Long-time asymptotics for the focusing NLS equation with time-periodic boundary condition

Abstract

We consider the focusing nonlinear Schrödinger equation on the quarter plane. Initial data vanish at infinity while boundary data are time-periodic ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>a</mml:mi> <mml:msup> <mml:mi mathvariant="normal">e</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi>ω</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> </mml:msup> </mml:math> ). The goal of this Note is to study the asymptotic behavior of the solution of this initial-boundary-value problem. The main tool is the asymptotic analysis of an associated matrix Riemann–Hilbert problem. We show that the solution of the IBV problem has different asymptotic behaviors in different regions. In the region <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>x</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>4</mml:mn> <mml:mi>b</mml:mi> <mml:mi>t</mml:mi> </mml:math> ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>b</mml:mi> <mml:mo>=</mml:mo> <mml:msqrt> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>a</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>−</mml:mo> <mml:mi>ω</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msqrt> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:math> ) the solution has the form of a Zakharov–Manakov vanishing asymptotics. In the region <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>4</mml:mn> <mml:mi>b</mml:mi> <mml:mi>t</mml:mi> <mml:mo>−</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>a</mml:mi> </mml:mrow> </mml:mfrac> <mml:mi>N</mml:mi> <mml:mi mathvariant="normal">log</mml:mi> <mml:mi>t</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>x</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>4</mml:mn> <mml:mi>b</mml:mi> <mml:mi>t</mml:mi> </mml:math> , where N is an integer, the solution behaves as a finite train of asymptotic solitons. In the region <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>4</mml:mn> <mml:mo stretchy="false">(</mml:mo> <mml:mi>b</mml:mi> <mml:mo>−</mml:mo> <mml:mi>a</mml:mi> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> <mml:mi>t</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>x</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>4</mml:mn> <mml:mi>b</mml:mi> <mml:mi>t</mml:mi> </mml:math> the solution is a modulated elliptic wave. Finally, in the sector <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>x</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>4</mml:mn> <mml:mo stretchy="false">(</mml:mo> <mml:mi>b</mml:mi> <mml:mo>−</mml:mo> <mml:mi>a</mml:mi> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> <mml:mi>t</mml:mi> </mml:math> the solution is a plane wave.