Abstract
In recent years, a vast amount of work has been done on initial value problems for important nonlinear evolution equations like the nonlinear Schrödinger equation (NLS) and the Korteweg-de Vries equation (KdV). No comparable attention has been given to mixed initial-boundary value problems for these equations, i.e. forced nonlinear systems. But in many cases of physical interest, the mathematical model leads precisely to the forced problems. For example, the launching of solitary waves in a shallow water channel, the excitation of ion-acoustic solitons in a double plasma machine, etc. In this article, we present the PDE (Partial Differential Equation) method to study the following iut = uxx - g|u|pu, g ∈ R, p > 3, x ∈ Ω = [0,L], 0 ≤ t u (x,0) = u0 (x) ∈ H2 (Ω) and Robin inhomogeneous boundary condition ux (0,t) + αu (0,t) = R1(t), t ≥ 0 and ux (L,t) + αu (L,t) = R2 (t), t ≥ 0 (here α is a real number). The equation is posed in a semi-infinite strip on a finite domain Ω. Such problems are called forced problems and have many applications in other fields like physics and chemistry. The main tool of PDE method is semi-group theory. We are able to prove local existence and uniqueness theorem for the nonlinear Schrödinger equation under initial condition and Robin inhomogeneous boundary condition.
Highlights
A vast amount of work has been done on initial value problems for important nonlinear evolution equations like the nonlinear Schrödinger equation (NLS) and the Korteweg-de Vries equation (KdV)
We present the PDE (Partial Differential Equation) method to study the following iu=t uxx − g u p u, g ∈ R, p > 3, x ∈ Ω= [0, L], 0 ≤ t < ∞ with initial condition u ( x, 0)= u0 ( x) ∈ H 2 (Ω) and Robin inhomogeneous boundary condition ux (0,t ) + αu (0,t ) =R1 (t ),t ≥ 0 and ux ( L,t ) + αu ( L,t ) =R2 (t ),t ≥ 0
We are able to prove local existence and uniqueness theorem for the nonlinear Schrödinger equation under initial condition and Robin inhomogeneous boundary condition
Summary
This paper is the continuation of an earlier one [1] where local existence and uniqueness theorem was presented for a one-dimensional nonlinear Schrödinger. C. Bu equation with initial condition and Dirichlet type inhomogeneous boundary condition on a finite domain Ω =[0, L]. The following nonlinear Schrödinger equation (NLS) posed in the quarter plane with Dirichlet inhomogeneous condition (k is a real constant) has been studied by the author: iu=t uxx + k u 2 u (1.1). Existence and uniqueness of a global classical solution were proved via PDE method provided that the initial-boundary data are “nice” (cf [12]). For the NLS posed in the quarter plane with Robin inhomogeneous condition ( k ∈ R ):. Similar results were available [14] [15] Solving such problems has important physical and mathematical implications. Robin type inhomogeneous boundary conditions are imposed on both endpoints. Using PDE method, we will prove the existence of a unique local classical solution
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