Diverse soliton solutions and dynamical analysis of the discrete coupled mKdV equation with 4×4 Lax pair

Under consideration in this study is the discrete coupled modified Korteweg-de Vries(mKdV)equation with 4×4 Lax pair.Firstly,through using continuous limit technique,this discrete equation can be mapped to the coupled KdV and mKdV equations,which may depict the development of shallow water waves,the optical soliton propagation in cubic nonlinear media and the Alfven wave in a cold collision-free plasma.Secondly,the discrete generalized(r,N-r)-fold Darboux transformation is constructed and extended to solve this discrete coupled equation with the fourth-order linear spectral problem,from which diverse exact solutions including usual multi-soliton and semi-rational soliton solutions on the vanishing background,higher-order rational soliton and mixed hyperbolic-rational soliton solutions on the non-vanishing background are derived,and the limit states of some soliton and rational soliton solutions are analyzed by the asymptotic analysis technique.Finally,the numerical simulations are used to explore the dynamical behaviors of some exact soliton solutions.These results may be helpful for understanding some physical phenomena in fields of shallow water wave,optics,and plasma physics.

New exact solitary wave solutions to generalized mKdV equation and generalized Zakharov-Kuzentsov equation

In this paper, based on hyperbolic tanh-function method and homogeneous balance method, and auxiliary equation method, some new exact solitary solutions to the generalized mKdV equation and generalized Zakharov-Kuzentsov equation are constructed by the method of auxiliary equation with function transformation with aid of symbolic computation system Mathematica. The method is of important significance in seeking new exact solutions to the evolution equation with arbitrary nonlinear term.

A connection between the(G'/G)-expansion method and the truncated Painlevé expansion method and its application to the mKdV equation

Recently the （G′/G）-expansion method was proposed to find the traveling wave solutions of nonlinear evolution equations. This paper shows that the （G′/G）-expansion method is a special form of the truncated Painlev＇e expansion method by introducing an intermediate expansion method. Then the generalized （G′/G）-（G/G′） expansion method is naturally derived from the standpoint of the nonstandard truncated Painlev＇e expansion. The application of the generalized method to the mKdV equation shows that it extends the range of exact solutions obtained by using the （ G′/ G）-expansion method.

Riemann theta function periodic wave solutions for the variable-coefficient mKdV equation

In this paper, a variable-coefficient modified Korteweg-de Vries （vc-mKdV） equation is considered. Bilinear forms are presented to explicitly construct periodic wave solutions based on a multidimensional Riemann theta function, then the one and two periodic wave solutions are presented~ and it is also shown that the soliton solutions can be reduced from the periodic wave solutions.

mKdV Equation for Solitary Rossby Waves with Linear Topography Effect in Barotropic Fluids

The modify Korteweg-de Vries(mKdV) equations,governing the evolution of the amplitude of solitary Rossby waves,are derived from quasi-geostrophic vorticity equation by using the perturbation method.The result manifests that the linear topography effect with the change of latitude can induce solitary Rossby wave.

Discrete doubly periodic and solitary wave solutions for the semi-discrete coupled mKdV equations

In this paper, the improved Jacobian elliptic function expansion approach is extended and applied to constructing discrete solutions of the semi-discrete coupled modified Korteweg de Vries （mKdV） equations with the aid of the symbolic computation system Maple. Some new discrete Jacobian doubly periodic solutions are obtained. When the modulus m →1, these doubly periodic solutions degenerate into the corresponding solitary wave solutions, including kink-type, bell-type and other types of excitations.

mKdV Equation for the Amplitude of Solitary Rossby Waves in Stratified Shear Flows with a Zonal Shear Flow

In this paper,modified Korteweg-de Vries (mKdV) equations for the amplitude of solitary Rossby waves in stratified fluids with a zonal shear flow are derived by using a weakly nonlinear method.It is found that the coefficients of mKdV equations depend not only on the β effect and the Visl-Brunt frequency,but also on the basic shear flow.

Approximate analytic solutions for a generalized Hirota—Satsuma coupled KdV equation and a coupled mKdV equation

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.

Exponential-fraction trial function method to the 5th-order mKdV equation

This paper obtains some solutions of the 5th-order mKdV equation by using the exponential-fraction trial function method, such as solitary wave solutions, shock wave solutions and the hopping wave solutions. It successfully shows that this method may be valid for solving other nonlinear partial differential equations.

