A semi-discrete modified KdV equation

Fractal fractional analysis of modified KdV equation under three different kernels

The theoretical analysis of the lattice hydrodynamic models for traffic flow theory

KdV and kink–antikink solitons in car-following models

A transformation is introduced for generalized mKdV (GmKdV for short) equation and Jacobi elliptic function expansion method is applied to solve it. It is shown that GmKdV equation with a real number parameter can be solved directly by using Jacobi elliptic function expansion method when this transformation is introduced, and periodic solution and solitary wave solution are obtained. Then the generalized solution to GmKdV equation deduces to some special solutions to some well-known nonlinear equations, such as KdV equation, mKdV equation, when the real parameter is set specific values.

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

We address a specific but possible situation in natural water bodies when the three-layer stratification has a symmetric nature, with equal depths of the uppermost and the lowermost layers. In such case, the coefficients at the leading nonlinear terms of the modified Korteweg-de Vries (mKdV) equation vanish simultaneously. It is shown that in such cases there exists a specific balance between the leading nonlinear and dispersive terms. An extension to the mKdV equation is derived by means of combination of a sequence of asymptotic methods. The resulting equation contains a cubic and a quintic nonlinearity of the same magnitude and possesses solitary wave solutions of different polarity. The properties of smaller solutions resemble those for the solutions of the mKdV equation, whereas the height of the taller solutions is limited and they become table-like. It is demonstrated numerically that the collisions of solitary wave solutions to the resulting equation are weakly inelastic: the basic properties of the counterparts experience very limited changes but the interactions are certainly accompanied by a certain level of radiation of small-amplitude waves.

Numerical solutions of KDV and mKDV equations: Using sequence and multi-core parallelization implementation

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.

The effect of changing the direction of motion of a defect (a soliton of small amplitude) in soliton lattices described by the Korteweg–de Vries and modified Korteweg–de Vries integrable equations (KdV and mKdV) was studied. Manifestation of this effect is possible as a result of the negative phase shift of a small soliton at the moment of nonlinear interaction with large solitons, as noted in [1], within the KdV equation. In the recent paper [2], an expression for the mean soliton velocity in a “cold” KdV-soliton gas has been found using kinetic theory, from which this effect also follows, but this fact has not been mentioned. In the present paper, we will show that the criterion of negative velocity is the same for both the KdV and mKdV equations and it can be obtained using simple kinematic considerations without applying kinetic theory. The averaged dynamics of the “smallest” soliton (defect) in a soliton gas consisting of solitons with random amplitudes has been investigated and the average criterion of changing the sign of the velocity has been derived and confirmed by numerical solutions of the KdV and mKdV equations.

Solitary wave solutions for the KdV and mKdV equations by differential transform method

Bifurcation, bistability and solitary waves of the combined KdV and mKdV equation are investigated systematically. At first, bifurcation and bistability are analyzed by selecting an integral constant as the bifurcation parameter. Then, different conditions expressed in terms of the bifurcation parameter are obtained for the existence of breather-like, algebraic, pulse-like solitary waves, and shock waves. All types of the solitary wave and shock wave solutions are given by direct integration. Finally, an approximate analytic method by employing the interpolation polynomials is utilized to give simpler forms for the pulse-like solitary wave solutions. In view of the references, our results are the most complete and the theoretical methods are the simplest hitherto.

In this paper, by using the extended tanh-function method,the general variable coefficient KdV and MKdV equations are reduced to first-order variable coefficient nonlinear ordinary differential equations,and then the new exact solutions for these equations,which include exact soliton-like,rational formal and triangle function solutions,are obtained through solving these ordinary differential equations-

Compound KdV equation and MKdV equation are investigated in the presented work, with the aid of Matlab software, using the trigonometric function transform method and the Wu elimination method, and more exact traveling wave solutions are obtained for compound KdV equation and MKdV equation.

An infinite number of form-invariant symmetries is obtained and a one-to-one correspondence between symmetries and conservation laws is established for the non-isospectral and variable coefficient generalizations of both the KdV and the MKdV equations. Two families of symmetries and their Lie algebraic structures are constructed. Some interesting facts about their Hamiltonian structures are presented.

Lax representations and integrable time discretizations of the DDKdV, DDmKdV, and DDHOKdV