A. Geyer, R.H. Martins, F. Natali, and D.E. Pelinovsky
Stability of smooth periodic traveling waves in the Camassa-Holm equation
Stud Appl Math. 148 (2022) 27-61
Abstract:
Smooth periodic travelling waves in the Camassa-Holm (CH) equation are
revisited. We show that these periodic waves can be characterized in two different ways
by using two different Hamiltonian structures. The standard formulation, common to the
Korteweg-de Vries (KdV) equation, has several disadvantages, e.g., the period function
is not monotone and the quadratic energy form may have two rather than one negative
eigenvalues. We explore the nonstandard formulation and prove that the period function
is monotone and the quadratic energy form has only one simple negative eigenvalue. We
deduce a precise condition for the spectral and orbital stability of the smooth periodic
travelling waves and show numerically that this condition is satisfied in the open region
where the smooth periodic waves exist.
Keywords:
Camassa-Holm equation; smooth periodic waves; spectral stability; orbital stability, Hamiltonian structure.