论文标题
复杂的立方camassa-holm方程的长期渐近学
Long-time asymptotics for a complex cubic Camassa-Holm equation
论文作者
论文摘要
在本文中，我们调查了以下复杂的立方Camassa-Holm（CCCH）方程的Cauchy问题$$ m_ {t} = b u_ {x}+\ frac {1} {2} \ left [m \ left（| U |^{2} - \ left | u_ {x} \ right |^{2} \ right |^{2} \ right） \ bar {u} _ {x} -u_ {x} \ bar {u} \ right），\ quad m = u-u_ {x x}，$ b> 0 $是任意正常的真实常数。方程式的长期渐近学是通过$ \ bar {\ partial} $ - 最陡的下降方法获得的。首先，基于LAX对和散射矩阵的光谱分析，可以通过求解相应的Riemann-Hilbert（RH）问题来构建方程的解。然后，我们在不同的时空孤子区域$ u（y，t）$呈现$ u（y，t）$的不同长时间渐近扩展。半平面$ {（y，t）： - \ Infty <y <\ iftty，t> 0} $分为四个渐近区域：$ξ\ in（ - \ infty，-1）$，$ξ\ in（-1,1,0）$，$ （\ frac {1} {8}，+\ infty）$。当$ξ$落入$（ - \ infty，-1）\ cup（\ frac {1} {8}，+\ infty）$时，在时空区域的跳转配置文件上不存在相位函数$θ（z）$的固定相位点。在这种情况下，可以用$ n（λ）$ - 孤子子具有不同的残留错误顺序$ O（t^{ - 1+2 \ varepsilon}）$来表征相应的渐近近似值。跳跃曲线上有四个固定相点和八个固定相点，分别为$ξ\ in（-1,0）$和$ξ\ in（0，\ frac {1} {8}）$。相应的渐近形式伴随着剩余错误顺序$ O（t^{ - \ frac {3} {4}}}）$。
In this paper, we investigate the Cauchy problem of the following complex cubic Camassa-Holm (ccCH) equation $$m_{t}=b u_{x}+\frac{1}{2}\left[m\left(|u|^{2}-\left|u_{x}\right|^{2}\right)\right]_{x}-\frac{1}{2} m\left(u \bar{u}_{x}-u_{x} \bar{u}\right), \quad m=u-u_{x x},$$ where $b>0$ is an arbitrary positive real constant. Long-time asymptotics of the equation is obtained through the $\bar{\partial}$-steepest descent method. Firstly, based on the spectral analysis of the Lax pair and scattering matrix, the solution of the equation is able to be constructed via solving the corresponding Riemann-Hilbert (RH) problem. Then, we present different long time asymptotic expansions of the solution $u(y,t)$ in different space-time solitonic regions of $ξ=y/t$. The half-plane ${(y,t):-\infty <y< \infty, t > 0}$ is divided into four asymptotic regions: $ξ\in(-\infty,-1)$, $ξ\in (-1,0)$, $ξ\in (0,\frac{1}{8})$ and $ξ\in (\frac{1}{8},+\infty)$. When $ξ$ falls in $(-\infty,-1)\cup (\frac{1}{8},+\infty)$, no stationary phase point of the phase function $θ(z)$ exists on the jump profile in the space-time region. In this case, corresponding asymptotic approximations can be characterized with an $N(Λ)$-solitons with diverse residual error order $O(t^{-1+2\varepsilon})$. There are four stationary phase points and eight stationary phase points on the jump curve as $ξ\in (-1,0)$ and $ξ\in (0,\frac{1}{8})$, respectively. The corresponding asymptotic form is accompanied by a residual error order $O(t^{-\frac{3}{4}})$.
