The Cauchy problem of the Camassa-Holm equation in a weighted Sobolev space: Long-time and Painlev\'e asymptotics

Based on the $\overline\partial$-generalization of the Deift-Zhou steepest descent method, we extend the long-time and Painlev\'e asymptotics for the Camassa-Holm (CH) equation to the solutions with initial data in a weighted Sobolev space $H^{4,2}(\mathbb{R})$. With a new scale $(y,t)$ and a RH problem associated with the initial value problem,we derive different long time asymptotic expansions for the solutions of the CH equation in different space-time solitonic regions. The half-plane $\{ (y,t): -\infty 0\}$ is divided into four asymptotic regions: 1. Fast decay region, $y/t \in(-\infty,-1/4)$ with an error $\mathcal{O}(t^{-1/2})$; 2. Modulation-solitons region, $y/t \in(2,+\infty)$, the result can be characterized with an modulation-solitons with residual error $\mathcal{O}(t^{-1/2 })$; 3. Zakhrov-Manakov region,$y/t \in(0,2)$ and $y/t \in(-1/4,0)$. The asymptotic approximations is characterized by the dispersion term with residual error $\mathcal{O}(t^{-3/4})$; 4. Two transition regions, $|y/t|\approx 2$ and $|y/t| \approx -1/4$, the results are describe by the solution of Painlev\'e II equation with error order $\mathcal{O}(t^{-1/2})$.Comment: 61 page

Long time asymptotics of the Camassa–Holm equation on the half-line

We study the long-time behavior of solutions of the initial-boundary value (IBV) problem for the Camassa–Holm (CH) equation ut − utxx + 2ux + 3uux = 2uxuxx + uuxxx on the half-line x ≥ 0. The paper continues our study of the IBV problems for the CH equation [9], the key tool of which is the formulation and analysis of the associated Riemann-Hilbert factorization problem. We specify the regions in the quarter space-time plane x> 0, t> 0 having qualitatively different asymptotic pictures, and give the main terms of the asymptotics in terms of the spectral data associated with the initial and boundary values

Generalized Camassa-Holm Equations: Symmetry, Conservation Laws and Regular Pulse and Front Solutions

In this paper, we consider a member of an integrable family of generalized Camassa-Holm (GCH) equations. We make an analysis of the point Lie symmetries of these equations by using the Lie method of infinitesimals. We derive nonclassical symmetries and we find new symmetries via the nonclassical method, which cannot be obtained by Lie symmetry method. We employ the multiplier method to construct conservation laws for this family of GCH equations. Using the conservation laws of the underlying equation, double reduction is also constructed. Finally, we investigate traveling waves of the GCH equations. We derive convergent series solutions both for the homoclinic and heteroclinic orbits of the traveling-wave equations, which correspond to pulse and front solutions of the original GCH equations, respectively

On critical behaviour in systems of Hamiltonian partial differential equations

We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlev\ue9-I (PI) equation or its fourth-order analogue P2I. As concrete examples, we discuss nonlinear Schr\uf6dinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture

On universality of critical behavior in the focusing nonlinear Schr\uf6dinger equation, elliptic umbilic catastrophe and the Tritronqu\ue9e solution to the Painlev\ue9-I equation

We argue that the critical behavior near the point of "gradient catastrophe" of the solution to the Cauchy problem for the focusing nonlinear Schrodinger equation i epsilon Psi(t) + epsilon(2)/2 Psi(xx) + vertical bar Psi vertical bar(2)Psi = 0, epsilon << 1, with analytic initial data of the form Psi( x, 0; epsilon) = A(x)e(i/epsilon) (S(x)) is approximately described by a particular solution to the Painleve-I equation