Evolution theorem for a class of perturbed envelope soliton solutions. (English) Zbl 0548.35101

Laedke, E. W.; Spatschek, K. H.; Stenflo, L.

J. Math. Phys. 24, 2764-2769 (1983).

Summary: Envelope soliton solutions of a class of generalized nonlinear Schrödinger equations are investigated. If the quasiparticle number N is conserved, the evolution of solitons in the presence of perturbations can be discussed in terms of the functional behavior of \(N(\eta^ 2)\), where \(\eta^ 2\) is the nonlinear frequency shift. For \(\partial_{\eta^ 2}N>0\), the system is stable in the sense of Lyapunov, whereas, in the opposite region, instability occurs. The theorem is applied to various types of envelope solitons such as spikons, relations, and others.

MSC:

35Q99	Partial differential equations of mathematical physics and other areas of application
35J10	Schrödinger operator, Schrödinger equation
35B35	Stability in context of PDEs

Keywords:

Lyapunov stability; Envelope soliton solutions; nonlinear Schrödinger equations; quasiparticle number

References:

[1]	DOI: 10.1016/0378-4363(76)90273-4
[2]	Zakharov V. E., Zh. Eksp. Teor. Fiz. 62 pp 1745– (1972)
[3]	Zakharov V. E., Sov. Phys. JETP 35 pp 908– (1972)
[4]	DOI: 10.1098/rspa.1972.0074
[5]	DOI: 10.1103/PhysRevD.13.2739
[6]	DOI: 10.1016/0370-1573(78)90074-1
[7]	DOI: 10.1016/0167-2789(82)90019-7
[8]	Kolokolov A. A., Izv. Vyssh. Uchebn. Zaved. Radiofiz. 17 pp 1332– (1974)
[9]	DOI: 10.1063/1.861553
[10]	DOI: 10.1088/0032-1028/19/9/008
[11]	DOI: 10.1016/0375-9601(77)90432-7
[12]	Litvak A. G., Pis’ma Zh. Eksp. Teor. Fiz. 27 pp 549– (1978)
[13]	Litvak A. G., JETP Lett. 27 pp 517– (1978)
[14]	DOI: 10.1103/PhysRevA.18.1591
[15]	DOI: 10.1016/0375-9601(78)90395-X
[16]	DOI: 10.1007/BF00639526
[17]	DOI: 10.1016/0375-9601(79)90771-0
[18]	DOI: 10.1063/1.523737 · Zbl 0383.35015
[19]	Berezhiani V. I., Sov. J. Plasma Phys. 7 pp 365– (1981)
[20]	DOI: 10.1063/1.863836 · Zbl 0495.76108
[21]	DOI: 10.1063/1.525370
[22]	DOI: 10.1017/S0022377800000428