Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
1997-12-27
Nonlinear Sciences
Exactly Solvable and Integrable Systems
38 pages, 1 figure, LaTeX
Scientific paper
Using the matrix Riemann-Hilbert factorization approach for nonlinear evolution systems which take the form of Lax-pair isospectral deformations and whose corresponding Lax operators contain both discrete and continuous spectra, the leading-order asymptotics as $t \to \pm \infty$ of the solution to the Cauchy problem for the modified nonlinear Schr\"{o}dinger 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 gauge-equivalent nonlinear evolution equations; in particular, the derivative nonlinear Schr\"{o}dinger 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.
Kitaev Alexander V.
Vartanian A. H.
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