Deriving conservation laws for ABS lattice equations from Lax pairs

Zhang, Da‐jun; Cheng, Jun-wei; Sun, Yingying

doi:10.1088/1751-8113/46/26/265202

Cited by 10 publications

(19 citation statements)

References 31 publications

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“…It is evident that Zh = hZ due to (2.7) and (2.11). The conjugation transformation T → ZT Z −1 can be expressed in an explicit block form 14) where γ ij = γ i − γ j . Formula (2.14) shows that the block entry T ij of the matrix T is given by T ij = (λ − λ 0 ) γ ij T ij where T ij is the block element of T .…”

Section: Asymptotic Diagonalization Of the Discrete Operator Around Amentioning

confidence: 99%

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Asymptotic diagonalization of the discrete Lax pair around singularities and conservation laws for dynamical systems

Habibullin¹,

Poptsova²

2015

J. Phys. A: Math. Theor.

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Abstract. A method of the formal diagonalization of the discrete linear operator with a parameter is studied. In the case when the operator provides a Lax operator for a nonlinear quad system the formal diagonalization method allows one to describe effectively conservation laws and generalized symmetries for this system. Asymptotic representation of the Lax operators eigenfunctions are constructed and infinite series of conservation laws are described for the quad system connected with A (1) 3 affine Lie algebra, for the modified discrete Boussinesq system and for the discrete Tzitzeica equation. For a newly found multiquadratic discrete model conservation laws and several generalized symmetries are presented.

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Section: Asymptotic Diagonalization Of the Discrete Operator Around Amentioning

confidence: 99%

“…A method based on reducing the Lax equation to the Riccati type nonlinear system is suggested in [14,15].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic diagonalization of the discrete Lax pair around singularities and conservation laws for dynamical systems

Habibullin¹,

Poptsova²

2015

J. Phys. A: Math. Theor.

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“…In the fully discrete case, multi-dimensional consistency [1][2][3] provides a kind of strong integrability, under which and extra conditions lattice equations defined on quadrilateral were classified [3] and discrete Boussinesq(DBSQ)-type equations were found [4] as multi-component models. For these multi-dimensional consistent equations, infinitely many conservation laws were already constructed in several different ways [5][6][7][8][9][10][11][12][13][14]. However, these known conservation laws are only formal ones -the nonzero boundary condition at infinity (asymptotic behavior) of the solutions of the considered lattice equations was neglected.…”

Section: Introductionmentioning

confidence: 99%

“…With regard to discrete conservation laws, as an example, let us take a look at the second conservation law of the lpKdV equation derived from its Lax pair (cf. [12]), which is in the form (1.1) with…”

Section: Introductionmentioning

confidence: 99%

Spectrum transformation and conservation laws of lattice potential KdV equation

2016

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Many multi-dimensional consistent discrete systems have soliton solutions with nonzero backgrounds, which brings difficulty in the investigation of integrable characteristics. In this letter we derive infinitely many conserved quantities for the lattice potential Korteweg-de Vries equation. The derivation is based on the fact that the scattering data a(z) is independent of discrete space and time and the analytic property of Jost solutions of the discrete Schrödinger spectral problem. The obtained conserved densities are different from those in the known literatures. They are asymptotic to zero when |n| (or |m|) tends to infinity. To obtain these results, we reconstruct a discrete Riccati equation by using a conformal map which transforms the upper complex plane to the inside of unit circle. Series solution to the Riccati equation is constructed based on the analytic and asymptotic properties of Jost solutions.

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“…К динамическим системам, содержащим дефекты, он применялся так-же в работах [9], [10]. В недавних работах [11], [12] такой способ был адаптирован к полностью дискретным моделям. Следующий важный шаг в изучении дискретной линейной задачи был сделан А. В. Михайловым.…”

Section: Introductionunclassified