In this paper, we investigate the instability of one-dimensionally stable periodic traveling wave solutions of the generalized Korteweg-de Vries equation to long wavelength transverse perturbations in the generalized Zakharov-Kuznetsov equation in two space dimensions. By deriving appropriate asymptotic expansions of the periodic Evans function, we derive an index which yields sufficient condit...

Using the KAM method, we exhibit some solutions of a finite dimensional approximation of the Zakharov Hamiltonian formulation of gravity water waves (Zakharov, 1968), which are spatially periodic, quasi-periodic in time, and not permanent form travelling waves. For this Hamiltonian, which is the total energy of the waves, the canonical variables are some complex quantities an and a ∗ n ( n ∈ Z)...

As studied by Jaulent in 1982, the inverse problem of lossy electric transmission lines is closely related to the inverse scattering of Zakharov-Shabat equations with two potential functions. Focusing on the numerical solution of this inverse scattering problem, we develop a fast one-shot algorithm based on the Gelfand-LevitanMarchenko equations and on some differential equations derived from t...

The inverse scattering problem for coupled wave equations has various applications such as waveguide filter design and electric transmission line fault diagnosis. In this paper, an efficient numerical algorithm is presented for solving the inverse scattering problem related to generalized Zakharov-Shabat equations with two potential functions. Inspired by the work of Xiao and Yashiro on Zakharo...

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.

We show how the Witten-Dijkgraff-Verlinde-Verlinde (WDVV) equations and the Darboux-Zakharov-Manakov (DZM) system can be characterized via a background family of functions.

Nonlinear Schrödinger (NLS) equations have attracted the attention of the physical and mathematical community for over four decades. NLS equations arise in such diverse fields as deep water waves [4, 25], plasma physics [24], fiber optics [11], and Bose-Einstein condensation [17]. The basic method for solving the NLS initial-value problem is the inverse scattering transform (IST) method [2–4, 6...

In this paper we construct and study discretizations of a nonlinear Zakharov-wave system occurring in plasma physics. These systems are generalizations of the Zakharov system that can be recovered by taking a singular limit. We introduce two numerical schemes that take into account this singular limit process. One of the scheme is conservative but sensible to the polarization of the initial dat...

We prove that the weak solution of the Cauchy problem for the KleinGordon-Zakharov system and for the Zakharov system is unique in the energy space for the former system, and in H×L for the latter system, in dimensions three or lower. These are the largest Sobolev spaces (in three dimensional case) where the local wellposedness has been proven so far. Our proof uses an infinite iteration, where...

We calculate the Lyapunov exponent for the non-Hermitian Zakharov-Shabat eigenvalue problem corresponding to the attractive non-linear Schrödinger equation with a Gaussian random pulse as initial value function. Using an extension of the Thouless formula to non-Hermitian random operators, we calculate the corresponding average density of states. We also calculate the distribution of a set of sc...