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Painlevé analysis of the damped, driven nonlinear Schrödinger equation

Published online by Cambridge University Press:  14 November 2011

Peter A. Clarkson
Peter A. Clarkson
Affiliation:
Mathematics Department, Birmingham University, Birmingham B15 2TT, U.K.
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Synopsis

In this paper we apply the Painlevé tests to the damped, driven nonlinear Schrödinger equation

where a(x, t) and b(x, t) are analytic functions of x and t, todetermine under what conditions the equation might be completely integrable. It is shown that (0.1) can pass the Painlevé tests only if

where α0(t),α1(t) and β(t) are arbitrary, real analytic functions of time. Furthermore, it is shown that in this special case, (0.1) may be transformed into the original nonlinear Schrödinger equation, which is known to be completely integrable.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 109 , Issue 1-2 , 1988 , pp. 109 - 126
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

1Ablowitz, M. J., Kaup, D. J., Newell, A. C. and Segur, H.. The inverse scattering transformFourier analysis for nonlinear problems. Stud. Appl. Math. 53 (1974), 249–315.CrossRefGoogle Scholar
2Ablowitz, M. J., Ramani, A. and Segur, H.. Nonlinear evolution equations and ordinary differential equations of Painlevé type. Lett. Nuovo Cimento 23 (1978), 333–338.CrossRefGoogle Scholar
3Ablowitz, M.J., Ramani, A. and Segur, H.. A connection between nonlinear evolution equations and ordinary differential equations of P-type I. J. Math. Phys. 21 (1980), 715–721.CrossRefGoogle Scholar
4Ablowitz, M. J. and Segur, H.. The inverse scattering transform: semi-infinite interval. J. Math. Phys. 16 (1975), 1054–1056.CrossRefGoogle Scholar
5Ablowitz, M. J. and Segur, H.. Solitons and the Inverse Scattering Transform (Philadelphia: SIAM, 1981).CrossRefGoogle Scholar
6Balakrishnan, R.. Dynamics of a generalised classical Heisenberg chain. Phys. Lett. A 92 (1982), 243–246.CrossRefGoogle Scholar
7Balakrishnan, R.. Soliton propagation in nonuniform media. Phys. Rev. A 32 (1985), 1114–1149.CrossRefGoogle ScholarPubMed
8Balakrishnan, R.. Inverse spectral method analysis of a nonlinear Schrödinger equation with x-dependent coefficients. Phys. D 16 (1985), 403–413.CrossRefGoogle Scholar
9Balakrishnan, R.. Non-perturbative solutions of a nonlinear Schrödinger equation in a quadratic potential. Phys. D 18 (1986), 303–304.CrossRefGoogle Scholar
10Beals, R. and Coifman, R. R.. Scattering and inverse scattering for first order systems. Comm. Pure Appl. Math. 37 (1984), 39–90.CrossRefGoogle Scholar
11Calogero, F. and Degasperis, A.. Spectral transform and Solitons I (Amsterdam: North-Holland, 1982).Google Scholar
12Chen, H. H. and Liu, C. S.. Solitons in nonuniform media. Phys. Rev. Lett. 37 (1976), 693–696.CrossRefGoogle Scholar
13Chen, H. H. and Liu, C. S.. Nonlinear wave and soliton propagation in media with arbitrary inhomogenities. Phys. Fluids 21 (1978), 337–380.CrossRefGoogle Scholar
14Fokas, A. S.. Inverse scattering transform on the half-line - the nonlinear analogueof the Sine transform. Preprint, I.N.S. Report #75, Clarkson University, Potsdam, New York, U.S.A.Google Scholar
15Grimshaw, R.. Slowly varying solitary waves. II. Nonlinear Schrodinger equation. Proc. Roy. Soc. London Ser. A 368 (1979), 377–388.Google Scholar
16Gupta, M. R.. Exact inverse scattering solution of a nonlinear evolution equation in a non-uniform medium. Phys. Lett. A 72 (1979), 420–422.CrossRefGoogle Scholar
17Gupta, M. R. and Ray, J.. Extension of inverse scattering method to nonlinear evolution equations in a nonuniform medium. J. Math. Phys. 22 (1981), 2180–2183.CrossRefGoogle Scholar
18Gupta, M. R., Som, B. K. and Dasgupta, B.. Exact solution of damped nonlinear Schrödinger equation for a parabolic density profile. Phys. Lett. A 69 (1978), 172–174.CrossRefGoogle Scholar
19Hastings, S. P. and McLeod, J. B.. A boundary value problem associated with the second Painlevé transcendent and the Kortweg-de Vries equation. Arch. Rational Mech. Anal. 73 (1980), 31–51.CrossRefGoogle Scholar
20Herrera, J. J. E.. Envelope solitons in inhomogeneous media. J. Phys. A 17 (1984), 95–98.CrossRefGoogle Scholar
21Joshi, N.. Painlevé property of general variable coefficient versions of the Korteweg-de Vries and nonlinear Schrodinger equations. Phys. Lett. A 125 (1986), 456–460.CrossRefGoogle Scholar
22Karpman, V. I.. Soliton evolution in the presence of perturbation. Phys. Scripta 20 (1979), 462–478.CrossRefGoogle Scholar
23Karpman, V. I. and Maslov, E. M.. Perturbation theory for solitons and solitary waves. Soviet Phys. JETP 46 (1978), 281–291.Google Scholar
24Kaup, D. J. and Newell, A. C.. Theory of nonlinear oscillating dipolar excitations in one-dimensional condensates. Phys. Rev. B 18 (1978), 5162–5167.CrossRefGoogle Scholar
25Kaup, D. J. and Newell, A. C.. Solitons as particles, oscillators, and in slowly varyingmedia: a singular perturbation theory. Proc. Roy. Soc. London Ser. A 361 (1978), 413–446.Google Scholar
26Keener, J. P. and McLaughlin, D. W.. Solitons under perturbations. Phys. Rev. A 16 (1977), 777–790.CrossRefGoogle Scholar
27Kodama, Y. and Ablowitz, M. J.. Perturbations of solitons and solitary waves. Stud. Appl. Math. 64 (1981), 225–245.CrossRefGoogle Scholar
28Leclert, G. P., Karney, C. F. F., Bers, A. and Kaup, D. J.. Two-dimensional self-modulation of lower hybrid waves in inhomogeneous plasmas. Phys. Fluids 22 (1979), 1545–1553.CrossRefGoogle Scholar
29McLeod, J. B. and Olver, P. J.. The connection between partial differential equations soluble by inverse scattering and ordinary differential equations of Painleve type. SIAMJ. Math. Anal. 14 (1983), 488–506.CrossRefGoogle Scholar
30Newell, A. C.. Nonlinear tunelling. J. Math. Phys. 19 (1978), 1126–1133.CrossRefGoogle Scholar
31Rice, M. J., Bishop, A. R., Krumhausel, J. A. and Tullinger, S. E.. Weakly pinned Frohlichcharge-density-wave condensates: A new nonlinear, current-carrying elementary excitation. Phys. Rev. Lett. 36 (1976), 432–435.CrossRefGoogle Scholar
32Weiss, J., Tabor, M. and Carnevale, G.. The Painleve property for partial differential equations. J. Math. Phys. 24 (1983), 522–526.CrossRefGoogle Scholar
33Zakharov, V. E. and Shabat, A. B.. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Soviet Phys. IETP 34 (1972), 62–69.Google Scholar