Abstract
The most challenging problem in the implementation of the so-called unified transform to the analysis of the nonlinear Schrödinger 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 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 τ-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 τ-periodic functions, which includes {a exp(iωt)|a > 0, ω ≥ a2}, then the large t behavior of the Neumann value is given by a τ-periodic function which can be computed explicitly.
Highlights
Let q(x, t) satisfy the nonlinear Schrodinger (NLS) on the half-line with a given initial condition, iqt + qxx − 2λ|q|2q = 0, q(x, 0) = q0(x),0 < x < ∞, t > 0, λ = ±1, (1) 0 < x < ∞, (2)where q0(x) has sufficient decay as x → ∞.In the above setting, the so-called unified transform [8] can be used to analyze problems with either a linearizable boundary condition or with a non-linearizable boundary condition which decays for large t.1.1
We discuss briefly an extension of the formalism introduced in [3, 4, 5, 6], which suggests that using the technique of finite gap integration, it is possible to obtain a large class of τ periodic functions which have the crucial property that if they are assigned as the Dirichlet data for the NLS on the half line, the associated function g1(t) has the property that it asymptotes as t → ∞ to a τ -periodic function and this function can be computed explicitly
It is natural to ask the following question: Given the asymptotic form g0B(t) of the Dirichlet datum, can we find the asymptotic form g1B(t) of the Neumann value? Theorem 2.2 below provides a constructive algorithm for computing the asymptotic form of the Neumann value from the asymptotic form of the Dirichlet data in a perturbative expansion
Summary
We discuss briefly an extension of the formalism introduced in [3, 4, 5, 6], which suggests that using the technique of finite gap integration, it is possible to obtain a large class of τ periodic functions which have the crucial property that if they are assigned as the Dirichlet data for the NLS on the half line, the associated function g1(t) has the property that it asymptotes as t → ∞ to a τ -periodic function and this function can be computed explicitly. Theorem 2.2 below provides a constructive algorithm for computing the asymptotic form of the Neumann value from the asymptotic form of the Dirichlet data in a perturbative expansion.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
