Exact elliptic compactons in generalized Korteweg-De Vries equations

Abstract

Using the action principle, and assuming a solitary wave of the generic form u(x,t) = AZ(β(x + q(t)), we derive a general theorem relating the energy, momentum, and velocity of any solitary wave solution of the generalized Korteweg-De Vries equation K^∗ (l,p). Specifically we find that q=r(l,p)H/P where l,p are nonlinearity parameters. We also relate the amplitude, width, and momentum to the velocity of these solutions. We obtain the general condition for linear and Lyapunov stability. We then obtain a two-parameter family of exact solutions to these equations, which include elliptic and hyper-elliptic compacton solutions. For this general family we explicitly verify both the theorem and the stability criteria.