Stability of solitons in elastic composite media

Problems of dynamical stability and instability were considered for soliton solutions of an infinite-dimensional Hamiltonian system of equations describing plane waves in non-linear elastic composite media in both the presence and absence of anisotropy. In the anisotropic case there are fast and slow soliton families branching off from the zero solution corresponding to a rest state in the absence of preliminary deformations. In the isotropic case these two families coalesce and form a unique three-parameter family. It is shown that the solitons in the slow family in an anisotropic compositemedium and the solitons in an isotropic composite medium are dynamically stable if their velocities lie in a certain range. The proof of stability is based on a verification that a soliton solution provides a local minimum to the Hamiltonian under the condition of a fixed value of a functional that is preserved in view of the translation invariance of the system. The conditionalminimum is valid under the non-negative definiteness of the self-adjoint operator that is the second variational derivative of the Hamiltonian, on a certain closed linear subspace of the fundamental space of solutions.