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Nonlinear dispersion and compact structures. (English) Zbl 0953.35501
Summary:
Relaxing the distinguished ordering that underlies the derivation of soliton supporting equations leads to new equations endowed with nonlinear dispersion crucial for the formation and coexistence of compactons, solitons with a compact support, and conventional solitons. Vibrations of the anharmonic mass-spring chain lead to a new Boussinesq equation admitting compactons and compact breathers. The model equation \(u_t+[\delta u+3\gamma u^2/2+u^{1-\omega}(u^\omega u_x)_x]_x+\nu u_{txx}=0\) \((\omega,\nu,\delta,\gamma \text{const})\) admits compactons and for \(2\omega=\nu\gamma=1\) has a bi-Hamiltonian structure. The infinite sequence of commuting flows generates an integrable, compacton’s supporting variant of the Harry Dym equation.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
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[1] P. Rosenau, Phys. Rev. Lett. 70 pp 564– (1993) · Zbl 0952.35502 · doi:10.1103/PhysRevLett.70.564
[2] P. Rosenau, in: Nonlinear Coherent Structures in Physics and Biology (1994) · Zbl 0953.35501
[3] V.I. Petviashvili, Sov. Phys. Dokl. 231 pp 17– (1978)
[4] S. Kichenassamy, Siam J. Math. Anal. 23 pp 1141– (1992) · Zbl 0755.76023 · doi:10.1137/0523064
[5] P. Rosenau, Prog. Theor. Phys. 79 pp 1028– (1988) · doi:10.1143/PTP.79.1028
[6] P. Rosenau, Physica (Amsterdam) 27D pp 224– (1987)
[7] P. Rosenau, Phys. Rev. B 36 pp 5868– (1987) · doi:10.1103/PhysRevB.36.5868
[8] P. Rosenau, Phys. Rev. A 39 pp 6614– (1989) · doi:10.1103/PhysRevA.39.6614
[9] R. Camassa, Phys. Rev. Lett. 71 pp 1661– (1993) · Zbl 0972.35521 · doi:10.1103/PhysRevLett.71.1661
[10] P.J. Olver, in: Applications of Lie Groups to Differential Equations (1986) · Zbl 0588.22001 · doi:10.1007/978-1-4684-0274-2
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