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Stability of cnoidal waves. (English) Zbl 1147.35079

Summary: This paper is concerned with the stability of periodic travelling-wave solutions of the Korteweg–de Vries equation \[ u_{t} + uu_{x} + u_{xxx} = 0. \] Here, \(u\) is a real valued function of the two variables \(x,t \in \mathbb R\) and subscripts connote partial differentiation. These special solutions were termed cnoidal waves by Korteweg and de Vries. They also appear in earlier work of Boussinesq. It is shown that these solutions are stable to small, periodic perturbations in the context of the initial-value problem. The approach is that of the modern theory of stability of solitary waves, but adapted to the periodic context. The theory has prospects for the study of periodic travelling-wave solution of other partial differential equations.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35Q51 Soliton equations
35B10 Periodic solutions to PDEs
35B35 Stability in context of PDEs
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
76B25 Solitary waves for incompressible inviscid fluids
76E99 Hydrodynamic stability
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