Asymptotics of Solutions to the Modified Nonlinear Schrödinger Equation: Solitons on a NonVanishing Continuous Background
Abstract
Using the matrix RiemannHilbert factorization approach for nonlinear evolution systems which take the form of Laxpair isospectral deformations and whose corresponding Lax operators contain both discrete and continuous spectra, the leadingorder asymptotics as $t \to \pm \infty$ of the solution to the Cauchy problem for the modified nonlinear Schrödinger equation, $i \partial_{t} u + {1/2} \partial_{x}^{2} u +  u ^{2} u + i s \partial_{x} ( u ^{2} u) = 0$, $s \in \Bbb R_{>0}$, which is a model for nonlinear pulse propagation in optical fibers in the subpicosecond time scale, are obtained: also derived are analogous results for two gaugeequivalent nonlinear evolution equations; in particular, the derivative nonlinear Schrödinger equation, $i \partial_{t} q + \partial_{x}^{2} q  i \partial_{x} ( q ^{2} q) = 0$. As an application of these asymptotic results, explicit expressions for position and phase shifts of solitons in the presence of the continuous spectrum are calculated.
 Publication:

arXiv eprints
 Pub Date:
 December 1997
 DOI:
 10.48550/arXiv.solvint/9801001
 arXiv:
 arXiv:solvint/9801001
 Bibcode:
 1998solv.int..1001K
 Keywords:

 Exactly Solvable and Integrable Systems;
 Nonlinear Sciences  Exactly Solvable and Integrable Systems
 EPrint:
 38 pages, 1 figure, LaTeX