An initial-value problem for the defocusing modified Korteweg–de Vries equation

In this paper we address an initial-value problem for the defocusing modified Korteweg–de Vries (mKdV−) equation. The normalized modified Korteweg–de Vries equation considered is given byuτ−γu2ux+uxxx=0,−∞0, where x and τ   represent dimensionless distance and time respectively and γ(>0) is a constant. We consider the case when the initial data has a discontinuous step, where u(x,0)=u0(>0) for x⩾0x⩾0 and u(x,0)=−u0u(x,0)=−u0 for x<0x<0. The method of matched asymptotic coordinate expansions is used to obtain the complete large-τ asymptotic structure of the solution to this problem, which exhibits the formation of a permanent form travelling wave (kink) solution propagating in the −x   direction with speed −u02γ3 and connecting u=u0u=u0 to u=−u0u=−u0, while the solution is oscillatory in x<−γu02τ as τ→∞τ→∞ (oscillating about u=−u0u=−u0), with the oscillatory envelope being of O(τ−12) as τ→∞τ→∞. The asymptotic correction to the propagation speed of the travelling wave solution is given by 12u032γ1τ as τ→∞τ→∞, and the rate of convergence of the solution of the initial-value problem to the travelling wave solution is found to be algebraic in τ  , as τ→∞τ→∞, being of O(1τ).A brief discussion of the structure of the large-time solution to the mKdV− equation when the initial data is given by the general discontinuous step, u(x,0)=u+u(x,0)=u+ for x⩾0x⩾0 and u(x,0)=u−(≠u+) for x<0x<0, is also given.