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Register a new userAn integrable generalization of the nonlinear Schrodinger equation on the half-line and solitons
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We analyze initial-boundary value problems for an integrable generalization of the nonlinear Schrodinger equation formulated on the half-line. In particular, we investigate the so-called linearizable boundary conditions, which in this case are of Robin type. Furthermore, we use a particular solution to verify explicitly all the steps needed for the solution of a well-posed problem.
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We consider an integrable generalization of the nonlinear Schrodinger (NLS) equation that was recently derived by one of the authors using bi-Hamiltonian methods. This equation is related to the NLS equation in the same way that the Camassa Holm equation is related to the KdV equation. In this paper we: (a) Use the bi-Hamiltonian structure to write down the first few conservation laws. (b) Derive a Lax pair. (c) Use the Lax pair to solve the initial value problem. (d) Analyze solitons.
A new integrable (2+1)-dimensional nonlocal nonlinear Schrodinger equation is proposed. The $N$-soliton solution is given by Gram type determinant. It is found that the localized N-soliton solution has interesting interaction behavior which shows change of amplitude of localized pulses after collisions.
We derive asymptotic formulas for the solution of the derivative nonlinear Schrodinger equation on the half-line under the assumption that the initial and boundary values lie in the Schwartz class. The formulas clearly show the effect of the boundary on the solution. The approach is based on a nonlinear steepest descent analysis of an associated Riemann-Hilbert problem.
The most challenging problem in the implementation of the so-called textit{unified transform} to the analysis of the nonlinear Schrodinger equation on the half-line is the characterization of the unknown boundary value in terms of the given initial and boundary conditions. For the so-called textit{linearizable} boundary conditions this problem can be solved explicitly. Furthermore, for non-linearizable boundary conditions which decay for large $t$, this problem can be largely bypassed in the sense that the unified transform yields useful asymptotic information for the large $t$ behavior of the solution. However, for the physically important case of periodic boundary conditions it is necessary to characterize the unknown boundary value. Here, we first present a perturbative scheme which can be used to compute explicitly the asymptotic form of the Neumann boundary value in terms of the given $tau$-periodic Dirichlet datum to any given order in a perturbation expansion. We then discuss briefly an extension of the pioneering results of Boutet de Monvel and co-authors which suggests that if the Dirichlet datum belongs to a large class of particular $tau$-periodic functions, which includes ${a exp(i omega t) , | , a>0, , omega geq a^2}$, then the large $t$ behavior of the Neumann value is given by a $tau$-periodic function which can be computed explicitly.
It has been conjectured that the defocusing nonlinear Schrodinger (NLS) equation on the half-line does not admit solitons. We give a proof of this conjecture.
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