On the Stability of the Compacton Solutions

The stability of the recently discovered compacton solutions is studied by means of both linear stability analysis as well as Lyapunov stability criteria. From the results obtained it follows that, unlike solitons, all the allowed compacton solutions are stable, as the stability condition is satisfied for arbitrary values of the nonlinearity parameter. The results are shown to be true even for the higher order nonlinear dispersion equations for compactons. Some new conservation laws for the higher order nonlinear dispersion equations are also presented. PACS: 3.40.Kf, 52.35.Sb, 63.20.Ry Typeset using REVTEX [email protected] , † [email protected] 1 The observed stationary and dynamical patterns in nature are usually finite in extent. However, most of the weakly nonlinear and linear dispersion equations so far studied admit solitary waves that are infinite in extent, although of a localised in nature. Therefore, the recently discovered compacton solutions (i.e. solitary waves with compact support), of the nonlinear dispersive K(m,n) equations have become very important from the point of study of the effect of nonlinear dispersion on pattren formation as well as the formation of nonlinear structures like liquid drops etc. The compacton speed depends on its height, but unlike the solitons, its width is independent of the speed, a fact, which seems to play a very crucial role in its stability property. Compactons have remarkable soliton like property that they collide elastically. However, unlike soliton collisions in an integrable systems, the point where two compactons collide is marked by the creation of low amplitude compactonanticompacton pairs [1,2]. In fact, it is now known that the K(m,n) system of equations are not integrable [1,2]. This suggest that the observed almost elastic collisions of the compactons is probably not due to the integrability and thus the mechanism responsible for the coherence and robustness of the compactons remains a mystery. Stability analysis of the compacton solutions may provide some clues in this direction. As has been said above, the stability of the compactons is crucial in the context of its applications in the study of pattern formation. Beside, the stability problem of the K(m,n) equations is interesting because, for such equations with higher power of nonlinearity and nonlinear dispersion, the phenomena of collapse is possible. Also, in the context of soliton equations, from the stability analysis it has been shown that the higher order linear dispersion term stabilises the solitons [3,4]. In this regard it will be of interest to see what role does the higher order nonlinear dispersion term plays with respect to the stability of the compacton solutions of theK(m,n) type equations. In this communication we report on the stability analysis of the compacton solutions of the nonlinear dispersion K(m,n) type equations as considered by Cooper et al [2]. We use both the linear stability analysis and the Lyapunov stability criteria to analyse the problem. We start with the K(l, p) equations 2 ut + uxu l−2 + α[2u3xu p + 4puuxu2x + p(p− 1)u ux] = 0 (1) These equations have the same terms as the K(m,n) equations considered by Rosenau et al [1] but the relative weights of the terms are different, leading to the fact that, whereas the K(l, p) equation (Eqn.(1)) can be derived from a Lagrangian, the K(m,n) equation considered in [1] donot have a Lagrangian. For the sake of comparison, it may be noted that the set of parameters (m,n) in [1] corresponds to the set (l−1, p+1) in Eqn.(1) [2]. Assuming a solution to Eqn.(1) in the form of a travelling wave u(x, t) = u(ξ), where ξ = x − Dt, Eqn.(1) reduces to the same K(l, p) equations considered by Cooper et al (Eqn.(7) in [2]) for the compactons solutions. The compacton solutions to Eqn.(1) are given by [2] u(ξ) = [ D 2 (p+ 1)(p+ 2)] 1 p cos 2 p [ pξ 2 √