solitons

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1: Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate
Sidebar 22.SB1: Decay of a Soliton in a Bose–Einstein Condensate
For details see the NIST news item Decay of a dark soliton into vortex rings in a Bose–Einstein condensate. … Cornell, Watching Dark Solitons Decay into Vortex Rings in a Bose–Einstein Condensate, Phys. Rev. Lett. 86, 2926–2929 (2001)
2: Mark J. Ablowitz
for appropriate data they can be linearized by the Inverse Scattering Transform (IST) and they possess solitons as special solutions. …Some of the relationships between IST and Painlevé equations are discussed in two books: Solitons and the Inverse Scattering Transform and Solitons, Nonlinear Evolution Equations and Inverse Scattering. …
4: 9.16 Physical Applications
These first appeared in connection with the equation governing the evolution of long shallow water waves of permanent form, generally called solitons, and are predicted by the Korteweg–de Vries (KdV) equation (a third-order nonlinear partial differential equation). The KdV equation and solitons have applications in many branches of physics, including plasma physics lattice dynamics, and quantum mechanics. …
5: 23.21 Physical Applications
§23.21(ii) Nonlinear Evolution Equations
For applications to soliton solutions of the Korteweg–de Vries (KdV) equation see McKean and Moll (1999, p. 91), Deconinck and Segur (2000), and Walker (1996, §8.1). …
6: 18.38 Mathematical Applications
Integrable Systems
It has elegant structures, including $N$-soliton solutions, Lax pairs, and Bäcklund transformations. …However, by using Hirota’s technique of bilinear formalism of soliton theory, Nakamura (1996) shows that a wide class of exact solutions of the Toda equation can be expressed in terms of various special functions, and in particular classical OP’s. …
7: 22.19 Physical Applications
§22.19(iii) Nonlinear ODEs and PDEs
Such solutions include standing or stationary waves, periodic cnoidal waves, and single and multi-solitons occurring in diverse physical situations such as water waves, optical pulses, quantum fluids, and electrical impulses (Hasegawa (1989), Carr et al. (2000), Kivshar and Luther-Davies (1998), and Boyd (1998, Appendix D2.2)). …
8: Bibliography
• M. J. Ablowitz and P. A. Clarkson (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Mathematical Society Lecture Note Series, Vol. 149, Cambridge University Press, Cambridge.
• M. J. Ablowitz and H. Segur (1981) Solitons and the Inverse Scattering Transform. SIAM Studies in Applied Mathematics, Vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
• 9: Bibliography D
• B. Deconinck and H. Segur (1998) The KP equation with quasiperiodic initial data. Phys. D 123 (1-4), pp. 123–152.
• P. G. Drazin and R. S. Johnson (1993) Solitons: An Introduction. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge.
• 10: Bibliography H
• A. Hasegawa (1989) Optical Solitons in Fibers. Springer-Verlag, Berlin, Germany.