In this paper, we have developed Cauchy matrix approach to construct the q-difference two-dimensional Toda lattice (q-2DTL) and q-difference sine-Gordon (q-sG) equation, and explore their integrability such as Lax pair and explicit solutions. By leveraging specific dispersion relations pertaining to r and s of the Sylvester equation <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11005_2025_1990_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="134" /> </InlineMediaObject> <EquationSource Format="TEX">$KM + ML = rs^\top $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mi>M</mi> <mo>+</mo> <mi>M</mi> <mi>L</mi> <mo>=</mo> <mi>r</mi> <msup> <mi>s</mi> <mi>⊤</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, we establish the q-2DTL and derive its Lax pair. We also clarify the connection of the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11005_2025_1990_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="11" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau $</EquationSource> <EquationSource Format="MATHML"><math> <mi>τ</mi> </math></EquationSource> </InlineEquation> function of the q-2DTL with Cauchy matrix approach. Besides, explicit solutions of the q-2DTL are formulated and classified by comprehensively investigating its underlying systems of linear q-difference equations. As typical examples, the dynamical behaviors of both soliton solutions and a double-pole solution are simulated numerically. Under the assumption <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11005_2025_1990_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$K = L$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>=</mo> <mi>L</mi> </mrow> </math></EquationSource> </InlineEquation>, we demonstrate how to reduce the q-sG equation from the q-2DTL both by Cauchy matrix approach and by 2-periodic reductions. Besides, the bilinear representation for the q-sG equation is reported for the first time. Furthermore, rich solutions such as kink solutions and breathers are explicitly presented and graphically illustrated for the q-sG equation.

The q-difference 2D Toda lattice, the q-difference sine-Gordon equation and classifications of solutions

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发表日期：

2025年9月9日

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Anhui Yan,
Chunxia Li

摘要

In this paper, we have developed Cauchy matrix approach to construct the q-difference two-dimensional Toda lattice (q-2DTL) and q-difference sine-Gordon (q-sG) equation, and explore their integrability such as Lax pair and explicit solutions. By leveraging specific dispersion relations pertaining to r and s of the Sylvester equation $KM + ML = rs^\top $ $K M + M L = r s^{⊤}$ , we establish the q-2DTL and derive its Lax pair. We also clarify the connection of the $\tau $ $τ$ function of the q-2DTL with Cauchy matrix approach. Besides, explicit solutions of the q-2DTL are formulated and classified by comprehensively investigating its underlying systems of linear q-difference equations. As typical examples, the dynamical behaviors of both soliton solutions and a double-pole solution are simulated numerically. Under the assumption $K = L$ $K = L$ , we demonstrate how to reduce the q-sG equation from the q-2DTL both by Cauchy matrix approach and by 2-periodic reductions. Besides, the bilinear representation for the q-sG equation is reported for the first time. Furthermore, rich solutions such as kink solutions and breathers are explicitly presented and graphically illustrated for the q-sG equation.