Approximate direct reduction method:infinite series reductions to the perturbed mKdV equation

The approximate direct reduction method is applied to the perturbed mKdV equation with weak fourth order dispersion and weak dissipation. The similarity reduction solutions of different orders conform to formal coherence, accounting for infinite series reduction solutions to the original equation and general formulas of similarity reduction equations. Painleve Ⅱ type equations, hyperbolic secant and Jacobi elliptic function solutions are obtained for zeroorder similarity reduction equations. Higher order similarity reduction equations are linear variable coefficient ordinary differential equations.

EXPLICIT EXACT SOLUTIONS FOR THE GENERALIZED COMBINED KdV AND MKdV EQUATION

With the aid of Mathematica and Wu elimination method,via using a new generalized ansatz and well known Riccati equation,thirty two families of explicit and exact solutions for the generalized combined KdV and mKdV equation are obtained,which contain new solitary wave solutions and periodic wave solutions.This approach can also be applied to other nonlinear evolution equations.

Novel Solutions of MKdV Equation with the Modified Bcklund Transformation

A modified version of the bilinear Bcklund transformation for the MKdV equation was given, with which some new solutions of the MKdV equation are obtained. The approach used here is general and can be applied to other soliton equations.

Explicit Solutions of the Coupled mKdV Equation by the Dressing Method via Local Riemann-Hilbert Problem

We study the coupled mKdV equation by the dressing method via local Riemann-Hilbert problem. With the help of the Lax pairs, we obtain the matrix Riemann-Hilbert problem with zeros. The explicit solutions for the coupled mKdV equation are derived with the aid of the regularization of the Riemann-Hilbert problem.

Approximate Analytical Solutions of Fractional Coupled mKdV Equation by Homotopy Analysis Method

In this paper, the approximate analytical solutions of the fractional coupled mKdV equation are obtained by homotopy analysis method (HAM). The method includes an auxiliary parameter which provides a convenient way of adjusting and controlling the convergence region of the series solution. The suitable value of auxiliary parameter is determined and the obtained results are presented graphically.

New travelling wave solutions for combined KdV-mKdV equation and （2＋1）-dimensional Broer- Kaup- Kupershmidt system

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.

A New Periodic Solution to Jacobi Elliptic Functions of MKdV Equation and BBM Equation

Based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equations. New exact solutions to the Jacobi elliptic function of MKdV equations and Benjamin-Bona-Mahoney （BBM） equations are obtained with the aid of computer algebraic system Maple. The method is also valid for other （l＋l）-dimensional and higher dimensional systems.

Soliton molecules and the CRE method in the extended mKdV equation

The soliton molecules of the(1+1)-dimensional extended modified Korteweg–de Vries(mKdV)system are obtained by a new resonance condition,which is called velocity resonance.One soliton molecule and interaction between a soliton molecule and one-soliton are displayed by selecting suitable parameters.The soliton molecules including the bright and bright soliton,the dark and bright soliton,and the dark and dark soliton are exhibited in figures 1–3,respectively.Meanwhile,the nonlocal symmetry of the extended mKdV equation is derived by the truncated Painlevémethod.The consistent Riccati expansion(CRE)method is applied to the extended mKdV equation.It demonstrates that the extended mKdV equation is a CRE solvable system.A nonauto-B?cklund theorem and interaction between one-soliton and cnoidal waves are generated by the CRE method.

Global Well-posedness for the Fifth-order mKdV Equation

We prove the global well-posedness for the Cauchy problem of fifth-order modified Korteweg-de Vries equation in Sobolev spaces H^3（R） for s〉-3/22.The main approach is the ＂I-method＂ together with the multilinear multiplier analysis.

Solutions to mKdV Eequation of Electromagnetic Wave Propagation in Cold Collisionless Plasma

The equation of electromagnetic wave propagation through cold collisionless plasma can be reduced to the modified Kortweg-de Vries （mKdV） equation. Using a new technique, whose keys are the trial solution in terms of the exponential function and the ideas of the like-terms＇ balance, some groups of accurate analytical solutions for this mKdV equation, such as solitary wave solutions, can be obtained. It is successfully shown that this method may be still valid for solving other nonlinear plasma equations.

An Eight Component Integrable Hamiltonian Hierarchy from a Reduced Seventh-Order Matrix Spectral Problem

We present an eight component integrable Hamiltonian hierarchy, based on a reduced seventh order matrix spectral problem, with the aim of aiding the study and classification of multicomponent integrable models and their underlying mathematical structures. The zero-curvature formulation is the tool to construct a recursion operator from the spatial matrix problem. The second and third set of integrable equations present integrable nonlinear Schrödinger and modified Korteweg-de Vries type equations, respectively. The trace identity is used to construct Hamiltonian structures, and the first three Hamiltonian functionals so generated are computed.