Conditional Orbital Stability of Soliton Solutions of the Module Schamel Equation

In this paper, we study the nonlinear orbital stability of soliton solutions of the Schamel equation with modular nonlinearity describing solitary waves of different polarities. The proof of the nonlinear stability of these solutions is established within the framework of the general theory for the stability of bound states that decrease rather rapidly on the real axis and correspond to soliton solutions of the translationally invariant infinite-dimensional Hamiltonian system (which is the Schamel equation under consideration). The required stability is established based on verification of the formulated stability conditions. These conditions, being sufficient, imply checking the spectral properties of the operator resulting from linearization of a functional that makes sense of the Lyapunov function. Generally speaking, for translationally invariant differential equations it is impossible to consider the usual stability, when a small perturbation of the solution remains small. The solutions of such equations are subject to investigation by the presence of orbital stability, i. e., stability “up to shear accuracy”. As a result, it is not possible to construct a Lyapunov function (functional) in the problems of stability of boundary states, which would have a local minimum determined by a neighborhood system of the basic functional space of the problem. If the conditions of orbital stability are met, the Lyapunov function has a conditional local minimum, i. e., a local minimum on some nonlinear submanifold of the basic functional space of the system of equations at a point defined by the solution to be investigated for stability. This nonlinear submanifold, as a rule, is determined by the condition of the constancy of the functional, the invariance of which, by virtue of the basic system of equations, is associated with the translational invariance of the problem.