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Bergé, L.; Couairon, A.

A variational method for extended nonlinear Schrödinger systems. (English) Zbl 0976.35076

Physica D 152-153, 752-762 (2001).
Summary: We develop a variational procedure for solving nonlinear Schrödinger equations in the form \[ i\partial_zu+\Delta u+ q|u|^2 u+{\mathcal F}(u)= 0, \] where \({\mathcal F}(u)\) is an arbitrary function of \(u\), being perturbative or not. This method provides a general dynamical system describing the typical length scale of localized solutions \(u\) and it includes a relation for the power lost by these solutions in dissipative systems. The complete set of dynamical equations is then applied to models describing the propagation of high-power beams in gases, which involve saturating nonlinearities, multiphoton sources and nonlinear dissipation as well. Theoretical results are confronted with numerical simulations.

Cited in 5 Documents

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35A15 Variational methods applied to PDEs
37L50 Noncompact semigroups, dispersive equations, perturbations of infinite-dimensional dissipative dynamical systems

Keywords:

nonlinear Schrödinger equations; dynamical system; propagation of high-power beams in gases; nonlinear dissipation
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References:

[1] Chiao, R. Y.; Garmire, E.; Townes, C. H., Phys. Rev. Lett., 13, 479 (1964)
[2] Kelley, P. L., Phys. Rev. Lett., 15, 1005 (1965)
[3] V.E. Zakharov, Zh. Eksp. Teor. Fiz. 62 (1972) 1745 [Sov. Phys. JETP 35 (1972) 908].; V.E. Zakharov, Zh. Eksp. Teor. Fiz. 62 (1972) 1745 [Sov. Phys. JETP 35 (1972) 908].
[4] Rasmussen, J. J.; Rypdal, K., Phys. Scripta, 33, 481 (1986) · Zbl 1063.35545
[5] V.E. Zakharov, A.B. Shabat, Zh. Eksp. Teor. Fiz. 61 (1971) 118 [Sov. Phys. JETP 34 (1972) 62].; V.E. Zakharov, A.B. Shabat, Zh. Eksp. Teor. Fiz. 61 (1971) 118 [Sov. Phys. JETP 34 (1972) 62].
[6] Anderson, D.; Bonnedal, M.; Lisak, M., Phys. Fluids, 22, 1838 (1979)
[7] Anderson, D.; Bonnedal, M., Phys. Fluids, 22, 105 (1979)
[8] Anderson, D., Phys. Rev. A, 27, 3135 (1983)
[9] Desaix, M.; Anderson, D.; Lisak, M., J. Opt. Soc. Am. B, 8, 2082 (1991)
[10] Marburger, J. H., Prog. Quant. Electron., 4, 35 (1975)
[11] Fibich, G.; Papanicolaou, G. C., Phys. Lett. A, 239, 167 (1998) · Zbl 0939.35174
[12] Fibich, G.; Papanicolaou, G. C., SIAM J. Appl. Math., 60, 183 (1999) · Zbl 1026.78013
[13] Weinstein, M. I., Commun. Math. Phys., 87, 567 (1983) · Zbl 0527.35023
[14] G.M. Fraiman, Zh. Eksp. Teor. Fiz. 88 (1985) 390 [Sov. Phys. JETP 61 (1985) 228].; G.M. Fraiman, Zh. Eksp. Teor. Fiz. 88 (1985) 390 [Sov. Phys. JETP 61 (1985) 228].
[15] Landman, M.; Papanicolaou, G. C.; Sulem, C.; Sulem, P. L., Phys. Rev. A, 38, 3837 (1988)
[16] LeMesurier, B.; Papanicolaou, G. C.; Sulem, C.; Sulem, P. L., Physica D, 32, 210 (1988) · Zbl 0694.35206
[17] Malkin, V. M., Phys. Lett. A, 151, 285 (1990)
[18] Bergé, L.; Pesme, D., Phys. Rev. E, 48, R684 (1993)
[19] Bergé, L., Phys. Rep., 303, 259 (1998)
[20] Bergé, L., Phys. Rev. E, 58, 6606 (1998)
[21] Malkin, V. M., Physica D, 64, 251 (1993) · Zbl 0772.35067
[22] Mlejnek, M.; Wright, E. M.; Moloney, J. V., Opt. Lett., 23, 382 (1998)
[23] Mlejnek, M.; Wright, E. M.; Moloney, J. V., Phys. Rev. E, 58, 4903 (1998)
[24] S.N. Vlasov, L.V. Piskunova, V.I. Talanov, Zh. Eksp. Teor. Fiz. 95 (1989) 1945 [Sov. Phys. JETP 68 (1989) 1125].; S.N. Vlasov, L.V. Piskunova, V.I. Talanov, Zh. Eksp. Teor. Fiz. 95 (1989) 1945 [Sov. Phys. JETP 68 (1989) 1125].
[25] Kosmatov, N. E.; Zakharov, V. E.; Shvets, V. F., Physica D, 52, 16 (1991) · Zbl 0735.76085
[26] Kath, W. L.; Smyth, N. F., Phys. Rev. E, 51, 1484 (1995)
[27] Esarey, E.; Sprangle, P., IEEE J. Quant. Electron., 33, 1879 (1997)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.