Positons: A New Concept in the Theory of Nonlinear Waves

Abstract

Positons can be characterized as slowly decaying oscillating solutions of the nonlinear completely integrable equations having the special property of being superreflectionless. This property means that the scattering operator associated with the auxiliary linear problem upon appropriate definition of the scattering data is an identical operator. It is well known that this situation cannot be realised for smooth initial data rapidly decreasing at infinity (exept in the trivial case of a potential equal to zero). Recall also that for the multi soliton solutions corresponding to the smooth reflectionless potentials the associated scattering matrix is diagonal but not equal to the unit matrix. The superreflectionless property leads to a fundamental difference between soliton-positon and soliton-soliton collisions: a soliton remains unchanged in a collision with a positon (see below). The positon however, while stable enough to preserve its individual structure after the collision, is more sensitive. For the positon the result of the collision with a soliton is described by two “phase shifts” which can be calculated analytically in terms of the spectral parameters of the soliton and positon. These properties have a natural extension for any number of interacting solitons and positons and the result of multi solitons-positons collision is completely described in terms of the pair interactions. First introduced for the KdV case [1], positon-like solutions have been studied in the meantime also for the modified KdV case, the sh-Gordon case and the nonlocal KdV case. For the KdV case the positon solution is singular. As a function of x it has a second order pole propagating from the right to the left. Its velocity is oscillating periodically around a constant average value fixed by the spectral parameter k of the positon. For the modified KdV equation the positon has two first order poles whose mutual distance ocillates periodically. For the sh-Gordon case the positon is also singular exept at a discrete sequence of values of time separated by the same interval. For the nonlocal KdV case however, the positon remains nonsingular almost all the time exept a discrete sequence of time-values. For some lattice models obtained by discretizing the continuum models both in space and time variables a nonsingular globaly bounded positon solution could be found [6].