,we point out that the author did not obtain so-called periodic two-soliton solutions in W. Long (in press) [1].     

Multiple soliton solutions and other exact solutions for a two‐mode KdV equation

In this work, we study the two‐mode Korteweg–de Vries (TKdV) equation, which describes the propagation of two different waves modes simultaneously. We show that the TKdV equation gives multiple soliton solutions for specific values of the nonlinearity and dispersion parameters involved in the equation. We also derive other distinct exact solutions for general values of these parameters. We apply the simplified Hirota's method to study the specific of the parameters, which gives multiple soliton solutions. We also use the tanh/coth method and the tan/cot method to obtain other set of solutions with distinct physical structures. Copyright © 2016 John Wiley & Sons, Ltd.     

Multiple soliton solutions for the sixth-order Ramani equation and a coupled Ramani equation

In this work, the completely integrable sixth-order nonlinear Ramani equation and a coupled Ramani equation are studied. Multiple soliton solutions and multiple singular soliton solutions are formally derived for these two equations. The Hirota’s bilinear method is used to determine the two distinct structures of solutions. The resonance relations for the three cases are investigated.     

New exact solutions for a generalization of the Korteweg-de Vries equation (KdV6)

The generalized tanh-coth method is used to construct periodic and soliton solutions for a new integrable system, which has been derived from an integrable sixth-order nonlinear wave equation (KdV6). The system is formed by two equations. One of the equations may be considered as a Korteweg-de Vries equation with a source and the second equation is a third-order linear differential equation.     

Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions

A broad set of sufficient conditions consisting of systems of linear partial differential equations is presented which guarantees that the Wronskian determinant solves the Korteweg-de Vries equation in the bilinear form. A systematical analysis is made for solving the resultant linear systems of second-order and third-order partial differential equations, along with solution formulas for their representative systems. The key technique is to apply variation of parameters in solving the involved non-homogeneous partial differential equations. The obtained solution formulas provide us with a comprehensive approach to construct the existing solutions and many new solutions including rational solutions, solitons, positons, negatons, breathers, complexitons and interaction solutions of the Korteweg-de Vries equation.

     

An extended simplest equation method and its application to several forms of the fifth-order KdV equation

In this paper, an extended simplest equation method is proposed to seek exact travelling wave solutions of nonlinear evolution equations. As applications, many new exact travelling wave solutions for several forms of the fifth-order KdV equation are obtained by using our method. The forms include the Lax, Sawada-Kotera, Sawada-Kotera-Parker-Dye, Caudrey-Dodd-Gibbon, Kaup-Kupershmidt, Kaup-Kupershmidt-Parker-Dye, and the Ito forms.     

一般变系数KdV方程的精确解

By asing the nonclassical method of symmetry reductions, the exact solutions for general variable-coefficient KdV equation with dissipative loss and nonuniformity terms are obtained. When the dissipative loss and nonuniformity terms don‘t exist, the multisoliton solutions are found and the corresponding Painleve II type equation for the variable-coefficient KdV equation is given.     

Exact solutions to mKdV equation with variable coefficients

In this paper a special mKdV with variable coefficients is considered. A transformation of variables is first applied in order to obtain a mKdV equation with constant coefficients. Some its one-, two- and three-soliton as well as breather-type soliton solutions are derived by using Hirorta’s bilinear approach.     

Binary Darboux transformation and soliton solutions for the coupled complex modified Korteweg-de Vries equations

This paper considers the coupled complex modified Korteweg-de Vries (mKdV) equations and presents a binary Darboux transformation for the equations. As a direct application, we give a classification of general soliton solutions derived from vanishing and non-vanishing backgrounds, on the basis of the dynamical behavior of the solutions. Special types of solutions in the presented solutions include breathers, bright-bright solitons, bright-dark solitons, bright-W-shaped solitons, and rogue wave solutions. Furthermore, dynamics and interactions of vector bright solitons are exhibited.     

Exp-function method for the exact solutions of fifth order KdV equation and modified Burgers equation

In this paper, we implemented the exp-function method for the exact solutions of the fifth order KdV equation and modified Burgers equation. By using this scheme, we found some exact solutions of the above-mentioned equations.     

