We investigate the large-time asymptotics of solution for Cauchy problem nonlocal focusing modified Kortweg-de Vries (MKdV) equation with step-like initial data, i.e., $u_0(x)\rightarrow 0$ as $x\rightarrow-\infty$, A$ $x\rightarrow+\infty$,where $A$ is an arbitrary positive real number. firstly develop direct scattering theory to establish basic Riemann-Hilbert (RH) associated data. Thanks sym...

In this paper, we obtain µ -symmetry and -conservation law of the extended mKdV equation. The equation dose not admit a variational problem since it is odd order. First in potential form because admits problem, using it, can

An action is constructed that gives an arbitrary equation in the KdV or MKdV hierarchies as equation of motion; the second Hamiltonian structure of the KdV equation and the Hamiltonian structure of the MKdV equation appear as Poisson bracket structures derived from this action. Quantization of this theory can be carried out in two different schemes, to obtain either the quantum KdV theory of Ku...

In this paper, we consider the complex modified Korteweg-de Vries (mKdV) equation as a model of few-cycle optical pulses. Using the Lax pair, we construct a generalized Darboux transformation and systematically generate the first-, second-, and third-order rogue wave solutions and analyze the nature of evolution of higher-order rogue waves in detail. Based on detailed numerical and analytical i...

Using the Inverse Scattering Method with a nonvanishing boundary condition, we obtain an explicit breather solution nonzero vacuum parameter b of focusing modified Korteweg–de Vries (mKdV) equation. Moreover, taking limiting case zero frequency, generalization double pole introduced by M. Wadati et al.

A new method of constructing elliptic finite-gap solutions of the stationary Korteweg-de Vries (KdV) hierarchy, based on a theorem due to Picard, is illustrated in the concrete case of the Lamé-Ince potentials −s(s+1)P(z), s ∈ N (P(.) the elliptic Weierstrass function). Analogous results are derived in the context of the stationary modified Korteweg-de Vries (mKdV) hierarchy for the first time.

Integrable equations in (1 + 1) dimensions have their own higher order integrable equations, like the KdV, mKdV and NLS hierarchies etc. In this paper we consider whether integrable equations in (2 + 1) dimensions have also the analogous hierarchies to those in (1 + 1) dimensions. Explicitly is discussed the Bogoyavlenskii-Schiff(BS) equation. For the BS hierarchy, there appears an ambiguity in...

We discuss the canonical structure of a class of integrable quantum mappings, i.e. iterative canonical transformations that can be interpreted as a discrete dynamical system. As particular examples we consider quantum map-pings associated with the lattice analogues of the KdV and MKdV equations. These mappings possess a non-ultralocal quantum Yang-Baxter structure leading to the existence of co...

A generalized ( G′ G ) -expansion method is devised to construct the exact traveling wave solutions of mKdV equation. With the aid of symbolic computation, many hyperbolic function solutions, trigonometric function solutions and rational function solutions are obtained. It is shown that the proposed method is more effective and powerful than the ( G′ G ) -expansion method in constructing the tr...