Spectral stability of periodic waves for the Zakharov system

What is it about?

The Zakharov system is a coupled system of partial differential equations, which was first obtained by Zakharov to describe the long wave Langmuir turbulence in a plasma. We are interested in coherent structures, and more specifically traveling wave solutions, which contain important clues about the global behavior of the solutions. We investigate periodic in space solutions, which are obtained in terms of Jacobi elliptic functions, which exist for a large set of parameters. Importantly, in the limit of large periods, these periodic waves converge to the solitons for this system. The stability of the dynamics near such periodic waves is an important question in mathematical physics, both from theoretical and practical point of view. In this work, we show that all periodic waves are stable.

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Photo by Alexander Grey on Unsplash

Photo by Alexander Grey on Unsplash

Why is it important?

We continue the long line of investigations of the stability of periodic waves using Hamiltonian index theory. Our main result removes important technical condition to allow all natural values of the parameters.