Algebro‐geometric constructions of the Wadati‐Konno‐Ichikawa flows and applications

Li, Zhu; Geng, Xianguo; Guan, L.

doi:10.1002/mma.3516

Cited by 13 publications

(4 citation statements)

References 50 publications

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“…It is interesting that the WKI equation (1.3) possesses rogue wave solution but possesses no modulation instability [57]. Additionally, the other aspects of the WKI equation (1.3) including the orbital stability [42], algebra-geometric constructions [26], the existence of global solution [38] and Riemann-Hilbert (RH) method also have been considered [31,56]. What's more, by introducing a hodograph transformation, the WKI equation (1.3) can be transformed into an equivalent form(called modified WKI equation) which has been studied with zero and nonzero boundary conditions by applying Darboux transformation [57].…”

Section: Introductionmentioning

confidence: 99%

Long time asymptotic for the Wadati-Konno-Ichikawa equation with finite density initial data

Li,

Tian,

Yang

2021

Preprint

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In this work, we investigate the Cauchy problem of the Wadati-Konno-Ichikawa (WKI) equation with finite density initial data. Employing the ∂-generalization of Deift-Zhou nonlinear steepest descent method, we derive the long time asymptotic behavior of the solution q(x, t) in spacetime soliton region. Based on the resulting asymptotic behavior, the asymptotic approximation of the WKI equation is characterized with the soliton term confirmed by N (I)-soliton on discrete spectrum and the t − 1 2 leading order term on continuous spectrum with residual error up to O(t − 3 4 ). Our results also confirm the soliton resolution conjecture for the WKI equation. Contents 1. Introduction 2. The spectral analysis of WKI equation 2.1. The singularity at k = 0 2.2. The singularity at k = ∞ 2.3. The scattering matrix 2.4. The connection between µ ± (x, t; z) and µ 0 ± (x, t; z) 3. The Riemann-Hilbert problem for WKI equation 4. Interpolation and conjugation 5. Opening ∂-lenses and mixed ∂-RH problem 6. Decomposition of the mixed ∂-RH problem 7. Asymptotic N -soliton solution 7.1. The error function Ẽ(z) between M sol and M sol I 8. Local solvable model near phase point z = z 0 9. The small norm RHP for error function E(z) 10. Analysis on pure ∂-Problem 11.

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Section: Introductionmentioning

confidence: 99%

Long time asymptotic for the Wadati-Konno-Ichikawa equation with finite density initial data

Li,

Tian,

Yang

2021

Preprint

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“…Exact stationary solution and the orbital stability for stationary solution of the WKI equation were studied in [22,23]. The existence of global solution for the WKI equation with small initial data was studied in [24] and the algebro-geometric construction of WKI flows were studied in [25]. In [26], the Darboux transformation was used to investigate the Wadati-Konno-Ichikawa system and the breathe and rogue wave solution were obtained.…”

Section: Introductionmentioning

confidence: 99%

Long-time asymptotics for the Wadati-Konno-Ichikawa equation with the Schwartz initial data

Wu,

Tian

2021

Preprint

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In this work, we investigate the long-time asymptotic behavior of the Wadati-Konno-Ichikawa equation with initial data belonging to Schwartz space at infinity by using the nonlinear steepest descent method of Deift and Zhou for the oscillatory Riemann-Hilbert problem. Based on the initial value condition, the original Riemann-Hilbert problem is constructed to express the solution of the Wadati-Konno-Ichikawa equation. Through a series of deformations, the original RH problem is transformed into a model RH problem, from which the long-time asymptotic solution of the equation is obtained explicitly.

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“…[18] Some related studies on the WKI system have been carried out, such as integrable extensions [19][20][21] and explicit solutions via different approaches. [22][23][24][25][26][27][28] In this article, we propose an integrable hierarchy of nonlinear evolution equations, where the first nontrivial member in the positive flows is a generalization of the WKI equation and that in the negative flows is a generalized Fokas-Lenells (FL) equation. Then we derive the infinite conservation laws of these two equations respectively.…”

Section: Introductionmentioning

confidence: 99%

Two integrable generalizations of WKI and FL equations: Positive and negative flows, and conservation laws*

Geng¹,

Guo²,

Zhai³

2020

Chinese Phys. B

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With the aid of Lenard recursion equations, an integrable hierarchy of nonlinear evolution equations associated with a 2 × 2 matrix spectral problem is proposed, in which the first nontrivial member in the positive flows can be reduced to a new generalization of the Wadati–Konno–Ichikawa (WKI) equation. Further, a new generalization of the Fokas–Lenells (FL) equation is derived from the negative flows. Resorting to these two Lax pairs and Riccati-type equations, the infinite conservation laws of these two corresponding equations are obtained.

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Algebro‐geometric constructions of the Wadati‐Konno‐Ichikawa flows and applications

Cited by 13 publications

References 50 publications

Long time asymptotic for the Wadati-Konno-Ichikawa equation with finite density initial data

Long time asymptotic for the Wadati-Konno-Ichikawa equation with finite density initial data

Long-time asymptotics for the Wadati-Konno-Ichikawa equation with the Schwartz initial data

Two integrable generalizations of WKI and FL equations: Positive and negative flows, and conservation laws*

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