An official website of the United States government
Abstract The inverse scattering transform for the focusing nonlinear Schrödinger equation is presented for a general class of initial conditions whose asymptotic behavior at infinity consists of counterpropagating waves. The formulation takes into account the branched nature of the two asymptotic eigenvalues of the associated scattering problem. The Jost eigenfunctions and scattering coefficients are defined explicitly as single‐valued functions on the complex plane with jump discontinuities along certain branch cuts. The analyticity properties, symmetries, discrete spectrum, asymptotics, and behavior at the branch points are discussed explicitly. The inverse problem is formulated as a matrix Riemann‐Hilbert problem with poles. Reductions to all cases previously discussed in the literature are explicitly discussed. The scattering data associated to a few special cases consisting of physically relevant Riemann problems are explicitly computed.
more »
« less
This content will become publicly available on February 18, 2026
Connection formulae for the radial Toda equations I
Abstract This paper is the first in a forthcoming series of works where the authors study the global asymptotic behavior of the radial solutions of the 2D periodic Toda equation of typeAn. The principal issue is the connection formulae between the asymptotic parameters describing the behavior of the general solution at zero and infinity. To reach this goal we are using a fusion of the Iwasawa factorization in the loop group theory and the Riemann-Hilbert nonlinear steepest descent method of Deift and Zhou which is applicable to 2D Toda in view of its Lax integrability. A principal technical challenge is the extension of the nonlinear steepest descent analysis to Riemann-Hilbert problems of matrix rank greater than 2. In this paper, we meet this challenge for the casen = 2 (the rank 3 case) and it already captures the principal features of the generalncase.
more »
« less
- Award ID(s):
- 1955265
- PAR ID:
- 10616291
- Publisher / Repository:
- London Mathematical Society
- Date Published:
- Journal Name:
- Nonlinearity
- Volume:
- 38
- Issue:
- 3
- ISSN:
- 0951-7715
- Page Range / eLocation ID:
- 035015
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract We obtain rigorous large time asymptotics for the Landau–Lifshitz (LL) equation in the soliton free case by extending the nonlinear steepest descent method to genus 1 surfaces. The methods presented in this paper pave the way to a rigorous analysis of other integrable equations on the torus and enable asymptotic analysis on different regimes of the LL equation.more » « less
-
Abstract Givenndisjoint intervals on together withnfunctions , , and an matrix , the problem is to find anL2solution , , to the linear system , where , is a matrix of finite Hilbert transforms with defined on , and is a matrix of the corresponding characteristic functions on . Since we can interpret , as a generalized multi‐interval finite Hilbert transform, we call the formula for the solution as “the inversion formula” and the necessary and sufficient conditions for the existence of a solution as the “range conditions”. In this paper we derive the explicit inversion formula and the range conditions in two specific cases: a) the matrix Θ is symmetric and positive definite, and; b) all the entries of Θ are equal to one. We also prove the uniqueness of solution, that is, that our transform is injective. In the case a), that is, when the matrix Θ is positive definite, the inversion formula is given in terms of the solution of the associated matrix Riemann–Hilbert Problem. In the case b) we reduce the multi interval problem to a problem onncopies of and then express our answers in terms of the Fourier transform. We also discuss other cases of the matrix Θ.more » « less
-
Abstract Gradient‐type iterative methods for solving Hermitian eigenvalue problems can be accelerated by using preconditioning and deflation techniques. A preconditioned steepest descent iteration with implicit deflation (PSD‐id) is one of such methods. The convergence behavior of the PSD‐id is recently investigated based on the pioneering work of Samokish on the preconditioned steepest descent method (PSD). The resulting non‐asymptotic estimates indicate a superlinear convergence of the PSD‐id under strong assumptions on the initial guess. The present paper utilizes an alternative convergence analysis of the PSD by Neymeyr under much weaker assumptions. We embed Neymeyr's approach into the analysis of the PSD‐id using a restricted formulation of the PSD‐id. More importantly, we extend the new convergence analysis of the PSD‐id to a practically preferred block version of the PSD‐id, or BPSD‐id, and show the cluster robustness of the BPSD‐id. Numerical examples are provided to validate the theoretical estimates.more » « less
-
Abstract Integrable standard and nonlocal derivative nonlinear Schrödinger equations are investigated. The direct and inverse scattering are constructed for these equations; included are both the Riemann–Hilbert and Gel’fand–Levitan–Marchenko approaches and soliton solutions. As a typical application, it is shown how these derivative NLS equations can be obtained as asymptotic limits from a nonlinear Klein–Gordon equation.more » « less