On the numerical solution of Korteweg-de Vries equation by the iterative splitting method

In this paper, we apply the method of iterative operator splitting on the Korteweg-de Vries (KdV) equation. The method is based on first, splitting the complex problem into simpler sub-problems. Then each sub-equation is combined with iterative schemes and solved with suitable integrators. Von Neumann analysis is performed to achieve stability criteria for the proposed method applied to the KdV equation. The numerical results obtained by iterative splitting method for various initial conditions are compared with the exact solutions. It is seen that they are in a good agreement with each other.     

Exact travelling wave solutions of the dissipative coupled Korteweg-de Vries equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative coupled Korteweg-de Vries equation. The possible kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful.     

Bell polynomial approach to an extended Korteweg–de Vries equation

Under investigation in this paper is an extended Korteweg–de Vries equation. Via Bell polynomial approach and symbolic computation, this equation is transformed into two kinds of bilinear equations by choosing different coefficients, namely KdV–SK‐type equation and KdV–Lax‐type equation. On the one hand, N‐soliton solutions, bilinear Bäcklund transformation, Lax pair, Darboux covariant Lax pair, and infinite conservation laws of the KdV–Lax‐type equation are constructed. On the other hand, on the basis of Hirota bilinear method and Riemann theta function, quasiperiodic wave solution of the KdV–SK‐type equation is also presented, and the exact relation between the one periodic wave solution and the one soliton solution is established. It is rigorously shown that the one periodic wave solution tend to the one soliton solution under a small amplitude limit. Copyright © 2013 John Wiley & Sons, Ltd.     

Bäcklund transformations and soliton solutions for the KdV6 equation

Using Hirota technique, a Bäcklund transformation in bilinear form is obtained for the KdV6 equation. Furthermore, we present a modified Bäcklund transformation by a dependent variable transformation, it is shown that a new representation of N-soliton solution and some novel solutions to the KdV6 equation are derived by performing an appropriate limiting procedure on the known soliton solutions.     

A study on two extensions of the Bogoyavlenskii-Schieff equation

In this work, two extensions of the Bogoyavlenskii-Schieff equation are examined. N-soliton solutions are formally determined for each extended equation. We show that the extension terms do not kill the integrability of the typical Bogoyavlenskii-Schieff equation. The simplified Hirota’s method established by Hereman and Nuseir is applied to achieve this goal.     

Sub-ODE method and soliton solutions for the variable-coefficient mKdV equation

In this work, an auxiliary equation is used for an analytic study on the time-variable coefficient modified Korteweg-de Vries (mKdV) equation. Five sets of new exact soliton-like solutions are obtained. The results show that the pulse parameters are time-dependent variable coefficients. Moreover, the basic conditions for the formation of derived solutions are presented.     

A non-homogeneous boundary-value problem for the Korteweg-de Vries equation in a quarter plane

The Korteweg-de Vries equation was first derived by Boussinesq and Korteweg and de Vries as a model for long-crested small-amplitude long waves propagating on the surface of water. The same partial differential equation has since arisen as a model for unidirectional propagation of waves in a variety of physical systems. In mathematical studies, consideration has been given principally to pure initial-value problems where the wave profile is imagined to be determined everywhere at a given instant of time and the corresponding solution models the further wave motion. The practical, quantitative use of the Korteweg-de Vries equation and its relatives does not always involve the pure initial-value problem. Instead, initial-boundary-value problems often come to the fore. A natural example arises when modeling the effect in a channel of a wave maker mounted at one end, or in modeling near-shore zone motions generated by waves propagating from deep water. Indeed, the initial-boundary-value problem

studied here arises naturally as a model whenever waves determined at an entry point propagate into a patch of a medium for which disturbances are governed approximately by the Korteweg-de Vries equation. The present essay improves upon earlier work on (0.1) by making use of modern methods for the study of nonlinear dispersive wave equations. Speaking technically, local well-posedness is obtained for initial data in the class for \frac34$"> and boundary data in , whereas global well-posedness is shown to hold for when , and for when . In addition, it is shown that the correspondence that associates to initial data and boundary data the unique solution of (0.1) is analytic. This implies, for example, that solutions may be approximated arbitrarily well by solving a finite number of linear problems.

     

Boundary controllability for the time-fractional nonlinear Korteweg-de Vries (KdV) equation

In this paper, we study the time-fractional nonlinear Korteweg-de Vries (KdV) equation. By using the theory of semigroups, we prove the well-posedness of the time-fractional nonlinear KdV equation. Moreover, we present the boundary controllability result for the problem.