Asymptotic Behavior for a Class of Solutions to the Critical Modified Zakharov-Kuznetsov Equation

Abstract

Abstract. We consider the initial value problem (IVP) associated to the modified Zakharov-Kuznetsov (mZK) equationwhich is known to have global solution for given data in u(x, y,, where φ is a solitary wave solution. In this work, the issue of the asymptotic behavior of the solutions of the modified Zakharov-Kuznetsov equation with negative energy is addressed. The principal tool to obtain the main result is the use of appropriate scaling argument from Angulo et al [4,5].

Decay of Small Solutions for the Zakharov-Kuznetsov Equation posed on a half-strip

“…It means that we could not prove the existence and uniqueness of global regular solutions without smallness restrictions for initial data similarly to the critical case for the KdV equation [8,10,11,24,31,32]. As far as the ZK equation is concerned, the results on both IVP and IBVP can be found in [5,6,8,27,28,30]. Our work has been inspired by [29] where critical KdV equation with internal damping posed on a bounded interval was considered and exponential decay of weak solutions has been established.…”

“…Our work has been inspired by [29] where critical KdV equation with internal damping posed on a bounded interval was considered and exponential decay of weak solutions has been established. We must note that solvability of initial-boundary value problems in classes of global regular solutions for the regular case of the 2D ZK equation (uu x ) has been established in [4,6,18,19,22,26,30,36,37] for arbitrary smooth initial data. On the other hand, for the 3D ZK equation, the convective term uu x , which is regular for the 2D ZK equation, corresponds to a critical case.…”

A note on the 2D generalized Zakharov–Kuznetsov equation: Local, global, and scattering results

“…Concerning other questions on the gZK equation we refer the reader to [2], [3], [11], [13], [14], and references therein.…”