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Long-time asymptotic behavior for the Gerdjikov-Ivanov type of derivative nonlinear Schrödinger equation with time-periodic boundary condition. (English) Zbl 1427.35259
Summary: 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.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B10 Periodic solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
35Q15 Riemann-Hilbert problems in context of PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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