Cauchy problem for a nonlinear Schrödinger equation with a large initial gradient in the weakly dispersive limit (Q6554771)

This paper aims to investigate the asymptotic behavior of the solutions to the cubic nonlinear Schrödinger equation on the real line \N\[\N\left\{ \begin{array}{rcl} i \epsilon u_t + \frac {\epsilon^2} 2 u_{xx} + \gamma |u|^2 u &= & 0, \\\Nu\big|_{t=0} & = & \Lambda(\frac \cdot \rho), \end{array} \right.\tag{1}\N\]\Nas the parameters \(\epsilon\) and \(\rho\) tend to \(0\), under the assumption that the function \(\Lambda\) admits no equal limits at infinity \( \Lambda (s)\stackrel{s\to\pm \infty}\longrightarrow \Lambda^0_\pm\), and has a continuous derivative tending to zero at infinity sufficiently rapidly. Based on the renormalization method, the author constructs an asymptotic solution, under constraints on the behavior of the problem parameters. In the particular focusing case, the resulting asymptotic solution expresses by means of known elliptic special functions.