Solitons Induced by Boundary Conditions

Abstract

Publisher Summary This chapter examines the solitons induced by boundary conditions. In water, solitary waves can be generated by three different methods, which include an initial profile evolving into one or many solitary waves, a moving ship or equivalent pressure source on the surface of water, or boundary excitation, such as a sluice opening or a wall pushing. The most exciting advances in soliton theory, such as soliton interaction and collision, and the inverse scattering transform have all come from pure initial value problems. The nonzero initial state has the effect of increasing the soliton maturing time, so that there are many solitons awaiting maturation, and only a few near the front have attained full maturity. The problem of existence and uniqueness has been addressed in detail. They showed that prescribing initial conditions and a single boundary condition with suitable smoothness and compatibility conditions results in a unique solution of the Korteweg–deVries equation in the quarterplane, with appropriate smoothness. With minimal assumptions on smoothness, the existence of a unique distribution solution is proved. The inverse scattering transform method is also summarized for pure initial value problems.