论文标题
在非零背景上修改的Camassa-Holm方程：Cauchy问题的大型渐近学
The modified Camassa-Holm equation on a nonzero background: large-time asymptotics for the Cauchy problem
论文作者
论文摘要
本文讨论了修改后的Camassa-Holm（MCH）方程\ BEGIN {ALIGNAT*} {4}＆M_T+\ left（（（U^2-u_x^2）m \ right）_x = 0，＆\ quad＆m：= 0， u-u_ {xx}，＆\ quad＆t> 0，＆\;＆\ infty <x <x <x <x <+\ infty，\\＆u（x，0）= u_0（x），x <x），&& && && && && && && uspry <x <x <x <x <x <+\ f in of the the Intiral {alignaT*}在case of Intiral ply $ __的$ _0（$ us）$ _0（x）假定将非零常数作为$ x \ to \ pm \ infty $。在最近的一篇论文中，我们为这个问题开发了Riemann-Hilbert形式主义，这使我们能够用相关的Riemann-Hilbert分解问题来表示Cauchy问题的解决方案。在本文中，我们基于这种riemann-hilbert形式主义应用了非线性最陡的下降方法，以研究该库奇问题解决方案的大型渐进性。我们介绍了两个部门的无孤子情况下的渐近分析结果，$ \ frac {3} {4} <\ frac {x} {x} {t} {t} <1 $和$ 1 <\ frac {x} {x} {x} {t} {t} {t} {是不平凡的：该术语由调制（根据$ \ frac {x} {t} $）给出（参数），衰减（AS $ t^{ - 1/2} $）三角振荡。
This paper deals with the Cauchy problem for the modified Camassa-Holm (mCH) equation \begin{alignat*}{4} &m_t+\left((u^2-u_x^2)m\right)_x=0,&\quad&m:= u-u_{xx},&\quad&t>0,&\;&-\infty<x<+\infty,\\ &u(x,0)=u_0(x),&&&&&&-\infty<x<+\infty, \end{alignat*} in the case when the initial data $u_0(x)$ as well as the solution $u(x,t)$ are assumed to approach a nonzero constant as $x\to\pm\infty$. In a recent paper we developed the Riemann--Hilbert formalism for this problem, which allowed us to represent the solution of the Cauchy problem in terms of the solution of an associated Riemann--Hilbert factorization problem. In this paper, we apply the nonlinear steepest descent method, based on this Riemann--Hilbert formalism, to study the large-time asymptotics of the solution of this Cauchy problem. We present the results of the asymptotic analysis in the solitonless case for the two sectors $\frac{3}{4}<\frac{x}{t}<1$ and $1<\frac{x}{t}<3$ (in the $(x,t)$ half-plane, $t>0$), where the leading asymptotic term of the deviation of the solution from the background is nontrivial: this term is given by modulated (with parameters depending on $\frac{x}{t}$), decaying (as $t^{-1/2}$) trigonometric oscillations.
