论文标题
在时空孤子区域中Camassa-Holm方程的长期渐近行为
On the long-time asymptotic behavior of the Camassa-Holm equation in space-time solitonic regions
论文作者
论文摘要
在这项工作中，我们致力于研究Camassa-Holm（CH）方程的Cauchy问题，并在时空孤子区域中使用加权的Sobolev初始数据\ BEGIN \ BEGIN {ALIGN*} M_T+2κQ_X+3QQQ_X+3QQ_X = 2Q_XQ_XQ_XQ_ {xx}+QQ_ {xx}+qq_ {xx+qq_ {xx}+} q（x，0）= q_0（x）\ in H^{4,2}（\ mathbb r），~~ x \ in \ mathbb r，~~ t> 0，\ end end {align*}，其中$κ$是一个正常数。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CH方程的解决方案具有初始边界值条件。此外，通过开发$ \ bar {\ partial} $ - deift-Zhou非线性陡峭下降方法的概括，得出了解决方案$ q（x，t）$的不同长期渐近扩展。在这项工作中分开了四个渐近区域：对于$之一，\左（ - \ infty， - \ frac {1} {1} {4} {4} \ right）\ cup（2，\ infty）$，相位函数$θ（z）$在跳高轮廓上没有固定点，并确认了$ nirdit of solitotic unigity $ nive y-nirdim nive nimit-n extim nive nimit-n extim nive nime nive n eartim niquit yourtimed（y）。在离散频谱上，剩余误差最高为$ o（t^{ - 1+2τ}）$;对于$ en \ left（ - \ frac {1} {4}，0 \右）$和$ξ\ in \ left（0,2 \ right）$，相位函数$θ（z）$具有四个和两个固定点，在跳跃轮廓上具有四个和两个固定点，并且可以用poltotic的$ n（$ n）（$ n）（$ n）（j_ nivit）（j。 $ t^{ - \ frac {1} {2}} $连续频谱上的订单项，残留错误最高为$ O（t^{ - 1}）$。我们的结果还证实了CH方程的孤子分辨率猜想，并在时空孤子区域中具有加权Sobolev初始数据。
In this work, we are devoted to study the Cauchy problem of the Camassa-Holm (CH) equation with weighted Sobolev initial data in space-time solitonic regions \begin{align*} m_t+2κq_x+3qq_x=2q_xq_{xx}+qq_{xx},~~m=q-q_{xx}+κ,\\ q(x,0)=q_0(x)\in H^{4,2}(\mathbb R),~~x\in\mathbb R, ~~t>0, \end{align*} where $κ$ is a positive constant. Based on the Lax spectrum problem, a Riemann-Hilbert problem corresponding to the original problem is constructed to give the solution of the CH equation with the initial boundary value condition. Furthermore, by developing the $\bar{\partial}$-generalization of Deift-Zhou nonlinear steepest descent method, different long-time asymptotic expansions of the solution $q(x,t)$ are derived. Four asymptotic regions are divided in this work: For $ξ\in\left(-\infty,-\frac{1}{4}\right)\cup(2,\infty)$, the phase function $θ(z)$ has no stationary point on the jump contour, and the asymptotic approximations can be characterized with the soliton term confirmed by $N(j_0)$-soliton on discrete spectrum with residual error up to $O(t^{-1+2τ})$; For $ξ\in\left(-\frac{1}{4},0\right)$ and $ξ\in\left(0,2\right)$, the phase function $θ(z)$ has four and two stationary points on the jump contour, and the asymptotic approximations can be characterized with the soliton term confirmed by $N(j_0)$-soliton on discrete spectrum and the $t^{-\frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-1})$. Our results also confirm the soliton resolution conjecture for the CH equation with weighted Sobolev initial data in space-time solitonic regions.
