Abstract. We apply the method of nonlinear steepest descent to compute the long-time asymptotics of the Korteweg-de Vries equation for decaying initial data in the soliton and similarity region. This paper can be viewed as an expository introduction to this method.

Abstract. We prove existence of a global conservative solution of the Cauchy problem for the two-component Camassa-Holm (2CH) system on the line, allowing for nonvanishing and distinct asymptotics at plus and minus infinity. The solution is proven to be smooth as long as the density is bounded away from zero. Furthermore, we show that by taking the limit of vanishing density in the 2CH system, we obtain the global conservative solution of the (scalar) Camassa-Holm equation, which provides a novel way to define and obtain these solutions. Finally, it is shown that while solutions of the 2CH system have infinite speed of propagation, singularities travel with finite speed.

We solve the Cauchy problem for the Korteweg-de Vries equation with initial conditions which are steplike Schwartz-type perturbations of finitegap potentials under the assumption that the respective spectral bands either coincide or are disjoint.

We introduce a novel solution concept, denoted α-dissipative solutions, that provides a continuous interpolation between conservative and dissipative solutions of the Cauchy problem for the twocomponent Camassa-Holm system on the line with vanishing asymptotics. All the α-dissipative solutions are global weak solutions of the same equation in Eulerian coordinates, yet they exhibit rather distinct behavior at wave breaking. The solutions are constructed after a transformation into Lagrangian variables, where the solution is carefully modified at wave breaking.

Abstract. We study stability of conservative solutions of the Cauchy problem for the periodic Camassa-Holm equation ut −uxxt +3uux −2uxuxx −uuxxx = 0 with initial data u 0 . In particular, we derive a new Lipschitz metric d D with the property that for two solutions u and v of the equation we have