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

Abstract

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 by u τ − γ u 2 u x + u x x x = 0 , − ∞ < x < ∞ , τ > 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 ) = u 0 ( > 0 ) for x ⩾ 0 and u ( x , 0 ) = − u 0 for x < 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 − u 0 2 γ 3 and connecting u = u 0 to u = − u 0 , while the solution is oscillatory in x < − γ u 0 2 τ as τ → ∞ (oscillating about u = − u 0 ), with the oscillatory envelope being of O ( τ − 1 2 ) as τ → ∞ . The asymptotic correction to the propagation speed of the travelling wave solution is given by 1 2 u 0 3 2 γ 1 τ as 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 → ∞ , being of O ( 1 τ ) . 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 + for x ⩾ 0 and u ( x , 0 ) = u − ( ≠ u + ) for x < 0 , is also given.