Asymptotic decay of a~one-dimensional wave packet in a~nonlinear dispersive medium

L. A. Kalyakin

Sbornik. Mathematics, Tome 60 (1988) no. 2, pp. 457-483

Résumé

The system of equations $$ \partial_tU+A(U)\partial_xU+B(U)U=0,\qquad x\in\mathbf{R}^1,\quad t>0\quad (U\in\mathbf R^m), $$ is considered with initial data in the form of a wave packet of small amplitude $$ U_{t=0}=\varepsilon\sum_{k=\pm1}\Phi_k(\xi)\exp(ikx),\quad \xi =\varepsilon x\quad(\Phi _k(\xi )=O((1+|\xi |)^{-N})\ \forall N). $$ The asymptotics of the solution $U(x,t,\varepsilon)$ as $\varepsilon\to0$ which is uniform in the strip $x\in\mathbf R^1$, $0\leqslant t\leqslant O(\varepsilon^{-2})$, is constructed by the method of multiscale expansions. The coefficients of the asymptotics are a system of wave packets traveling with group velocities; the leading term is determined from a system of nonlinear equations of Schrödinger type. Bibliography: 32 titles.