Long-time Asymptotic Behavior for the Derivative Schrödinger Equation with Finite Density Type Initial Data

Abstract

In this paper, the authors apply $\overline \partial $ steepest descent method to study the Cauchy problem for the derivative nonlinear Schrödinger equation with finite density type initial data $\matrix{{{\rm{i}}q + {q_{xx}} + {\rm{i}}{{\left( {{{\left| q \right|}^2}q} \right)}_x} = 0,} \hfill \cr {q\left( {x,0} \right) = {q_0}\left( x \right),} \hfill \cr } $ where $\mathop {\lim }\limits_{x \to \pm \infty } {q_0}\left( x \right) = {q_ \pm }\,{\rm{and}}\,\,\left| {{q_ \pm }} \right| = 1$. Based on the spectral analysis of the Lax pair, they express the solution of the derivative Schrödinger equation in terms of solutions of a Riemann-Hilbert problem. They compute the long time asymptotic expansion of the solution q(x, t) in different space-time regions. For the region $\xi = {x \over t}$ with ∣ξ + 2∣ < 1, the long time asymptotic is given by $q\left( {x,t} \right) = T{\left( \infty \right)^{ - 2}}q_\Lambda ^r\left( {x,t} \right) + {\cal O}\left( {{t^{ - {3 \over 4}}}} \right),$ in which the leading term is N(I) solitons, the second term is a residual error from a $\overline \partial $ equation. For the region ∣ξ + 2∣ > 1, the long time asymptotic is given by $q\left( {x,t} \right) = T{\left( \infty \right)^{ - 2}}q_\Lambda ^r\left( {x,t} \right) - {t^{ - {1 \over 2}}}{\rm{i}}{f_{11}} + {\cal O}\left( {{t^{ - {3 \over 4}}}} \right),$ in which the leading term is N(I) solitons, the second ${t^{ - {1 \over 2}}}$ order term is soliton-radiation interactions and the third term is a residual error from a $\overline \partial $ equation. These results are verification of the soliton resolution conjecture for the derivative Schrödinger equation. In their case of finite density type initial data, the phase function θ(z) is more complicated that in finite mass initial data. Moreover, two triangular decompositions of the jump matrix are used to open jump lines on the whole real axis and imaginary axis, respectively.

Keywords

Derivative Schrödinger equation / Riemann-Hilbert problem / $\overline \partial $ steepest descent method / Long-time asymptotics / Soliton resolution / Asymptotic stability

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