Let (L, M) be a Lax pair. A forced evolution equation (a partial differential equation) is of the form

L_t + LM - ML = f(x).

This is an asymmetric dynamical system and here f(x) is a given forcing function decaying to zero rapidly as |x| goes to infinity. Because the forcing term breaks those symmetries associated with the unforced systems, the traditional analytical methods, such as the inverse scattering method and Backlund transform, do not work any more. So far, one can only use numerical methods and asymptotic approximation to solve this type of forced evolution equations. In this talk the forced Korteweg-de Vries equation, forced cubic nonlinear Schrodinger equation and forced sine-Gordon equation are discussed. A user-friendly software has been developed, which solves these three types of forced evolution equations. The software is based upon the semi-implicit spectral method and its algorithm does not require artificial dispersion or dissipation terms, which are commonly used in algorithms. This algorithm is very accurate and efficient. Some conspicuous solution behavior of the forced systems will be shown, which does not occur in unforced systems, such as the generation and collision of uniform upstream advancing solitons in a channel flow of water over a bump. Another interesting result is that a stationary forced Korteweg-de Vries equation can have multiple solitary wave solutions. Bifurcation diagrams have been found analytically for some specific types of forcings. For this bifurcation problem, it will be demonstrated how to use our software to find out which branch of solutions are stable.

References:

1. S.S.P. Shen, Forced solitary waves and hydraulic falls in two-layer flows, J. Fluid Mech., 234, 583-612 (1992).

2. L. Gong and S.S.P. Shen, Multiple supercritical solitary wave solutions of the stationary forced Korteweg-de Vries equation and their stability, SIAM J. Appl. Math. 54, 1268-1290 (1994).

3. S.S.P. Shen, R.P. Manohar and L. Gong, Stability of the lower cusped solitary waves, Phys. Fluids A 7, 2507-2509 (1995).

4. S.S.P. Shen, Energy distribution for waves in transcritical flows over a bump, Wave Motion 23, 39-48(1996).