×

zbMATH — the first resource for mathematics

On nonanalytic solitary waves formed by a nonlinear dispersion. (English) Zbl 1052.35511
Summary: We study the prototypical, genuinely nonlinear, \(K(m,n)\) equation, \(u_t\pm a(u^m)_x+(u^n)_{xxx}=0\), \(a=\text{const}\), which exhibits a number of remarkable dispersive effects. In particular, the distinguished subclass wherein \(m=n+2\) is transformed into a new, purely dispersive equation free of convection. In addition to compactons, the \(K(m,n)\) can support both kinks and solitons with an infinite slope(s), periodic waves and dark solitons with cusp(s) all being manifestations of nonlinear dispersion in action. For \(n<0\) the enhanced dispersion at the tail may generate algebraically decaying patterns.

MSC:
35Q51 Soliton equations
PDF BibTeX Cite
Full Text: DOI
References:
[1] Rosenau, P.; Hyman, J.M., Phys. rev. lett., 70, 564, (1993)
[2] Rosenau, P., Phys. rev. lett., 73, 1737, (1994)
[3] Rosenau, P., Phys. lett. A, 211, 265, (1996)
[4] Ichikawa, Y.H.; Konno, K.; Wadati, M., J. phys. soc. jpn., 50, 1799, (1981)
[5] Camassa, R.; Holm, D., Phys. rev. lett., 71, 1661, (1993)
[6] Olver, P.J.; Rosenau, P., Phys. rev. E, 53, 1900, (1996)
[7] Fuchssteiner, B., Prog. theor. phys., 65, 861, (1981)
[8] Y.A. Li and P.J. Olver, submitted.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.