On the existence and uniqueness of smooth solution for a generalized Zakharov equation

Abstract

The paper deals with the existence and uniqueness of smooth solution for a generalized Zakharov equation. We establish local in time existence and uniqueness in the case of dimension d = 2, 3. Moreover, by using the conservation laws and Brezis-Gallouet inequality, the solution can be extended globally in time in two dimensional case for small initial data. Besides, we also prove global existence of smooth solution in one spatial dimension without any small assumption for initial data.

Well-posedness for a class of generalized Zakharov system

“…This system has been the subject of a large number of studies [3,[5][6][7][8][9][11][12][13][14][15][16]. In addition to the energy method, Zakharov type systems have also been studied by others using different approaches.…”

New Exact Explicit Solutions of the Generalized Zakharov Equation via the First Integral Method

“…Up to now, many methods have been used to solve the exact solution of the system (1) such as rational auxiliary equation method [2], F-expansion method [3], and Li et al obtained the generalized solitary solutions by exp-function method [4] [5], Hong got the doubly periodic solutions by the generalized Jacobi elliptic function expansion method [6], M. Javidi constructed dark and bright solitary wave solutions by a variational iteration method [7]. Besides, Guo discussed the existence and uniqueness of smooth solution [8], Gambo investigates the dynamical behavior [9], S. Abbasbandy solved the numerical solutions [10] of the system (1).…”

Exact 1-soliton solution of the Zakharov equation in plasmas withpower law nonlinearity

“…The study of nonlinear evolution equations (NLEES) appear in a lot of places in Applied Mathematics and Theoretical Physics [1][2][3][4][5][6][7][8][9][10]. They also appear in Mathematical Biology, Mathematical Chemistry, Nuclear physics, Fluid Dynamics, Plasma Physics, Nonlinear Optics and many other places.…”