Of Solitary Waves In Nonlinear Composite Material Instability

In this paper, we first establish some abstact results on the orbital stability and orbital instability for some abstract complex Hamitonian systems. Based on the abstract results obtained above, we investigate a class of complex Boussinesq-type equations arising in nonlinear composite media, and prove the orbital instability of two families of explicit solitary wave solutions (slow family in anisotropic case and solitary waves in isotropic case), which theoretically verify the numerical results and the guess in [11]. In this paper, we also prove the orbital stability of a class of peakon wave solutions for the generalized shallow water equation.1. In Chapter 2, we study complex Hamiltionian systems with the following formwhere the solitary waves have translation invariance and rotation invariance.Under the assumptions 1-4, we get the sufficient conditions for the orbital stability and instability of the solitary waves. As for the orbital stability, based on the ideas of [1], we prove that the solitary waves are orbitally stable if the scarlar function d(v)= E(v) + vQ(v) is convex at v, that is d"(v) > 0. Note that if the skew-symmetric linear operator J is not onto, then the abstract results on the orbital instability in [1] are not valid. Based on the main ideas in [3] [4] [5] [12], we extend the abstract results in [1] to the case where J = D, where D is a symmetric matrix of constant, which is not onto, and we prove that under some additional a priori estimates the solitary waves are orbitally unstable if d"(v) < 0.The main results in this chapter are as follows:Theorem 2.1 Under Asumptions 1-3, if d"(v) > 0, then the solitary waves are orbitally stable.Theorem 2.2 Under Asumptions 1-4, if d"(v) < 0, then the solitary waves are orbitally unstable.2. In Chapter 3, we investigate a class of complex Boussinesq-type equations whichFirstly, applying the classical continuous semigroup theory, we prove the localwell-posedness of the initial value problem of (2). The equations (2) can beabstract results obtained in Chapter 2, by detailed a priori estimates and complicated computations, the assumptions in Theorem 2.2 are verified, then orbital instability of solitary waves are obtained for (slow family in anisotropic case) and (solitry waves in isotropic case).The main results in this chapter are as follows:Theorem 3.1 Let , then there exists a unique local solution w(x,t) of (3.2.1) in Theorem 3.2 Let that and , where Assume satisfies (3.1.1) and for 0 < t < t0, where to denotes the maximum existence time for w and the constant C0 depends only on ||w0||x. Theorem 3.3 If d"(v) < 0, that isthen the solitary waves are orbitally unstable.3. In Chapter 4, we investigate the following generalized shallow water equatoin:A class of explicit peakons are obtained in [16]. Using special transformation in [15] and applying the abstract stability theory to (3), we prove the orbital stability of the explicit peakons.The main result in this chapter: Theorem 4.1 The solitary waves (4.1.3) are orbitally stable.