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Long-time asymptotic behavior of a mixed schrödinger equation with weighted Sobolev initial data

Qiaoyuan Cheng, Yiling Yang, Engui Fan

Abstract

We apply $\bar{\partial}$ steepest descent method to obtain sharp asymptotics for a mixed schrödinger equation $$ q_t+iq_{xx}-ia (\vert q \vert^2q)_x -2b^2\vert q \vert^2q=0,$$ $$q(x,t=0)=q_0(x),$$ under essentially minimal regularity assumptions on initial data in a weighted Sobolev space $q_0(x) \in H^{2,2}(\mathbb{R})$. In the asymptotic expression, the leading order term $\mathcal{O}(t^{-1/2})$ comes from dispersive part $q_t+iq_{xx}$ and the error order $\mathcal{O}(t^{-3/4})$ from a $\overline\partial$ equation

Long-time asymptotic behavior of a mixed schrödinger equation with weighted Sobolev initial data

Abstract

We apply steepest descent method to obtain sharp asymptotics for a mixed schrödinger equation under essentially minimal regularity assumptions on initial data in a weighted Sobolev space . In the asymptotic expression, the leading order term comes from dispersive part and the error order from a equation

Paper Structure

This paper contains 10 sections, 8 theorems, 148 equations, 6 figures.

Table of Contents

  1. Introction
  2. Spectral analysis
  3. Asymptotic
  4. Analyticity and symmetry
  5. The construction of a RH problem
  6. Triangular factorizations of jump matrix
  7. A hybrid $\overline{\partial}$ problem
  8. Analysis on a solvable model
  9. Analysis on a pure $\bar{\partial}$ problem
  10. Long time asymptotics of the mixed NLS

Key Result

Proposition 1

For $q(x) \in L^1(\mathbb{R} )$ and $t\in \mathbb{R}^+$, the eigenfunctions defined by (pioi) exist and are unique. Moreover, $\mu_{-,1}$ and $\mu_{+,2}$ are analytic in $D^{+}$; $\mu_{-,2}$ and $\mu_{+,1}$ are analytic in $D^{-}$.

Figures (6)

  • Figure 1: Analytical domains $D^+$, $D^-$ and boundary $\Sigma$ corresponding to the mixed NLS equation.
  • Figure 2: The jump matrices of $N$.
  • Figure 3: The jump matrix of $N^{(1)}(z)$.
  • Figure 4: $\mathcal{R}^{(2)}$ in each $\Lambda_j$.
  • Figure 5: The jump matrices $V_{N}^{(2)}$ for $N^{(2)}$. $\bar{\partial} R^{(2)}\not=0$ in pink domian; and $\bar{\partial} R^{(2)} =0$ in white domian
  • ...and 1 more figures

Theorems & Definitions (11)

  • Proposition 1
  • Proposition 2
  • Theorem 4.1
  • Lemma 5.1
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • ...and 1 more