论文标题
$ n $ -solitons的孤子分辨率和渐近稳定性在线上的degasperis-procesi方程
Soliton resolution and asymptotic stability of $N$-solitons to the Degasperis-Procesi equation on the line
论文作者
论文摘要
Degasperis-Procesi（DP）方程 \ begin {align} ＆u_t-u_ {txx}+3κU_x+4uu_x = 3U_x u_ {xx}+uu_ {xxx}，\ nonumber \ end {align} 作为描述浅水波的传播的模型，作为一个完全可以集成的系统，并接收了$ 3 \ times3 $矩阵lax对。在本手稿中，我们研究了schwarz空间中通用初始数据的cauchy问题解决方案的孤子分辨率和大量时间行为。我们采用了deift-Zhou非线性陡峭下降方法的$ \ bar \ partial $ - 赋形剂，我们在两种不同类型的时空区域中推断出解决方案$ u（x，t）$的不同长期渐近扩展。该结果验证了DP方程的$ N $ -Soliton解决方案的孤子分辨率猜想和渐近稳定性。
The Degasperis-Procesi (DP) equation \begin{align} &u_t-u_{txx}+3κu_x+4uu_x=3u_x u_{xx}+uu_{xxx}, \nonumber \end{align} serving as a model delineating the propagation of shallow water waves, stands as a completely integrable system and admits a $3\times3$ matrix Lax pair. In this manuscript, we study the soliton resolution and large time behavior of solutions to the Cauchy problem of the DP equation with generic initial data in Schwarz space. Employing the $\bar \partial$-generalization of the Deift-Zhou nonlinear steepest descent method, we deduce different long time asymptotic expansions of the solution $u(x,t)$ in two distinct types of space-time regions. This result verifies the soliton resolution conjecture and asymptotic stability of $N$-soliton solutions for the DP equation.
