Katrin Grunert

Long-Time Asymptotics for the Korteweg–de Vries Equation via Nonlinear Steepest Descent

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.

On the Cauchy problem for the Korteweg–de Vries equation with steplike finite-gap initial data: I. Schwartz-type perturbations

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

Global Solutions for the Two-Component Camassa–Holm System

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.

The general peakon–antipeakon solution for the Camassa–Holm equation

We compute explicitly the peakon–antipeakon solution of the Camassa–Holm (CH) equation ut − utxx + 3uux − 2uxuxx − uuxxx = 0 in the non-symmetric and α-dissipative case. The solution experiences wave breaking in finite time, and the explicit solution illuminates the interplay between the various variables.

A Lagrangian View on Complete Integrability of the Two-Component Camassa–Holm System

We show how the change from Eulerian to Lagrangian coordinates for the two-component Camassa–Holm system can be understood in terms of certain reparametrizations of the underlying isospectral problem. The respective coordinates correspond to different normalizations of an associated first order system. In particular, we will see that the two-component Camassa–Holm system in Lagrangian variables is completely integrable as well.

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

Abstract We investigate the kernels of the transformation operators for one-dimensional Schrodinger operators with potentials, which are asymptotically close to Bohr almost periodic infinite-gap potentials.

On the Equivalence of Eulerian and Lagrangian Variables for the Two-Component Camassa–Holm System

The Camassa–Holm equation and its two-component Camassa–Holm system generalization both experience wave breaking in finite time. To analyze this, and to obtain solutions past wave breaking, it is common to reformulate the original equation given in Eulerian coordinates, into a system of ordinary differential equations in Lagrangian coordinates. It is of considerable interest to study the stability of solutions and how this is manifested in Eulerian and Lagrangian variables. We identify criteria of convergence, such that convergence in Eulerian coordinates is equivalent to convergence in Lagrangian coordinates. In addition, we show how one can approximate global conservative solutions of the scalar Camassa–Holm equation by smooth solutions of the two-component Camassa–Holm system that do not experience wave breaking.