Consistency around a cube property of Hirota’s discrete KdV equation and the lattice sine-Gordon equation

• •

  For simplicity, we use the following shorthand notations:

  $$\overline{\rule{0.0pt}{5.0pt}\hskip 6.00006pt}:\,l\to l+1,\quad\widetilde{}\,:\,m\to m+1,\quad\widehat{}\,:\,n\to n+1.$$

  (1.10)

  For example,

  $$\overline{u}=u_{l+1,m},\quad\widetilde{u}=u_{l,m+1},\quad\widetilde{\overline{u}}=u_{l+1,m+1},\quad\overline{\overline{u}}=u_{l+2,m},\quad\widetilde{\widetilde{u}}=u_{l,m+2}$$

  (1.11)

  for $u=u_{l,m}$ and

  $$\overline{u}=u_{l+1,m,n},\quad\widetilde{u}=u_{l,m+1,n},\quad\widehat{u}=u_{l,m,n+1},\quad\widehat{\widetilde{\overline{u}}}=u_{l+1,m+1,n+1}$$

  (1.12)

  for $u=u_{l,m,n}$, and the like.

• •

  If there is a substitution notation (e.g., $l\to l+1$) in the subscript of an equation number, it means the equation with the substitution applied to all symbols in the corresponding equation. For example, (1.1)_l→l+1 implies the following equation:

  $$u_{l+2,m+1}-u_{l+1,m}=\dfrac{\beta_{m+1}-\alpha_{l+1}}{u_{l+1,m+1}}-\dfrac{\beta_{m}-\alpha_{l+2}}{u_{l+2,m}}.$$

• •

  The following terminologies are used in this paper.

  1. $(\mathrm{1})$

     A multivariate polynomial with complex coefficients that is affine linear with respect to each variable is called a multilinear polynomial. For example, the general form of a multilinear polynomial in four variables is given by

     	$\displaystyle A_{1}xyzw+A_{2}xyz+A_{3}xyw+A_{4}xzw+A_{5}yzw+A_{6}xy+A_{7}xz+A_{8}xw$
     	$\displaystyle\quad+A_{9}yz+A_{10}yw+A_{11}zw+A_{12}x+A_{13}y+A_{14}z+A_{15}w+A_{16},$		(1.13)

     where $A_{i}\in\mathbb{C}$ are parameters.

  2. $(\mathrm{2})$

     Let $Q(x,y,z,w)$ be a multilinear polynomial and $x=f(y,z,w)$ be the solution of $Q(x,y,z,w)=0$. We call $Q(x,y,z,w)$ an irreducible multilinear polynomial in four variables if the following holds:

     $$\dfrac{\partial}{\partial y}f(y,z,w)\neq 0,\quad\dfrac{\partial}{\partial z}f(y,z,w)\neq 0,\quad\dfrac{\partial}{\partial w}f(y,z,w)\neq 0.$$

     (1.14)

  3. $(\mathrm{3})$

     Consider a P$\Delta$E of the form

     $$Q(u,\overline{u},\widetilde{u},\widetilde{\overline{u}}\,)=0,$$

     (1.15)

     where $u=u_{l,m}$ and $Q$ is an irreducible multilinear polynomial. By putting the variable $u_{l,m}$ on the grid point $(l,m)$ of the integer lattice $\mathbb{Z}^{2}$, the P$\Delta$E (1.15) can be regarded as a relation on the quadrilateral given by

     $$(l,m),\quad(l+1,m),\quad(l,m+1),\quad(l+1,m+1)$$

     as shown in Figure 1.1. For this reason, a P$\Delta$E of the form (1.15) is called a quad-equation. Note that quad-equations are not necessarily autonomous but may contain parameters that evolve with $l$ or $m$.

This paper is organized as follows. In §2, using the CABC property of the dKdV equation (1.1) and the lsG equation (1.2), we show that they have the CAC property. In §3, we show how the P$\Delta$Es (1.1) and (1.2) can be extended to systems of P$\Delta$Es on the 3-dimensional integer lattice. Some concluding remarks are given in §4. In Appendix A, we recall the definitions of the CAC and CABC properties. In Appendix B, we list the Lax pairs of the P$\Delta$Es (1.1) and (1.2) constructed in this study. In Appendix C, we give the proof of Theorem 3.3.

