The inline movie below is a sequence of snapshots of the numerical solution to the partial differential equation

from t=0 to t=0.15 at 0.015 second time intervals. The feature of interest to us is the development of small scale structures which are clearly visible (see if you can find the oscillations in the amplitude of the solution). These oscillations are a manifestation of the equation's dislike for large derivatives.

The intuitive picture is that as time evolves, the "pulse" develops steep slopes, which is a completely non-linear phenomenon. These steep slopes are then "flattened" by a linear mechanism in the equation, which is sometimes called "dispersion".

Such an observable competition between nonlinear and linear mechanisms is always present in nonlinear partial differential equations, but is in fact a driving factor in this evolution problem when the coefficient in the nonlinear Schroedinger equation (1) is very very small.

The plots shown above are not for very small values of the parameter , in fact = 0.5840! Below are similar plots, with = 0.1, in which the oscillations are quite prominent.

What is the goal of our research? First, we would like provide a uniform asymptotic description of the solution to (1) which is valid for very very small, and estimate the error explicitly, so that we may use our approximation for more moderate values of (such as 0.5840).

Second, with this approximation, we would like to investigate parameter regimes in the equation, in order to MINIMIZE the size of the offending oscillations, and MAXIMIZE the time before these oscillations occur.

REASON:

The nonlinear Schroedinger equation is a model for the evolution of pulse-like signals in long-distance optical fiber communication systems, and the onset of steep gradients and their subsequent smoothing due to dispersion is a mechanism for degradation of these signals. Our purpose is to assist in minimizing these degradations, to increase the rate of transmission of information.

For details on the completed research see

Onset of oscillations in nonsoliton pulses in nonlinear dispersive fibers, by M. G. Forest and K. T-R McLaughlin,
Journal of Nonlinear Science, v. 7, 43-62 (1998)

Exact solution of the geometric optics approximation of the defocusing nonlinear Schroedinger equation, by O. Wright, M. G. Forest, and K. T-R McLaughlin
Physics Letters A, 257 (1999) 170-174.

Nonsoliton pulse evolution in normally dispersive fiber, by M. G. Forest, J. N. Kutz and K. T-R McLaughlin. J. Opt. Soc. Am. B., 16 (1999), No. 11, 1856-1862.

Thanks to David Adalsteinsson for teaching K. McLaughlin how to make the above inline movies.

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