Long-time asymptotic behavior for an extended modified Korteweg-de Vries equation

Abstract

We investigate an integrable extended modified Korteweg-de Vries equation on the line with the initial value belonging to the Schwartz space. By performing the nonlinear steepest descent analysis of an associated matrix Riemann-Hilbert problem, we obtain the explicit leading-order asymptotics of the solution of this initial value problem as time t goes to infinity. For a special case α = 0, we present the asymptotic formula of the solution to the extended modified Korteweg-de Vries equation in region P = {(x, t) ∈ R 2 |0 < x ≤ M t 1 5 , t ≥ 3} in terms of the solution of a fourth order Painlevé II equation.

Higher order Airy and Painlevé asymptotics for the mKdV hierarchy

“…Theorems 1.2 and 1.3 extend the results for the mKdV equation (1.1) in [9] to the mKdV hierarchy (1.3). For n = 2, the leading asymptotics in (1.13) is also known in [31].…”

“…As we will show later, the role played by the Ablowitz-Segur solution and Airy function in the mKdV equation will be replaced by their higher-order generalizations. In the literatures, we note that the Painlevé transcendents and their higher-order analogues are crucial in asymptotic analysis of many integrable nonlinear differential equations, as can be seen from their appearances in the focusing nonlinear Schrödinger equation [4,5], in critical asymptotics for Hamiltonian perturbations of hyperbolic and elliptic systems [10], in the Camassa-Holm equation [7], in the Sasa-Satsuma equation [24], in an extended mKdV equation [30,31] and in the sine-Gordon equation [32]. The higher order asymptotics in similarity region for other integrable equations can be found in [26,37].…”

Long‐time asymptotic behavior of the fifth‐order modified KdV equation in low regularity spaces

“…A few facts related to the RH problem associated with the fourth-order Painlevé II equation are collected in the Appendix, for the detailed proof one can see Ref. 23.…”

“…Numerous new significant results about asymptotics of solutions for the initial value or initial-boundary value problems to a lot of integrable systems were obtained under the assumptions that the initial or initial-boundary data belong to the Schwartz space. [17][18][19][20][21][22][23] To consider the asymptotic behavior of the solution in lower regularity spaces, we have to mention another meaningful result developed by Zhou 24 , that is, the 2 -Sobolev space bijectivity of the direct and inverse scattering of the 2 × 2 AKNS (Ablowitz-Kaup-Newell-Segur) system for the initial data 0 ( ) belonging to the weighted Solobev space , (ℝ), where ) 1 2 .…”

Soliton Molecules, Rational Positon Solution and Rogue Waves for the Extended Complex Modified KdV Equation

“…(3) are obtained in [26]. The long time asymptotic for the equation (3) with initial data or initial-boundary values are considered in [19,20]. In [17], the authors have obtained the explicit solitons and breather solutions for the equation ( 3) by the Riemann-Hilbert method.…”