The following system of equations is a new CABC-type BT from the P$\Delta$E (1.1) to Equation (1.6):

where $u=u_{l,m}$ and $v=v_{l,m}$. By direct calculation, we can verify that the system of equations (1.1) and (2.1) has the CABC and tetrahedron properties. Indeed, the following relations can be derived from the CABC-system (1.1) and (2.1):

As shown in [11, 15], the Lax pair (B.1) with (B.2a) of the P$\Delta$E (1.1) can be constructed from the equations (2.1b) and (2.1c) by setting

In this subsection, using the CABC-type BTs (1.5) and (1.7), we obtain a CAC-type BT from the P$\Delta$E (1.1) to the P$\Delta$E (1.2).

Eliminating $\widetilde{v}$ from the equations (1.5b) and (1.7b), we obtain

Let us assume that the following holds:

which satisfies Equation (2.4). Eliminating $v$ from Equation (1.5b) by using Equation (2.5) and $\widetilde{v}$ from Equation (1.5c) by using Equation (2.5)_m→m+1, we obtain

respectively. The equations above, together with the P$\Delta$Es (1.1) and (1.2), lead to the following theorem.

Theorem 2.1 gives the following corollary.

As shown in [3, 9, 16, 24], the Lax pair (B.1) with (B.2b) of the P$\Delta$E (1.1) can be constructed from the equations (2.7b) and (2.7c) by setting

In this subsection, using the CABC-type BTs (1.7) and (2.1), we obtain a CAC-type BT from the P$\Delta$E (1.1) to the P$\Delta$E (1.2). Since the process for demonstrating the result is the same as that for the CABC-type BTs (1.5) and (1.7) discussed in §2.2, we omit detailed arguments.

Setting

from the equations (2.12b) and (2.12c), we obtain the Lax pair (B.1) with (B.2c) of the P$\Delta$E (1.1).

In this subsection, using two CAC-systems (2.7) and (2.12), we obtain a CAC-system, which implies a CAC-type auto-BT of the P$\Delta$E (1.2).

Here, let us use the following CAC-system:

where $X=X_{l,m}$ and

instead of the CAC-system (2.7) with the variable $x_{l,m}$ and the parameters $p_{l}$, $q_{m}$ in the system (2.7) replaced by the variable $X_{l,m}$ and the parameters $P_{l}$, $Q_{m}$, respectively. Eliminating $\widetilde{u}u$ from the equations (2.12b) and (2.16b) and $\overline{u}u$ from the equations (2.12c) and (2.16c), we obtain

respectively. The equations above, together with the equations (2.12d) and (2.16d), lead to the following theorem.

Setting

from the equations (2.19b) and (2.19c), we obtain the Lax pair (B.4) of the P$\Delta$E (1.2).

In this subsection, using the three CABC-systems (1.5), (1.7) and (2.1), we construct an integrable system of P$\Delta$Es on the lattice $\mathbb{Z}^{3}$. We also show the $\tau$-function of the resulting system.

Let us denote the cubes for the three CABC-systems (1.5), (1.7) and (2.1) by the symbols $C^{(1)}_{l,m}$, $C^{(2)}_{l,m}$ and $C^{(3)}_{l,m}$, respectively. (See §A.2 for the correspondence between a CABC-system and a cube.) Overlap the cubes as in Figure 3.1 (Left), in the order $C^{(1)}_{l,m}$, $C^{(2)}_{l,m}$, $C^{(2)}_{l,m}$, $C^{(3)}_{l,m}$, so that the facing surfaces have the same equations. We here also denote the resulting rectangular constructed by the set of four stacked cubes by the symbol $C^{(1223)}_{l,m}$. Because both of the face-equations at the bottom and top of $C^{(1223)}_{l,m}$ are the dKdV equation, it can be placed repeatedly up and down as shown in Figure 3.1 (Right). Note that since the parameter $\gamma$ is not included in the dKdV equation, a different $\gamma$ can be chosen for each stack of $C^{(1223)}_{l,m}$. This repeated placement yields the following system of P$\Delta$Es on the lattice $\mathbb{Z}^{3}$:

where $u=u_{l,m,n}$, $v=v_{l,m,n}$, $x=x_{l,m,n}$ and

The $\tau$-function is one of the most important objects in the theory of integrable systems. We show the $\tau$-function of the system (3.1) in the following theorem.

In this subsection, we derive integrable systems on the lattice $\mathbb{Z}^{3}$ associated with the CAC property from the system (3.1).

As in §2.2 and §2.3, eliminating the $v$-variable from the system (3.1) we obtain the following system of P$\Delta$Es depending only on the $u$- and $x$-variables: