Painlevé transcendents in the defocusing mKdV equation with non-zero boundary conditions

2023/06/12 by Wang, Zhaoyu, Xu, Taiyang, Fan, Engui
#35B40 #35C20 #35P25 #35Q15 #35Q51 #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2306.07073

Abstract

We consider the Cauchy problem for the defocusing modified Korteweg-de Vries (mKdV) equation with non-zero boundary conditions amp;q_t(x,t)-6q²(x,t)q_x(x,t)+q_xxx(x,t)=0, amp;q(x,0)=q₀(x)→ ± 1, x→±∞, which can be characterized using a Riemann-Hilbert problem through the inverse scattering transform. Using the ∂-generalization of the Deift-Zhou nonlinear steepest descent approach, combined with the double scaling limit technique, we obtain the long-time asymptotics of the solution of the Cauchy problem for the defocusing mKdV equation in the transition region |x/t+6|t^2/3< C with C>0. The asymptotics can be expressed in terms of the solution of the second Painlevé transcendent.

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