Greg Lyng, Department of Mathematics, University of Wyoming
The secondary caustic in the semiclassical limit for the focusing nonlinear Schrödinger equation
We consider the cubic focusing nonlinear Schrödinger equation in one space dimension, with fixed initial data, in the semiclassical limit when a dispersion parameter analogous to Planck’s constant tends to zero. This problem is relevant in the theory of “supercontinuum generation” in which coherent white light is produced from a monochromatic source by propagation in an optical fiber with small dispersion. This is a highly unstable problem with limiting “dynamics” valid for analytic initial data being described by an initial-value problem for a nonlinear system of elliptic PDEs. Nonetheless, the assumption of analyticity of the initial data allows for detailed asymptotics to be obtained with the help of the solution of the nonlinear Schrödinger equation via the inverse-scattering transform. The solutions display remarkable structure consisting of regions of smoothly modulated quasiperiodic oscillations separated by asymptotically sharp “caustic” curves in the space/time plane. The first “primary” caustic curve has been explained by passage to an appropriate continuum limit of a dense distribution of discrete eigenvalues of an associated linear operator. This talk will describe recent joint work with Peter Miller (University of Michigan) in which the “secondary” caustic curve is studied, and a new mechanism is found to explain it that depends essentially on the discrete nature of the spectrum and (unlike the case of the primary caustic) cannot be obtained from a naive continuum limit.
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Name: Ian Cunningham