Painlevé-type asymptotics of an extended modified KdV equation in transition regions

Liu, Nan; Guo, Boling

doi:10.1016/j.jde.2021.01.020

Cited by 30 publications

(13 citation statements)

References 16 publications

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“…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].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Higher order Airy and Painlevé asymptotics for the mKdV hierarchy

Hua¹,

Zhang²

2021

Preprint

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In this paper, we consider Cauchy problem for the modified Korteweg-de Vries hierarchy on the real line with decaying initial data. Using the Riemann-Hilbert formulation and nonlinear steepest descent method, we derive a uniform asymptotic expansion to all orders in powers of t −1/(2n+1) with smooth coefficients of the variable (−1) n+1 x((2n + 1)t) −1/(2n+1) in the self-similarity region for the solution of n-th member of the hierarchy. It turns out that the leading asymptotics is described by a family of special solutions of the Painlevé II hierarchy, which generalize the classical Ablowitz-Segur solution for the Painelvé II equation and appear in a variety of random matrix and statistical physics models. We establish the connection formulas for this family of solutions. In the special case that the reflection coefficient vanishes at the origin, the solutions of Painlevé II hierarchy in the leading coefficient vanishes as well, the leading and subleading terms in the asymptotic expansion are instead given explicitly in terms of derivatives of the generalized Airy function.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Higher order Airy and Painlevé asymptotics for the mKdV hierarchy

Hua¹,

Zhang²

2021

Preprint

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“…The jump matrix defined in (22) involves the exponentials e ± Φ ; therefore, the sign structure of the quantity ReΦ( ) plays an important role as → ∞. In particular, in Regions I and II, it follows that there are two different real stationary points located at the points, where Φ = 0, namely, at…”

Section: Asymptotics In Regions I and Iimentioning

confidence: 99%

“…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 .…”

Section: Introductionmentioning

confidence: 99%

“…To implement thēapproach, our first step is to introduce a scale function ( ) conjugated to the solution ( , ; ) of the original RH problem 1.1. This operation is aimed to factorize the jump matrix (22) into a product of an upper/lower triangular and lower/upper triangular matrix, which is necessary for the following contour deformation. The second step is to deform the contour from ℝ to a new contour depicted in Figure 3 so that the jump matrix involves the exponential factor e − Φ on the parts of the contour where ReΦ is positive and the factor e Φ on the parts where ReΦ is negative.…”

Section: Introductionmentioning

confidence: 99%

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

2021

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Based on the nonlinear steepest descent method of Deift and Zhou for oscillatory Riemann-Hilbert problems and the Dbar approach, the long-time asymptotic behavior of solutions to the fifth-order modified KdV (Korteweg-de Vries) equation on the line is studied in the case of initial conditions that belong to some weighted Sobolev spaces. Using techniques in Fourier analysis and the idea of the -method, we give its global well-posedness in lower regularity Sobolev spaces and then obtain the asymptotic behavior in these spaces with weights.

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“…Thus the vectors |y 2i−1 and |y 2i can be chosen as Φ i and ΛΦ * i respectively. By using equation (30), one obtains (31) […”

Section: Darboux Transformationmentioning

confidence: 99%

Darboux transformation and solitonic solution to the coupled complex short pulse equation

Feng,

Ling

2021

Preprint

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The Darboux transformation (DT) for the coupled complex short pulse (CCSP) equation is constructed through the loop group method. The DT is then utilized to construct various exact solutions including bright soliton, dark-soliton, breather and rogue wave solutions to the CCSP equation. In case of vanishing boundary condition (VBC), we perform the inverse scattering analysis to understand the soliton solution better. Breather and rogue wave solutions are constructed in case of non-vanishing boundary condition (NVBC). Moreover, we conduct a modulational instability (MI) analysis based on the method of squared eigenfunctions, whose result confirms the condition for the existence of rogue wave solution.

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Painlevé-type asymptotics of an extended modified KdV equation in transition regions

Cited by 30 publications

References 16 publications

Higher order Airy and Painlevé asymptotics for the mKdV hierarchy

Higher order Airy and Painlevé asymptotics for the mKdV hierarchy

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

Darboux transformation and solitonic solution to the coupled complex short pulse equation

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