Algebro-geometric solutions to the lattice potential modified Kadomtsev–Petviashvili equation

Xu, Xiaoxue; Cao, Cungen; Zhang, Dajun

doi:10.1088/1751-8121/ac8252

Cited by 4 publications

(2 citation statements)

References 60 publications

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“…Ref. 31. For the terms in 𝐵, again, by using formula (A.1), the first two terms yield −ℎ𝑎 𝑛 𝑏 𝑛 (𝐸𝑎 𝑛 )𝐸 −𝑠−1 Δ 𝑠 𝑏 𝑛 = −ℎ𝑎 𝑛 𝑏 𝑛 𝐸𝑎 𝑛 𝐸 −𝑠−2 Δ 𝑠 𝑏 𝑛 .…”

Section: Discussionmentioning

confidence: 99%

Symmetries of the DΔmKP hierarchy and their continuum limits

Liu,

Zhang,

Zhao

2023

Stud Appl Math

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In the recent paper [Stud. App. Math. 147 (2021), 752], squared eigenfunction symmetry constraint of the differential‐difference modified Kadomtsev–Petviashvili (DΔmKP) hierarchy converts the DΔmKP system to the relativistic Toda spectral problem and its hierarchy. In this paper, we introduce a new formulation of independent variables in the squared eigenfunction symmetry constraint, under which the DΔmKP system gives rise to the discrete spectral problem and a hierarchy of the differential‐difference derivative nonlinear Schrödinger equation of the Chen–Lee–Liu type. In addition, by introducing nonisospectral flows, two sets of symmetries of the DΔmKP hierarchy and their algebraic structure are obtained. We then present a unified continuum limit scheme, by which we achieve the correspondence of the mKP and the DΔmKP hierarchies and their integrable structures.

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Section: Discussionmentioning

confidence: 99%

Symmetries of the DΔmKP hierarchy and their continuum limits

Liu,

Zhang,

Zhao

2023

Stud Appl Math

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“…In recent years, the algebro-geometric scheme has been extensively studied and some significant progress has been made [24,25]. Krichever [26,27] showed that when the fast oscillations of periodic solutions are averaged or smoothed, the Whitham equation appears as a modulation equation of the Riemann surface module; they also parameterized the algebro-geometric periodic solutions and constructed an algebro-geometric n-orthogonal curve coordinate system on the plane space.…”

Section: Introductionmentioning

confidence: 99%

The Whitham Modulation Solution of the Complex Modified KdV Equation

Zeng

Liu

2023

Mathematics

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This paper primarily concerns the Whitham modulation equation of the complex modified Korteweg–de Vries (cmKdV) equation with a step-like initial value. By utilizing the Lax pair, we derive the N-genus Whitham equations via the averaging method. The Whitham equation can be integrated using the hodograph transformation. We investigate Krichever’s algebro-geometric scheme to propose the averaging method for the cmKdV integrable hierarchy and obtain the Whitham velocities of the integrable hierarchy and the hodograph transformation. The connection between the equations of the Euler–Poisson–Darboux type linear overdetermined system, which determines the solutions of the hodograph transformation, is constructed through Riemann integration, which demonstrates that the Whitham equation can be solved. Finally, a step-like initial value problem is solved and an exotic wave pattern is discovered. The results of direct numerical simulation agree well with the Whitham theory solution, which shows the validity of the theory.

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The (2+1)‐dimensional Schwarzian Korteweg–de Vries equation and its generalizations with discrete Lax matrices

Liu,

Cao,

Yang

et al. 2024

Math Methods in App Sciences

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By using the discrete Lax matrices corresponding to and in the Adler–Bobenko–Suris list of quadrilateral lattice equations, we establish solutions of the (2+1)‐dimensional Schwarzian Korteweg–de Vries (SKdV) equation and its generalizations. According to the structure of integrable Hamiltonian systems provided by one discrete Lax matrix for the lattice, we construct a novel Lax representation for the (2+1)‐dimensional SKdV equation. On the basis of the Riemann surface and elliptic variables, the Hamiltonian phase flows related to the equation are linearized by the Abel–Jacobi coordinates. Consequently, we obtain finite genus solutions to the (2+1)‐dimensional SKdV equation. From the discrete Lax matrix of the lattice, we study the generalized (2+1)‐dimensional SKdV equation with a parameter , which can be reduced to a simple case of the Krichever–Novikov equation. Finally, we show that the case of can also be solved through a modified version of the other discrete Lax matrix for the equation.

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Algebro-geometric solutions to the lattice potential modified Kadomtsev–Petviashvili equation

Cited by 4 publications

References 60 publications

Symmetries of the DΔmKP hierarchy and their continuum limits

Symmetries of the DΔmKP hierarchy and their continuum limits

The Whitham Modulation Solution of the Complex Modified KdV Equation

The (2+1)‐dimensional Schwarzian Korteweg–de Vries equation and its generalizations with discrete Lax matrices

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