Riemann–Hilbert problem to the modified Korteveg–de Vries equation: Long-time dynamics of the steplike initial data

Abstract

We consider the modified Korteveg–de Vries equation on the line. The initial data are the pure step function, i.e., q(x,0)=0 for x≥0 and q(x,0)=c for x<0, where c is an arbitrary real number. The goal of this paper is to study the asymptotic behavior of the solution of the initial-value problem as t→∞. Using the steepest descent method and the so-called g-function mechanism, we deform the original oscillatory matrix Riemann–Hilbert problem to explicitly solving model forms and show that the solution of the initial-value problem has a different asymptotic behavior in different regions of the xt-plane. In the regions x<−6c2t and x>4c2t, the main terms of asymptotics of the solution are equal to c and 0, respectively. In the region −6c2t<x<4c2t, asymptotics of the solution takes the form of a modulated elliptic wave of finite amplitude.

Laguerre polynomials and transitional asymptotics of the modified Korteweg–de Vries equation for step-like initial data

“…Theorem 1. [30] In the region −6c 2 t < x < 4c 2 t the solution of the initial value problem (1), (2), (3) takes form of a modulated elliptic wave q(x, t) = q ell (x, t) + o (1) , t → ∞, where q ell (x, t) = c 2 − d 2 (ξ) Θ (πi + itB(d(ξ)) + i∆(d(ξ))|τ (d(ξ))) Θ (itB(d(ξ)) + i∆(d(ξ))|τ (d(ξ))) , ξ = x 12t .…”

“…If we substitute q(x, t) = c (constant) in (25), then the Lax pair equations (23), (24) admit explicit solutions [30] E 0 (x, t; k) = e −(ikx+4ik 3 t)σ3 , and E c (x, t; k) = K(k)e −(ixX(k)+2it(2k 2 −c 2 )X(k))σ3 , for c = 0 and c = 0, respectively. In the formula above we have set X(k) = √ k 2 + c 2 and…”

“…We have the following Proof. Using the relations (they follow from the symmetry and determinantal properties of the corresponding Jost solutions, see [30], chapter II, p.4-5)…”

“…The main result in this section is Theorem 2. It was shown in [30] that the RH problem 1 is equivalent to the following one: Figure 4), 2. asymptotics; we have M (3) (x, t; k) → I as k → ∞ and near the points ±ic we have that every entry is bounded by O(|z ∓ ic| − 1 4 ) (respectively).…”

A Matrix Baker-Akhiezer Function Associated with the Maxwell-Bloch Equations and their Finite-Gap Solutions

“…The matrix Baker-Akhiezer function will be useful for applying to Cauchy problems with periodic (quasi-periodic) finite-gap initial data as well as for the initial-boundary value problems with such type of initial and boundary functions. The suggested RH problem will be also useful for studying the long time/large space asymptotic behavior of solutions of different initialboundary value problems to the MB equations by the way as, for example, in [6,7,8,9,10,11,12,13,14,15,16,17,18,21,22,26,36,37,38,43,44,56,57,59,60]. The focusing nonlinear Schrödinger equation and its finite-gap solutions are widely used for modeling of the so-called rogue waves.…”