From the special 2 + 1 Toda lattice to the Kadomtsev-Petviashvili equation

Cao, Cungen; Geng, Xianguo; Wu, Yongtang

doi:10.1088/0305-4470/32/46/306

Cited by 109 publications

(66 citation statements)

References 16 publications

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“…Starting by spectral problem, we can can get many equation family and Hamilton system, such as, AKNS, TC, TA, BPT, Yang family [6,7]. The Lax nonlinear polarization method put forward by Cao Cewen is a effective method of structuring finite dimensional integrable Hamilton system which can nonlinearized the eigenvalue problem into finite dimensional completely integrable Hamilton system, such as, KdV, AKNS, Jaulent -Miodek, KaupNewell, clan [8][9][10][11], etc. Another important application of this method is transforming the solution of soliton equation connected with eigenvalue problem into into solving compatibility of ordinary differential equation [9][10][11].…”

Section: Integrable System Research Significance and Present Situationmentioning

confidence: 99%

“…The Lax nonlinear polarization method put forward by Cao Cewen is a effective method of structuring finite dimensional integrable Hamilton system which can nonlinearized the eigenvalue problem into finite dimensional completely integrable Hamilton system, such as, KdV, AKNS, Jaulent -Miodek, KaupNewell, clan [8][9][10][11], etc. Another important application of this method is transforming the solution of soliton equation connected with eigenvalue problem into into solving compatibility of ordinary differential equation [9][10][11]. Researching power system's complete integrability in the sense of Liouville, usually puts the system into Hamilton system, it's a good way of the theory of product to study integrability in today.…”

Section: Integrable System Research Significance and Present Situationmentioning

confidence: 99%

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The Integrable System on the Second-Order Eigenvalue Problem with the Potential and Speed

Sun¹,

Li²

2013

Appl. Math. Inf. Sci.

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Abstract:In this paper, the Bargmann system for the second-order eigenvalue problem with the energy is dependent on potential andFirst of all, some basic concepts are introduced. After that, by means of the auxiliary problem and the isospectral compatible condition, the reasonable bi-Hamilton operator K and J are defined, and the evolution equations related to the spectral problem are obtained. Using the functional gradient and Lenard recursive sequence, the Bargmann constraint is given. By the constraint relation between the potential function and the eigenvector, the associated Lax pairs are nonlineared, then the Bargmann system of the eigenvalue problem is found. According to the viewpoint of Hamilton mechanics, a reasonable coordinate system has been found. The Bargmann system is transformed into the Hamilton canonical equations in the coordinate system. Finally, based on the Liouville theorem and confocal involutive system, the integrability of the Hamilton system is proved. A new integrable system is found. Moreover, the involutive representations of the solutions for the evolution equations are generated.

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Section: Integrable System Research Significance and Present Situationmentioning

confidence: 99%

Section: Integrable System Research Significance and Present Situationmentioning

confidence: 99%

The Integrable System on the Second-Order Eigenvalue Problem with the Potential and Speed

Sun¹,

Li²

2013

Appl. Math. Inf. Sci.

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“…Cao and Geng made use of the nonlinearization technique of Lax pairs to generate algebro-geometric solutions of finite-dimensional integrable Hamiltonian systems and combined systems from lower-dimensions to higher-dimensions [16,17,18], and later, the nonlinearization technique was applied to constructing algebro-geometric solutions of a great number of soliton equations in both (1+1)-and (2+1)-dimensions (see, e.g., [19]- [27]). Gesztesy et al established an alternative approach for constructing quasi-periodic solutions to soliton hierarchies associated with 2 × 2 matrix spectral problems [3,28,29], and by this approach, quasi-periodic solutions to many continuous and discrete soliton hierarchies have been constructed within a different kind of formulation using the Riemann theta functions (see, e.g., [28,30,31]).…”

Section: Introductionmentioning

confidence: 99%

Trigonal curves and algebro-geometric solutions to soliton hierarchies I

2017

Proc. R. Soc. A.

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Using linear combinations of Lax matrices of soliton hierarchies, we introduce trigonal curves by their characteristic equations, and determine Dubrovin type equations for zeros and poles of meromorphic functions defined as ratios of the Baker-Akhiezer functions. We straighten out all flows in soliton hierarchies under the Abel-Jacobi coordinates associated with Lax pairs, and generate algebro-geometric solutions to soliton hierarchies in terms of the Riemann theta functions, through observing asymptotic behaviors of the Baker-Akhiezer functions. We analyze the four-component AKNS soliton hierarchy in such a way that it leads to a general theory of trigonal curves applicable to construction of algebro-geometric solutions of an arbitrary soliton hierarchy.

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“…With the help of the non-linearization of Lax pair and the Riemann-Jacobi inversion technique [2,3,6], the algebro-geometric solutions of the mKP equation (1.3) are obtained. Xu [16] studied the bifurcation behavior of traveling wave solutions of (1.3) and obtained some exact traveling wave solutions.…”

Section: Introductionmentioning

confidence: 99%

Smoothness property of traveling wave solutions in a modified Kadomtsev--Petviashvili equation

Zhang¹,

Li²,

Han³

2016

J. Nonlinear Sci. Appl.

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In this paper, dynamical systems theory is applied to investigate the smooth property of traveling wave solutions for a modified Kadomtsev-Petviashvili equation. The results of our study demonstrate that an abundant of smooth traveling waves arise when their corresponding orbits have intersection points with the singular straight line. In some conditions, exact parametric representations of these smooth waves in explicit or implicit forms are obtained.

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From the special 2 + 1 Toda lattice to the Kadomtsev-Petviashvili equation

Cited by 109 publications

References 16 publications

The Integrable System on the Second-Order Eigenvalue Problem with the Potential and Speed

The Integrable System on the Second-Order Eigenvalue Problem with the Potential and Speed

Trigonal curves and algebro-geometric solutions to soliton hierarchies I

Smoothness property of traveling wave solutions in a modified Kadomtsev--Petviashvili equation

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