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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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