Formation of shocklike modified Korteweg–de Vries solitons: Application to double layers

Abstract

In some commonly occurring situations, such as a stratified fluid with two layers or a plasma with a two-temperature electron component, the long waves are adequately described by a Korteweg–de Vries (KdV) equation having positive or negative solitary wave solutions, depending upon the sign of the quadratic term. For critical values of the physical parameters, the quadratic nonlinearity vanishes and the KdV equation is replaced by a modified KdV (or mKdV) equation. This paper concerns the mKdV equation having a cubic nonlinearity with a negative coefficient. The initial value problem with different asymptotic levels, u1 far to the left and u2 far to the right, is investigated both analytically and numerically, for this mKdV− equation. The necessary and sufficient condition u1u2<0 is demonstrated for a shocklike solution to form, also named double layer (dl) in plasma physics.