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

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.
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