Based on the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann--Hilbert problems Dbar approach, long-time asymptotic behavior solutions to fifth-order modified Korteweg-de Vries equation line is studied in case initial conditions that belong some weighted Sobolev spaces. Using techniques Fourier analysis idea $I$-method, we give its global well-posedness lower regulari...

In order to study the longtime behavior of a dissipative evolutionary equation, we generally aim to show that the dynamics of the equation is finite dimensional for long time. In fact, one possible way to express this fact is to prove that dynamical systems describing the evolutional equation comprise the existence of the global attractor 1 . The KDV equation without dissipative and forcing was...

Embedded solitons are solitary waves residing inside the continuous spectrum of a wave system. They have been discovered in a wide array of physical situations recently. In this article, we present the first comprehensive theory on the dynamics of embedded solitons and nonlocal solitary waves in the framework of the perturbed fifth-order Korteweg–de Vries (KdV) hierarchy equation. Our method is...

In this work, the generalized ( G′ G ) -expansion method is presented to seek some new exact solutions for generalized fifth order KdV equation with time-dependent coefficients. As a result, more explicit traveling wave solutions involving arbitrary parameters are found out, which are expressed in terms of hyperbolic functions, the trigonometric functions and rational functions. When the parame...

In this paper, the Newton-conjugate-gradient methods are developed for solitary wave computations. These methods are based on Newton iterations, coupled with conjugategradient iterations to solve the resulting linear Newton-correction equation. When the linearization operator is self-adjoint, the preconditioned conjugate-gradient method is proposed to solve this linear equation. If the lineariz...

which can be considered as a coupling between the KdV (with respect to u) and the mKdV (with respect to v) equations. The coupled KdV-mKdV equations were proposed by Kersten and Krasil’shchik [1] and originate from a supersymmetric extension of the classical KdV [2]. It also can be considered as a coupling between the KdV and mKdV equations: By setting v = 0 we obtain the KdV equation ut + uxxx...

We show that the integrable subclassess of the equations q,t = f(x, t) q,3 + H(x, t, q, q,1) are the same as the integrable subclassess of the equations q,t = q,3 + F (q, q,1).

We give a unified view of the relation between the SL(2) KdV, the mKdV, and the UrKdV equations through the Fréchet derivatives and their inverses. For this we introduce a new procedure of obtaining the Ur-KdV equation, where we require that it has no nonlocal operators. We extend this method to the SL(3) KdV equation, i.e., Boussinesq(Bsq) equation and obtain the hamiltonian structure of Ur-Bs...

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...