Two Kinds Of Coupling Bbm Equations Of Solitary Waves Existence And Orbital Stability And Instability

This paper is mainly concerned with the orbital stability and orbital instability of solitary wave solutions for two class of coupled BBM system, which can be written as the form of du/dt = JE'(u). Frstly, the existence of solitary waves with explicit form are obtained. Applying the abstract stability theory in [4] to these two kind of systems respectively, by detailed spectral analysis and com-putaion, for d" > 0, the orbital stability of solitary waves for system I and II are obtained. For the case d" < 0, due to J is not onto, the abstract instability theory in [4] can't be applied to such systems. Using the idea of [6]-[9] and additional invariant, by detailed a priori estimates and complicated computation, we can construct a Lyapunov function and prove the solitary waves for system II are orbitally unstable.The main results of this paper are as follows:Theorem 1.1. For c > 1 or c < 0,there exist solitary waves for system I of the formTheorem 1.2 For c < -1 or c>0, there exist solitary waves for system I of the formTheorem 1.3 For c > 1, there exist solitary waves for system II of theformTheorem 1.4 For c < - 1, there exist solitary waves for system II of theformTheorem 2.3 For c > 1, the solitary waves of system I obtained in Theorem 1.1 are orbitally stable.Theorem 2.4 For c < - 1, the solitary waves (x-ct) of system I obtained in Theorem 1.2 are orbitally stable.Theorem 2.5 For p > l,c > 1, if d"(c) > 0, then the solitary waves {x - ct) of system II obtained in Theorem 1.3 are orbitally stable. Especially, for p = 1 or p = 2, and for any c > 1, the solitary waves (x - ct) of system II obtained in Theorem 1.3 are all orbitally stable.Theorem3.2 For p > 1, c > 1, if d"(c) < 0, then the solitary waves (x - ct) of system II obtained in Theorem 1.3 are orbitally unstable.