Stability of solitons in nonlinear composite media

An analysis is made of the dynamic stability of soliton solutions of the Hamilton equations describing plane waves in nonlinear elastic composite media in the presence and absence of anisotropy. In the anisotropiccase two two-parameter soliton families, fast and slow, are obtained in analytic form; in the absence of anisotropy there is a single three-parameter soliton family. It is shown that solitons from the slow family in an anisotropic composite and solitons in an isotropic composite are dynamically stable if their velocities lie in a certain range known as the range of stability. The analysis of stability is based on the spectral properties of the “linearized Hamiltonian” ℋ. It is shown that the operator ℋ is positively semidefinite on some linear subspace of the main solution space from which stability follows. Problems of instability of the fast soliton family in the anisotropic case and representatives of soliton families whose velocities lie outside the range of stability in the presence and absence of anisotropy are discussed.