Painlevé analysis of the damped, driven nonlinear Schrödinger equation

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.

[1]  Nalini Joshi,et al.  Painlevé property of general variable-coefficient versions of the Korteweg-de Vries and non-linear Schrödinger equations , 1987 .

[2]  R. Balakrishnan.  Non-perturbative solutions of a nonlinear Schrödinger equation in a quadratic potential , 1986 .

[3]  Balakrishnan.  Soliton propagation in nonuniform media. , 1985, Physical review. A, General physics.

[4]  R. Balakrishnan.  Inverse spectral transform analysis of a nonlinear Schrödinger equation with x-dependent coefficients , 1985 .

[5]  J. Herrera.  Envelope solitons in inhomogeneous media , 1984 .

[6]  Ronald R. Coifman,et al.  Scattering and inverse scattering for first order systems , 1984 .

[7]  P. Olver,et al.  The Connection between Partial Differential Equations Soluble by Inverse Scattering and Ordinary Differential Equations of Painlevé Type , 1983 .

[8]  M. Tabor,et al.  The Painlevé property for partial differential equations , 1983 .

[9]  R. Balakrishnan.  Dynamics of a generalised classical Heisenberg chain , 1982 .

[10]  J. Ray,et al.  Extension of inverse scattering method to nonlinear evolution equation in nonuniform medium , 1981 .

[11]  Yuji Kodama,et al.  Perturbations of solitons and solitary waves , 1981 .

[12]  M. Ablowitz,et al.  A connection between nonlinear evolution equations and ordinary differential equations of P‐type. II , 1980 .

[13]  S. P. Hastings,et al.  A boundary value problem associated with the second painlevé transcendent and the Korteweg-de Vries equation , 1980 .

[14]  Roger H.J. Grimshaw,et al.  Slowly varying solitary waves. II. Nonlinear Schrödinger equation , 1979, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[15]  V. Karpman,et al.  Soliton Evolution in the Presence of Perturbation , 1979 .

[16]  M. R. Gupta,et al.  Exact inverse scattering solution of a non-linear evolution equation in a non-uniform medium , 1979 .

[17]  Charles F. F. Karney,et al.  Two-dimensional self-modulation of lower hybrid waves in inhomogeneous plasmas , 1979 .

[18]  B. Dasgupta,et al.  Exact solution of damped nonlinear Schrödinger equation for a parabolic density profile , 1978 .

[19]  M. Ablowitz,et al.  Nonlinear evolution equations and ordinary differential equations of painlevè type , 1978 .

[20]  D. J. Kaup,et al.  Solitons as particles, oscillators, and in slowly changing media: a singular perturbation theory , 1978, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[21]  H. H. Chen,et al.  Nonlinear wave and soliton propagation in media with arbitrary inhomogeneities , 1978 .

[22]  A. Newell,et al.  Theory of nonlinear oscillating dipolar excitations in one-dimensional condensates , 1978 .

[23]  James P. Keener,et al.  Solitons under perturbations , 1977 .

[24]  Chuan-Sheng Liu,et al.  Solitons in nonuniform media , 1976 .

[25]  A. Bishop,et al.  Weakly Pinned Fröhlich Charge-Density-Wave Condensates: A New, Nonlinear, Current-Carrying Elementary Excitation , 1976 .

[26]  M. Ablowitz,et al.  The inverse scattering transform: Semi‐infinite interval , 1975 .

[27]  M. Ablowitz,et al.  The Inverse scattering transform fourier analysis for nonlinear problems , 1974 .