Mahendra Panthee

On KP-II type equations on cylinders

Abstract In this article we study the generalized dispersion version of the Kadomtsev–Petviashvili II equation, on T × R and T × R 2 . We start by proving bilinear Strichartz type estimates, dependent only on the dimension of the domain but not on the dispersion. Their analogues in terms of Bourgain spaces are then used as the main tool for the proof of bilinear estimates of the nonlinear terms of the equation and consequently of local well-posedness for the Cauchy problem.

Higher-Order Hamiltonian Model for Unidirectional Water Waves

Formally second-order correct, mathematical descriptions of long-crested water waves propagating mainly in one direction are derived. These equations are analogous to the first-order approximations of KdV- or BBM-type. The advantage of these more complex equations is that their solutions corresponding to physically relevant initial perturbations of the rest state may be accurate on a much longer timescale. The initial value problem for the class of equations that emerges from our derivation is then considered. A local well-posedness theory is straightforwardly established by a contraction mapping argument. A subclass of these equations possess a special Hamiltonian structure that implies the local theory can be continued indefinitely.

A Note on Local Well-Posedness of Generalized KdV Type Equations with Dissipative Perturbations

In this note we report local well-posedness results for the Cauchy problems associated to generalized KdV type equations with dissipative perturbation for given data in the low regularity

A SYSTEM OF COUPLED SCHRÖDINGER EQUATIONS WITH TIME-OSCILLATING NONLINEARITY

L^2

Well-Posedness for Multicomponent Schrödinger–gKdV Systems and Stability of Solitary Waves with Prescribed Mass

-based Sobolev spaces. The method of proof is based on the {\em contraction mapping principle} employed in some appropriate time weighted spaces.

On well-posedness of the third-order nonlinear Schrödinger equation with time-dependent coefficients

This paper is concerned with the initial value problem (IVP) associated to the coupled system of supercritical nonlinear Schrodinger equations

On the Cauchy problem for a coupled system of KdV equations

\left\{\begin{array}{@{}l@{}}iu_{t}+\Delta u+\theta_{1}(\omega t)(\vert u\vert^{2p}+\beta \vert u\vert^{p-1}\vert v \vert^{p+1})u=0,\\ iv_{t}+\Delta v+\theta_2(\omega t)(\vert v \vert^{2p}+\beta \vert v \vert^{p-1}\vert u \vert^{p+1})v=0,\end{array}\right.

Unique continuation property for a higher order nonlinear Schrödinger equation

where θ1 and θ2 are periodic functions, which has applications in many physical problems, especially in nonlinear optics. We prove that, for given initial data φ, ψ ∈ H1(ℝn), as |ω| → ∞, the solution (uω, vω) of the above IVP converges to the solution (U, V) of the IVP associated to