where Î±, Î³, Ï‰ are given real constants. Equation (1) was first introduced as a model describing propagation of unidirectional gravitational waves in a shallow water approximation over a flat bottom,â€¦ (More)

Combining algebro-geometric methods and factorization techniques for finite difference expressions we provide a complete and self-contained treatment of all real-valued quasi-periodic finite-gapâ€¦ (More)

The Camassaâ€“Holm equation utâˆ’uxxt +3uuxâˆ’2uxuxxâˆ’uuxxx = 0 enjoys special solutions of the form u(x, t) = Pn i=1 pi(t)e âˆ’|xâˆ’qi(t)|, denoted multipeakons, that interact in a way similar to that ofâ€¦ (More)

We extend the trace formula recently proven for general one-dimensional SchrÃ¶dinger operators which obtains the potential V (x) from a function Î¾(x,Î») by deriving trace relations computing moments ofâ€¦ (More)

We provide a detailed treatment of real-valued, smooth and bounded algebro-geometric solutions of the Camassa-Holm (CH) hierarchy and describe the associated isospectral torus. We employâ€¦ (More)

We review a variety of recently obtained trace formulas for oneand multidimensional SchrÃ¶dinger operators. Some of the results are extended to Sturm-Liouville and matrix-valued SchrÃ¶dinger operators.â€¦ (More)

We show how to construct globally defined multipeakon solutions of the Camassaâ€“Holm equation. The construction includes in particular the case with peakonantipeakon collisions. The solutions areâ€¦ (More)

We prove that a certain finite difference scheme converges to the weak solution of the Cauchy problem on a finite interval with periodic boundary conditions for the Camassaâ€“Holm equation ut âˆ’ uxxt +â€¦ (More)

Abstract. We prove existence of global and conservative solutions of the Cauchy problem for the nonlinear partial differential equation ut âˆ’ uxxt + f(u)x âˆ’ f(u)xxx + (g(u) + 1 2 f (u)(ux))x = 0 whereâ€¦ (More)