Interaction of ‘solitons’ in a collisionless plasma and the recurrence of initial states. (English) Zbl 1201.35174

Zabusky, N. J.; Kruskal, M. D.

Phys. Rev. Lett. 15, 240-243 (1965).

From the text: We have observed unusual nonlinear interactions among “solitary-wave pulses” propagating in nonlinear dispersive media. These phenomena were observed in the numerical solutions of the Korteweg–de Vries equation \[ u_t+uu_x+\delta^2u_{xxx}=0. \tag{1} \] This equation can be used to describe the one-dimensional, long-time asymptotic behavior of small, but finite amplitude: shallow-water waves, collisionless plasma magnetohydrodynamic waves, and long waves in anharmonic crystals. Furthermore, the interaction and “focusing” in space-time of the solitary-wave pulses allows us to give a phenomenolocigal description of the near recurrence to the initial state in numerical calculations for a discretized weakly-nonlinear string made by Fermi, Pasta, and Ulam (FPU).

MSC:

35Q53	KdV equations (Korteweg-de Vries equations)
35Q51	Soliton equations
82D05	Statistical mechanics of gases
76X05	Ionized gas flow in electromagnetic fields; plasmic flow

Keywords:

unusual nonlinear interactions among solitary-wave pulses; nonlinear dispersive media; numerical solutions of Korteweg–de Vries equation

References:

[1]	J. J. Stoker, in: Water Waves (1957)
[2]	N. J. Zabusky, in: Proceedings of the Conference or Mathematical Models in the Physical Sciences (1963)
[3]	M. D. Kruskal, in: Progress on the Fermi, Pasta, Ulam Problem