University of Alberta
Edmonton, Alberta, Canada
Let $(L, M)$ be a Lax pair. The forced evolution equations are of the form: \[ L_t + LM - ML = f(x) \] where $f(x)$ is a given forcing function decaying to zero rapidly as $|x| \to \infty$. Because the forcing term breaks those symmetries associated with the unforced systems, the traditional analytical methods, such as the inverse scattering method and B\"{a}cklund 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 Schr\"{o}dinger equation and forced sine-Gordon equation are discussed. A user-friendly software has been developed, which solves these three types of forced evolution equations. I will show conspicuous solution behavior of the forced systems which does not exist in unforced systems, such as the generation of uniform upstream advancing solitons in a channel flow of water over a bump. I will also demonstrate the collision of two groups of uniform solitons in a forced Korteweg-de Vries system. 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.Shen, Forced solitary waves and hydraulic falls in two-layer flows, {\it J. Fluid Mech.} {\bf 234}, 583-612 (1992)
[2] S.S. Shen, R.P. Manohar and L. Gong, Stability of the lower cusped solitary waves, {\it Phys. Fluids} A {\bf 7}, 2507-2509 (1995).