论文标题
具有有限密度类型初始数据的非局部非线性schrödinger方程的长期渐近行为
Long-time asymptotic behavior of the nonlocal nonlinear Schrödinger equation with finite density type initial data
论文作者
论文摘要
在这项工作中，我们采用$ \ bar {\ partial} $ - 最陡的下降方法来研究非局部非线性schrödinger（nnls）方程的cauchy问题，具有有限密度类型的初始条件，在加权Sobolev Space $ \ Mathcal {H}（H}（H}（H}（\ Mathbb r}））中。基于LAX频谱问题，构建了与原始问题相对应的Riemann-Hilbert问题，以使NNLS方程的解决方案具有有限密度类型的初始边界值条件。通过开发$ \ bar {\ partial} $ - deift-zhou非线性陡峭下降方法的概括，我们将领先顺序近似近似于解决方案$ q（x，t）$在时空区域中，$ \ left（\ frac {x}} {x} {2t} {2t} {2t} {2t} {2t} {2t} {2t} $ $ a $ a $ a $ a $ a $ k）真正的常数），并为错误衰减的界限作为$ | t | \ rightarrow \ infty $。 Based on the resulting asymptotic behavior, the asymptotic approximation of the NNLS equation is characterized with the soliton term confirmed by $N(Λ)$-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}}）$。
In this work, we employ the $\bar{\partial}$-steepest descent method to investigate the Cauchy problem of the nonlocal nonlinear Schrödinger (NNLS) equation with finite density type initial conditions in weighted Sobolev space $\mathcal{H}(\mathbb{R})$. Based on the Lax spectrum problem, a Riemann-Hilbert problem corresponding to the original problem is constructed to give the solution of the NNLS equation with the finite density type initial boundary value condition. By developing the $\bar{\partial}$-generalization of Deift-Zhou nonlinear steepest descent method, we derive the leading order approximation to the solution $q(x,t)$ in soliton region of space-time, $\left(\frac{x}{2t}\right)=ξ$ for any fixed $ξ=\in (1,K)$($K$ is a sufficiently large real constant), and give bounds for the error decaying as $|t|\rightarrow\infty$. Based on the resulting asymptotic behavior, the asymptotic approximation of the NNLS equation is characterized with the soliton term confirmed by $N(Λ)$-soliton on discrete spectrum and the $t^{-\frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-\frac{3}{4}})$.
