Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function

Kodama, Yuji; Matsutani, Shigeki; Превиато, Эмма

doi:10.5802/aif.2772

Cited by 8 publications

(3 citation statements)

References 40 publications

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“…These properties are given as differential equations and related to the integrable system as in [B2,Mu,Pr1,Pr2,EEMOP2]. Since the additional structure of the hyperelliptic curves [EEMOP1] is closely related to Toda lattice equations and classical Poncelet's problem, the additional structures were also revealed as dynamical systems [KMP1].…”

Section: Introductionmentioning

confidence: 99%

Jacobi inversion formulae for a curve in Weierstrass normal form

Komeda¹,

Matsutani²

2018

Preprint

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We consider a pointed curve (X, P ) which is given by the Weierstrass normal form,where x is an affine coordinate on P 1 , the point ∞ on X is mapped to x = ∞, and each A j is a polynomial in x of degree ≤ js/r for a certain coprime positive integers r and s (r < s) so that its Weierstrass non-gap sequence at ∞ is a numerical semigroup. It is a natural generalization of Weierstrass' equation in the Weierstrass elliptic function theory. We investigate such a curve and show the Jacobi inversion formulae of the strata of its Jacobian using the result of Jorgenson [Jo].

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Section: Introductionmentioning

confidence: 99%

Jacobi inversion formulae for a curve in Weierstrass normal form

Komeda¹,

Matsutani²

2018

Preprint

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“…In Ref. [49], some quasi-periodic and periodic solutions in terms of hyperelliptic σ functions for arbitrary genus have been given by Toda's original approach. In Ref.…”

Section: Introductionmentioning

confidence: 99%

Dynamics of bright and dark multi-soliton solutions for two higher-order Toda lattice equations for nonlinear waves

Liu

Wen

2018

Adv Differ Equ

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Discrete N-fold Darboux transformation (DT) is used to derive new bright and dark multi-soliton solutions of two higher-order Toda lattice equations. Propagation and elastic interaction structures of such soliton solutions are shown graphically. The details of their evolutions are studied via numerical simulations. Numerical results show the accuracy of our numerical scheme and the stable evolutions of such bright and dark multi-solitons without a noise. To compare the numerical evolution results with the classical Toda lattice equation, we also investigate the dynamical behaviors of the multi-soliton solutions for Toda lattice equation via numerical simulations, and we find that the multi-soliton solutions of Toda lattice equation have better stability and are more robust against a big noise than its two corresponding higher-order equations. The same small noise has different effect on the evolutions of the multi-soliton solutions for three different equations in the same hierarchy. The possible reason is that the higher-order nonlinear terms of the higher-order equation cause the instability of the wave propagation. The discrete generalized (n, N-n)-fold DTs are constructed and used to derive some discrete rational solutions of three equations, and a few mathematical features for such rational solutions are also discussed. Results might be helpful for understanding the propagation of nonlinear waves in soliton theory.

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“…Moreover, we will note that as a characterization of sigma, the sigma function could be said to be "algebraic" due to the studies [B1,B2,B3,BLE1,BLE2,EEG,EEL,EEMOP,KMP,M,MP1,MP2,Ô1,Ô2]. In particular, in contrast to the theta functions, the Taylor series of the sigma function about the origin has coefficients which are polynomials in the parameters of the curve.…”

Section: Introductionmentioning

confidence: 99%

Relationship between the prime form and the sigma function for some cyclic (r,s) curves

Gibbons¹,

Matsutani²,

Ônishi³

2013

J. Phys. A: Math. Theor.

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In this article, we study some cyclic (r, s) curves X given byWe give an expression for the prime form E(P, Q), where (P, Q ∈ X), in terms of the sigma function for some such curves, specifically any hyperelliptic curve (r, s) = (2, 2g+1) as well as the cyclic trigonal curve (r, s) = (3, 4),where ♮ r is a certain multi-index of differentials. Here u 1 and v 1 are respectively the first components of u = w(P ) and v = w(Q) which are given by the Abel map w : X → C g , where g is the genus of X. These explicit formulae are useful in applications, for instance to the problem of constructing classes of Schwarz-Christoffel maps to slit domains.

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Quasi-periodic and periodic solutions of the Toda lattice via the hyperelliptic sigma function

Cited by 8 publications

References 40 publications

Jacobi inversion formulae for a curve in Weierstrass normal form

Jacobi inversion formulae for a curve in Weierstrass normal form

Dynamics of bright and dark multi-soliton solutions for two higher-order Toda lattice equations for nonlinear waves

Relationship between the prime form and the sigma function for some cyclic (r,s) curves

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