论文标题
具有加权Sobolev初始数据的耦合通用非线性schrödinger方程的孤子分辨率
Soliton resolution for a coupled generalized nonlinear Schrödinger equations with weighted Sobolev initial data
论文作者
论文摘要
在这项工作中，我们采用$ \ bar {\ partial} $最陡的下降方法，以研究CGNLS方程的Cauchy问题，具有最初条件的sobolev Space $ H^{1,1}（\ Mathbb {r}） l^{2}（\ mathbb {r}）\} $。解决方案$ u（x，t）$和$ v（x，t）$的较大时间渐近行为是在固定时空锥$ s（x_ {1}，x_ {2}，v_ {1}，v_ {1}，v_ {2}，v_ {2}）= \ {（x，x，x，x，x，x，x，x，x）= _ \ in \ mathbb {2} {2} {2} {2} {2} {2} {2} {2} {2} {2} {2} { 〜x_ {0} \ in [x_ {1}，x_ {2}]，〜v \ in [v_ {1}，v_ {2}] \} $。基于由此产生的渐近行为，我们证明了解决方案的cgnls方程的解决方案，其中包含由$ | \ m natercal {z}确认的孤子术语（\ nathcal {i}）| $ -Soliton在离散的频谱上，$ t^{ - \ frac contran contry of nive spectrum and contry of contry of clange and \ frac { - \ frac { - \ \ frac {2} $ {2} $ {2}} $ O（t^{ - \ frac {3} {4}}）$。
In this work, we employ the $\bar{\partial}$ steepest descent method in order to study the Cauchy problem of the cgNLS equations with initial conditions in weighted Sobolev space $H^{1,1}(\mathbb{R})=\{f\in L^{2}(\mathbb{R}): f',xf\in L^{2}(\mathbb{R})\}$. The large time asymptotic behavior of the solution $u(x,t)$ and $v(x,t)$ are derived in a fixed space-time cone $S(x_{1},x_{2},v_{1},v_{2})=\{(x,t)\in\mathbb{R}^{2}: x=x_{0}+vt, ~x_{0}\in[x_{1},x_{2}], ~v\in[v_{1},v_{2}]\}$. Based on the resulting asymptotic behavior, we prove the solution resolution conjecture of the cgNLS equations which contains the soliton term confirmed by $|\mathcal{Z}(\mathcal{I})|$-soliton on discrete spectrum and the $t^{-\frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-\frac{3}{4}})$.
