The large-time asymptotic solution of the mKdV equation

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

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

In this paper, we consider an initial-value problem for the Korteweg–de Vries equation. The normalized Korteweg–de Vries equation considered is given by where x and τ represent dimensionless distance and time, respectively. In particular, we consider the case when the initial data has a discontinuous expansive step, where u(x, 0) = u0 for x ⩾ 0 and u(x, 0) = 0 for x < 0. The method of matched asymptotic coordinate expansions is used to obtain the large-τ asymptotic structure of the solution to this problem, which exhibits the formation of an expansion wave in x ⩾ 0, while the solution is oscillatory in x < 0, with the oscillatory envelope being of as τ → ∞.

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.

In this paper, we present an integrable semi-discretization of the modified Korteweg-deVries (mKdV) equation. We discretize the “time” variable of the mKdV equation and get an integrable differential-difference system. Under a standard limit, the differential-difference system converges to the continuous mKdV equation. By Hirota’s bilinear method, we find some explicit solutions including solitons and breather solutions. From the semi-discrete system, we design a numerical scheme to the mKdV equation and carry out several numerical experiments with the 3-soliton solution and breather solution.

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

Generalized inversion of the Korteweg–de Vries equation

KdV and kink–antikink solitons in car-following models

Fractal fractional analysis of modified KdV equation under three different kernels

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.

In this paper, existing classical Korteweg‐de Vries (KdV) equations are converted into the corresponding time‐fractional KdV equations by using Caputo‐Fabrizio fractional derivative and then solved with appropriate initial conditions by implementing semi‐numerical technique, that is, Laplace transform together with an iterative scheme. The obtained solutions are novel, and previous literature lacks such derivations. The stability of implemented technique is analyzed by applying Banach contraction principle and S‐stable mapping. Efficiency of Caputo‐Fabrizio fractional derivative is exhibited through graphical illustrations, and fractional results are drafted in tabular form for specific values of fractional parameter to validate the numerical investigation.

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

The initial value problems for the Korteweg-de Vries (KdV) and modified KdV (mKdV) equations under periodic and decaying boundary conditions are considered. These initial value problems are shown to be globally well-posed in all L 2 L^2 -based Sobolev spaces H s H^s where local well-posedness is presently known, apart from the H 1 4 ( R ) H^{\frac {1}{4}} (\mathbb {R} ) endpoint for mKdV and the H − 3 4 H^{-\frac {3}{4}} endpoint for KdV. The result for KdV relies on a new method for constructing almost conserved quantities using multilinear harmonic analysis and the available local-in-time theory. Miura’s transformation is used to show that global well-posedness of modified KdV is implied by global well-posedness of the standard KdV equation.

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.