In this article we derive the Marchenko integral equations for solving the inverse scattering problem for the matrix Zakharov-Shabat system with a potential without symmetry properties and having L1 entries under a technical hypothesis preventing the accumulation of discrete eigenvalues on the continuous spectrum. Wederive additional symmetry properties in the focusing case. The norming constan...

We consider the initial value problem (IVP) associated to the modified Zakharov-Kuznetsov (mZK) equation ut + 6uux + uxxx + uxyy = 0, (x, y) ∈ R, t ∈ R, which is known to have global solution for given data in u(x, y, 0) = u0(x, y) ∈ H(R) satisfying ‖u0‖L2 < √ 3‖φ‖L2 , where φ is a solitary wave solution. In this work, the issue of the asymptotic behavior of the solutions of the modified Zakhar...

The reductions of the integrable N-wave type equations solvable by the inverse scattering method with the generalized Zakharov-Shabat systems L and related to some simple Lie algebra g are analyzed. The Zakharov-Shabat dressing method is extended to the case when g is an orthogonal algebra. Several types of one soliton solutions of the corresponding N-wave equations and their reductions are stu...

In [2], F.Linares, G.Ponce, J-C.Saut have proved that a non-fully dispersive Zakharov system arising in the study of Laser-plasma interaction, is locally well posed in the whole space, for fields vanishing at infinity. Here we show that in the periodic case, seen as a model for fields non-vanishing at infinity, the system develops strong instabilities of Hadamard’s type, implying that the Cauch...

in this paper an approximate analytical solution of the fractional zakharov-kuznetsov equations will be obtained with the help of the reduced differential transform method (rdtm). it is in-dicated that the solutions obtained by the rdtm are reliable and present an effective method for strongly nonlinear fractional partial differential equations.

In this paper, we prove the nonlinear orbital stability of the stationary traveling wave of the one-dimensional Gross-Pitaevskii equation by using Zakharov-Shabat’s inverse scattering method.

by using bifurcation theory of planar ordinary differential equations all different bounded travelling wave solutions of the generalized zakharov equation are classified in different parametric regions . in each of these parametric regions the exact explicit parametric representation of all solitary , kink (anti kink) and periodic wave solutions as well as their numerical simulation and thei...

In this article we prove that the discrete eigenvalues of the Zakharov-Shabat system belong to certain neighborhoods of the endpoints of the spectral gap and the discrete eigenvalue of the free Hamiltonian.

This paper is a continuation of our previous work in which we studied sl(3) Zakharov-Shabat type auxiliary linear problem with reductions Mikhailov and the integrable hierarchy nonlinear evolution equations associated it. Now, shall demonstrate how one can construct special solutions over constant background through Zakharov-Shabat's dressing technique. That approach will be illustrated on exam...

This paper continues the study, initiated in [28, 8], of the validity of the Zakharov model describing Langmuir turbulence. We give an existence theorem for a class of singular quasilinear equations. This theorem is valid for well-prepared initial data. We apply this result to the Euler-Maxwell equations describing laser-plasma interactions, to obtain, in a high-frequency limit, an asymptotic e...