NONTRIVIAL PERIODIC SOLUTIONS OF THE SIN-GORDON EQUATION
Abstract
In this paper, we study the problem of time-periodic solutions of the sin-Gordon equation with Neumann and Dirichlet boundary conditions on an interval. The novelty of the paper lies in the fact that in previous papers the existence of periodic solutions of the sin-Gordon equation on an interval was proved for the case of Dirichlet and third kind boundary conditions. In the study of the equation, a variational method is used. Periodic solution of the problem is found as a critical point of the energy functional. To prove the existence of a critical point, the functional is limited to finite-dimensional subspaces and a kind of “pass” theorem is used, which allows finding saddle stationary points. Using the features of the spectrum of the differential operator and the nonlinear term in these subspaces, meshing surfaces are found that satisfy the conditions of the “pass” theorem. To implement the passage to the limit, when the dimension of the subspaces tends to infinity, we prove uniform estimates for the sequence of functions that are stationary points of the functional on these subspaces. The passage to the limit uses the compactness method. The proof of the smoothness of the generalized solution is carried out with the help of Fourier series. To prove the convergence of the Fourier series and their derivatives, we study the eigenvalues of the differential operator corresponding to the linear part of the equation.
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