On the Burgers-Poisson Equation

In this paper, we prove the existence and uniqueness of weak entropy solutions to the Burgers-Poisson equation for initial data in L^1(R). Additional an Oleinik type estimate is established and some criteria on local smoothness and wave breaking for weak entropy solutions are provided.Comment: 22 page

The transformation operator for Schrödinger operators on almost periodic infinite-gap backgrounds

AbstractWe investigate the kernels of the transformation operators for one-dimensional Schrödinger operators with potentials, which are asymptotically close to Bohr almost periodic infinite-gap potentials

Global dissipative solutions of the two-component Camassa-Holm system for initial data with nonvanishing asymptotics

We show existence of a global weak dissipative solution of the Cauchy problem for the two-component Camassa-Holm (2CH) system on the line with nonvanishing and distinct spatial asymptotics. The influence from the second component in the 2CH system on the regularity of the solution, and, in particular, the consequences for wave breaking, is discussed. Furthermore, the interplay between dissipative and conservative solutions is treated.Comment: arXiv admin note: text overlap with arXiv:1111.318

Uniqueness of dissipative solutions for the Camassa-Holm equation

We show that the Cauchy problem for the Camassa-Holm equation has a unique, global, weak, and dissipative solution for any initial data $u_0\in H^1(\mathbb{R})$, such that $u_{0,x}$ is bounded from above almost everywhere. In particular, we establish a one-to-one correspondence between the properties specific to the dissipative solutions and a solution operator associating to each initial data exactly one solution.Comment: 41 pages, 3 figure

Periodic conservative solutions for the two-component Camassa-Holm system

We construct a global continuous semigroup of weak periodic conservative solutions to the two-component Camassa-Holm system, $u_t-u_{txx}+\kappa u_x+3uu_x-2u_xu_{xx}-uu_{xxx}+\eta\rho\rho_x=0$ and $\rho_t+(u\rho)_x=0$, for initial data $(u,\rho)|_{t=0}$ in $H^1_{\rm per}\times L^2_{\rm per}$. It is necessary to augment the system with an associated energy to identify the conservative solution. We study the stability of these periodic solutions by constructing a Lipschitz metric. Moreover, it is proved that if the density $\rho$ is bounded away from zero, the solution is smooth. Furthermore, it is shown that given a sequence $\rho_0^n$ of initial values for the densities that tend to zero, then the associated solutions $u^n$ will approach the global conservative weak solution of the Camassa-Holm equation. Finally it is established how the characteristics govern the smoothness of the solution.Comment: To appear in Spectral Analysis, Differential Equations and Mathematical Physics, Proc. Symp. Pure Math., Amer. Math. So