Breather collapse for a dispersive effective equation: Asymptotics on the well-posedness boundary

Abstract

For the solution of the Cauchy problem for the equation $$ u_{tt} = u_{xx} + i2u_{ttx} + u_{ttxx} $$ with discontinuous initial data, asymptotic formulas as t → ∞ are derived, which agree well with numerical results. The stability of the numerical methods used is analyzed. Other results are presented concerning nonstandard linear equations produced by homogenizing the equations describing wave processes in periodic stratified media.