On the generation of solitons and breathers in the modified Korteweg-de Vries equation

We consider the evolution of an initial disturbance described by the modifi ed Korteweg-de Vries equation with a positive coefficient of the cubic nonl inear term, so that it can support solitons. Our primary aim is to determin e the circumstances which can lead to the formation of solitons and/or brea thers. We use the associated scattering problem and determine the discrete spectrum, where real eigenvalues describe solitons and complex eigenvalues describe breathers. For analytical convenience we consider various piecewis e-constant initial conditions. We show how complex eigenvalues may be gener ated by bifurcation from either the real axis, or the imaginary axis; in th e former case the bifurcation occurs as the unfolding of a double real eige nvalue. A bifurcation from the real axis describes the transition of a soli ton pair with opposite polarities into a breather, while the bifurcation fr om the imaginary axis describes the generation of a breather from the conti nuous spectrum. Within the class of initial conditions we consider, a distu rbance of one polarity, either positive or negative, will only generate sol itons, and the number of solitons depends on the total mass. On the other h and, an initial disturbance with both polarities and very small mass will f avor the generation of breathers, and the number of breathers then depends on the total energy. Direct numerical simulations of the modified Korteweg- de Vries equation confirms the analytical results, and show in detail the f ormation of solitons, breathers, and quasistationary coupled soliton pairs. Being based on spectral theory, our analytical results apply to the entire hierarchy of evolution equations connected with the same eigenvalue proble m. (C) 2000 American Institute of Physics. [S1054- 1500(00)01202-7].