The   ) ( exp    -expansion method is a promising method for finding exact traveling wave solutions to nonlinear evolution equations in physical sciences. In this article, we use the   ) ( exp    -expansion method to find the exact solutions for the nonlinear Zakharov-Kuznetsov-Benjamin-Bona-Mahony equation and the good Boussinesq equations. Many solitary wave solutions are formally d...

We study the corresponding scattering problem for Zakharov and Shabat compatible differential equations in two-dimensions, the representation for a solution of the nonlinear Schrödinger equation is formulated as a variational problem in two-dimensions. We extend the derivation to the variational principle for the Zakharov and Shabat equations in one-dimension. We also developed an approximate a...

In this article, we derive a viscous Boussinesq system for surface water waves from Navier-Stokes equations. So, we use neither the irrotationality assumption, nor the Zakharov-Craig-Sulem formulation. During the derivation, we find the bottom shear stress, and also the decay rate for shallow (and not deep) water. In order to justify our derivation, we check it by deriving the viscous Korteweg-...

and Applied Analysis 3 Substituting 2.1 into 1.3 , we have k2φ′′ 2φψ − ( p2 q ) φ − 2λφ2n 1 0, k2 ( 4p2 − 1 ) ψ ′′ k2 ( φ2n )′′ 0. 2.2 Integrating the second equation of 2.2 twice and letting the first integral constant be zero, we have ψ φ2n 1 − 4p2 g, p / 1 2 , 2.3 where g is the second integral constant. Substituting 2.3 into the first equation of 2.2 , we have k2φ′′ ( 2g − p2 − q ) φ 2 ( 1 ...

The modied Zakharov–Kuznetsov (mZK) equation, ut + uux + uxxx + uxyy = 0, (1) represents an anisotropic two-dimensional generalization of the Korteweg–de Vries equation and can be derived in a magnetized plasma for small amplitude Alfvén waves at a critical angle to the undisturbed magnetic field, and has been studied by many authors because of its importance [1–5]. However, Eq. (1) possesses m...

We prove that the initial value problem for the two-dimensional modified ZakharovKuznetsov equation is locally well-posed for data in H(R), s > 3/4. Even though the critical space for this equation is L(R) we prove that well-posedness is not possible in such space. Global well-posedness and a sharp maximal function estimate are also established.