Jonatan Lenells

Conservation laws of the Camassa–Holm equation

We use the bi-Hamiltonian structure of the Camassa-Holm equation to show that its conservation laws H-n[m] are homogeneous with respect to the scaling m --> lambdam. Moreover, a direct argument is presented proving that H-1, H-2,..., are of local character. Finally, simple representations of the conservation laws in terms of their variational derivatives are derived and used to obtain a constructive scheme for computation of the H(n)s.

GENERALIZED HUNTER-SAXTON EQUATION AND THE GEOMETRY OF THE GROUP OF CIRCLE DIFFEOMORPHISMS

We study an equation lying ‘mid-way’ between the periodic Hunter–Saxton and Camassa–Holm equations, and which describes evolution of rotators in liquid crystals with external magnetic field and self-interaction. We prove that it is an Euler equation on the diffeomorphism group of the circle corresponding to a natural right-invariant Sobolev metric. We show that the equation is bihamiltonian and admits both cusped and smooth traveling-wave solutions which are natural candidates for solitons. We also prove that it is locally well-posed and establish results on the lifespan of its solutions. Throughout the paper we argue that despite similarities to the KdV, CH and HS equations, the new equation manifests several distinctive features that set it apart from the other three.

A Variational Approach to the Stability of Periodic Peakons

Abstract The peakons are peaked traveling wave solutions of an integrable shallow water equation. We present a variational proof of their stability.

Stability of periodic peakons

The peakons are peaked traveling wave solutions of a nonlinear integrable equation modeling shallow water waves. We give a simple proof of their stability.

The Scattering Approach for the Camassa–Holm equation

Abstract We present an approach proving the integrability of the Camassa–Holm equation for initial data of small amplitude.

Inverse scattering transform for the Degasperis–Procesi equation

We develop the inverse scattering transform (IST) method for the Degasperis–Procesi equation. The spectral problem is an Zakharov–Shabat problem with constant boundary conditions and finite reduction group. The basic aspects of the IST, such as the construction of fundamental analytic solutions, the formulation of a Riemann–Hilbert problem, and the implementation of the dressing method, are presented.

Integrable Evolution Equations on Spaces of Tensor Densities and Their Peakon Solutions

We study a family of equations defined on the space of tensor densities of weight λ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct “peakon” and “multi-peakon” solutions for all λ ≠ 0, 1, and “shock-peakons” for λ = 3. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold’s approach to Euler equations on Lie groups.

On a novel integrable generalization of the nonlinear Schrödinger equation

We consider an integrable generalization of the nonlinear Schrodinger (NLS) equation that was recently derived by one of the authors using bi-Hamiltonian methods. This equation is related to the NLS equation in the same way as the Camassa-Holm equation is related to the KdV equation. In this paper we (a) use the bi-Hamiltonian structure to write down the first few conservation laws, (b) derive a Lax pair, (c) use the Lax pair to solve the initial value problem and (d) analyse solitons.

The Hunter–Saxton Equation: A Geometric Approach

We provide a rigorous foundation for the geometric interpretation of the Hunter–Saxton equation as the equation describing the geodesic flow of the

The unified method : I. Nonlinearizable problems on the half-line

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