Long-time asymptotic behavior for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition

Abstract:

The Gerdjikov-Ivanov (GI) type of derivative nonlinear Schrödinger equation is considered on the quarter plane whose initial data vanish at infinity while boundary data are time-periodic, of the form $ae^{i\delta }e^{2i\omega t}$. The main purpose of this paper is to provide the long-time asymptotics of the solution to the initial-boundary value problems for the equation. For $\omega <a^{2}(\frac {1}{4}a^{2}+3b-1)$ with $0<b<\frac {a^{2}}{4}$, our results show that different regions are distinguished in the quarter plane $\Omega =\{(x,t)\in \mathbb {R}^{2}|x>0, t>0\}$, on which the asymptotics admit qualitatively different forms. In the region $x>4tb$, the solution is asymptotic to a slowly decaying self-similar wave of Zakharov-Manakov type. In the region $0< x <4t\left (b-\sqrt {2a^{2}\left (\frac {a^{2}}{4}-b\right )}\right )$, the solution takes the form of a plane wave. In the region $4t\left (b-\sqrt {2a^{2}\left (\frac {a^{2}}{4}-b\right )}\right )<x<4tb$, the solution takes the form of a modulated elliptic wave.