On the Fourth-Order Lattice Gel'fand-Dikii Equations

Da-jun Zhang; Guesh Yfter Tela; Song-lin Zhao

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On the Fourth-Order Lattice Gel'fand-Dikii Equations

Guesh Yfter Tela, Song-Lin Zhao, Da-Jun Zhang

Exactly Solvable and Integrable Systems · Submitted 29 Jun 2022

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The authors' main claim Utilizing the direct linearization approach, we present some equations of the extended lattice Gel'fand-Dikii type.

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The fourth-order lattice Gel'fand-Dikii equations in quadrilateral form are investigated. Utilizing the direct linearization approach, we present some equations of the extended lattice Gel'fand-Dikii type. These equations are related to a quartic discrete dispersion relation and can be viewed as higher-order members of the extended lattice Boussinesq type equations. The resulting lattice equations given here are in five-component form, and some of them are multi-dimensionally consistent by introducing extra equations. Lax integrability is discussed both by direct linearization scheme and also through multi-dimensional consistent property. Some reductions of the five-component lattice equations to the four-component forms are considered.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The fourth-order lattice Gel'fand-Dikii equations in quadrilateral form are investigated.	contribution 0.55	0.74	0.07	asserted 0.96
s1	Utilizing the direct linearization approach, we present some equations of the extended lattice Gel'fand-Dikii type.	contribution 0.91	0.95	0.20	asserted 1.00
s2	These equations are related to a quartic discrete dispersion relation and can be viewed as higher-order members of the extended lattice Boussinesq type equations.	interpretation 0.87	0.68	0.21	asserted 0.91
s3	The resulting lattice equations given here are in five-component form, and some of them are multi-dimensionally consistent by introducing extra equations.	result 0.90	0.91	0.18	asserted 0.95
s4	Lax integrability is discussed both by direct linearization scheme and also through multi-dimensional consistent property.	method 0.69	0.85	0.16	asserted 0.99
s5	Some reductions of the five-component lattice equations to the four-component forms are considered.	contribution 0.82	0.89	0.08	asserted 0.95

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Original abstract

The fourth-order lattice Gel'fand-Dikii equations in quadrilateral form are investigated. Utilizing the direct linearization approach, we present some equations of the extended lattice Gel'fand-Dikii type. These equations are related to a quartic discrete dispersion relation and can be viewed as higher-order members of the extended lattice Boussinesq type equations. The resulting lattice equations given here are in five-component form, and some of them are multi-dimensionally consistent by introducing extra equations. Lax integrability is discussed both by direct linearization scheme and also through multi-dimensional consistent property. Some reductions of the five-component lattice equations to the four-component forms are considered.

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Uses the method or tool of 1 paper

Besides, one can choose specific measures $d_j(k)$ and integration contour $Γ_j$ in [ref] to get soliton solutions, which can be operated along the lines of [ref] .

Direct Linearization of extended lattice BSQ systems · Conclusions and remarks · 0.82

Compares with 3 papers

We will see that these lattice equations (see [ref], [ref], [ref]) are the counterparts of the Hietarinta's (A-2), (B-2) and (C-3) equations of the DBSQ type [ref] .

Boussinesq-like multi-component lattice equations and multi-dimensional consistency · Lattice GD-4 type equations · 0.96
We will see that these lattice equations (see [ref], [ref], [ref]) are the counterparts of the Hietarinta's (A-2), (B-2) and (C-3) equations of the DBSQ type [ref] .

Direct Linearization of extended lattice BSQ systems · Lattice GD-4 type equations · 0.96
First of all, compared with the DBSQ case ($N=3$, see [ref] and also Appendix [ref]) which are all MDC, the GD-4 type lattice equations have something special.

Discrete Boussinesq-type equations · Conclusions and remarks · 0.97

Cites as background 16 papers

We first reviewed the DL scheme for the extended generic lattice GD type equations, which appeared in [ref] .

Direct Linearization of extended lattice BSQ systems · Conclusions and remarks · 0.84
One can refer to [ref] for the DBSQ case and [ref] for the elliptic solution case.] In an ideal multicomponent quadrilateral equation, all dependent variables should evolve under certain boundary conditions.

Elliptic solutions of Boussinesq type lattice equations and the elliptic Nth root of unity · Lattice GD-4 type equations · 0.86
In [ref] , Hietarinta obtained DBSQ type equations that are quadrilateral and multidimensionally consistent.

Boussinesq-like multi-component lattice equations and multi-dimensional consistency · Deformation of the lattice GD-4 type equations · 0.86

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