On solitons, compactons, and Lagrange maps

doi:10.1016/0375-9601(95)00933-7

Physics Letters A

Volume 211, Issue 5, 26 February 1996, Pages 265-275

https://doi.org/10.1016/0375-9601(95)00933-7 Get rights and content

Abstract

Two local conservation laws of the K(m, n) equation, u_t ± (u^m)_x + (uⁿ)_xxx = 0, are used to define two Lagrange-type transformations into mass and momentum space. These mappings help to identify new integrable cases (K(−1, −2), K(−2, −2), $K(32, −1 2)$ ), transform conventional solitary waves into compactons - solitary waves on compactum - and relate certain soliton-carrying systems with compacton-carrying systems. Integrable equations are transformed into new integrable equations and interaction of N-solitons of the, say, m-KdV (m = 3, n = 1) is thus projected into an interaction in a compact domain from which N ordered stationary compactons emerge. The interaction of traveling compactons is the image of super-imposed equilibria of the corresponding soliton equation. For m = n + 2, the potential form of the K(m, n) equation may also be cast into a conserved form and thus transformed, yielding generalized Dym and Wadati equations and two new integrable cases. It is shown that $r_{t} + (1−r^{2})^{32} (r_{xx} + r)_{x} = 0$ is integrable and supports compact kinds.

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Somewhat surprisingly, numerical simulations show that even in a nonintegrable case the compacton solutions recover their shapes after the collisions, yet the dynamics of interaction slightly differs from that of KdV solitons. In this connection note that compactons, i.e. soliton-like solutions with compact support, see [1,3,16] and references therein, exist for a number of physically relevant models and possess several interesting features making them a subject of intense research, cf. e.g. [17–22] and references therein. The paper is organized as follows.
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On solitons, compactons, and Lagrange maps

Abstract

Phys. Rev. Lett.

Phys. Rev. Lett.

J. Phys. Soc. Japan

Phys. Rev. Lett.