Exact solutions for a class of variable coefficient nonlinear Schr¨odinger equations

Abstract

The nonlinear Schr¨oodinger equation (NLS) is a widely applicable model for wave packets dynamics [1]. For wave propagation in an inhomogeneous medium, e.g. a fluid with variable depth or an optical fiber with spatially dependent dispersion, variable coefficient NLS equation (VCNLS) or variable coefficient Korteweg-de Vries models [2], are relevant. Here VCNLS with real dispersion in the presence of linear and/or nonlinear gain/loss is solved exactly. A modified Hirota bilinear method which has been used earlier in the literature to treat the complex Ginzburg Landau equation is employed [3]. An additional ingredient is the usage of time- or space-dependent wave numbers [4]. One-soliton solution of such VCNLS is obtained in an analytical form. One restriction of the present algorithm is that the coefficient of the second-order dispersion must be real. A simple example of an exponentially modulated dispersion profile is worked out in detail to illustrate the principle. The competition between the linear gain and nonlinear loss, and vice versa, is investigated. The stability of the solitary waves is tested in direct simulations. They appear to be very robust objects. References: 1. C.C. Mei, The Applied Dynamics of Ocean Waves, Wiley, New York, (1983). 2. S.R. Clarke, and R.H.J. Grimshaw, Journal of Fluid Mechanics, 415, 323-345, (2000). 3. K. Nozaki and N. Bekki, Journal of the Physical Society of Japan, 53, 1581-1582, (1984). 4. C.C. Mak, K.W. Chow and K. Nakkeeran, Journal of the Physical Society of Japan, 72, 3070-3074, (2005).