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- W2005709344 abstract "Weak interactions of solitary waves in the generalized nonlinear Schrodinger equations are studied. It is first shown that these interactions exhibit similar fractal dependence on initial conditions for different nonlinearities. Then by using the Karpman-Solov'ev method, a universal system of dynamical equations is derived for the velocities, amplitudes, positions, and phases of interacting solitary waves. These dynamical equations contain a single parameter, which accounts for the different forms of nonlinearity. When this parameter is zero, these dynamical equations are integrable, and the exact analytical solutions are derived. When this parameter is nonzero, the dynamical equations exhibit fractal structures which match those in the original wave equations both qualitatively and quantitatively. Thus the universal nature of fractal structures in the weak interaction of solitary waves is analytically established. The origin of these fractal structures is also explored. It is shown that these structures bifurcate from the initial conditions where the solutions of the integrable dynamical equations develop finite-time singularities. Based on this observation, an analytical criterion for the existence and locations of fractal structures is obtained. Lastly, these analytical results are applied to the generalized nonlinear Schrodinger equations with various nonlinearities such as the saturable nonlinearity, and predictions on their weak interactions of solitary waves are made." @default.
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- W2005709344 date "2007-03-12" @default.
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- W2005709344 title "Universal fractal structures in the weak interaction of solitary waves in generalized nonlinear Schrödinger equations" @default.
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- W2005709344 doi "https://doi.org/10.1103/physreve.75.036605" @default.
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