Weak Interactions Of Solitary Waves In Generalized Nonlinear Schr(?)dinger Equations

Weak interactions of two solitary waves in the generalized nonlinear Schro¨dingerequation are studied in this article. It's first shown that these weak interactions exhibitsimilar fractal dependence on initial conditions for di?erent nonlinearities, i.e. thegraphs of exit velocity versus initial phase di?erence are fractals. In order to explainthis phenomenon, by using the Karpman-Solov'ev's direct perturbation method, a uni-versal system of dynamical equations is derived for the parameters of the interactingsolitary waves. After variable rescaling, these dynamical equations can be expressed asa two-degree-of-freedom Hamiltonian system. This system contains a single parameterwhich accounts for the di?erent forms of nonlinearity. When this parameter is positive,the dynamical equations exhibit fractal structures which match those in the originalwave equations both qualitatively and quantitatively. Thus the universal nature of frac-tal structures in the weak interactions of solitary waves is analytically established. Tounderstand the fractal regime further, using asymptotic methods along separatrix orbits,a simple second-order map is derived. This map describes the momentum changes ofthe dynamical equations and it does not have any free parameters after variable rescal-ings, thus is universal for all weak solitary-wave interactions in generalized nonlinearSchro¨dinger equations. Comparison with direct computations from the wave equationsand their reduced dynamical equations confirms that the map completely captures allmajor aspects of fractal scattering in the wave equations and their reduced dynamicalequations both qualitatively and quantitatively. By analyzing the map deeply, we cananalytically explain all important features of weak interactions of solitary waves suchas the criterion for the existence of fractal structures, scaling laws and so on.