Characteristic integrals and general solution of the Ferapontov-Shabat-Yamilov lattice

Dmitry K. Demskoi School of Computing and Mathematics, Charles Sturt University, NSW 2678, Australia ddemskoy@csu.edu.au

Abstract

For the finite (non-periodic) systems obtained from a lattice introduced by Ferapontov and independently by Shabat and Yamilov, we present a quadrature-free general solution and a recurrent formula for the characteristic integrals. The derivation of these formulae relies on the underlying determinantal equations. We illustrate the results using a two-component system.

1 Introduction

The lattice equation

\partial^{2}_{tx}v_{n}=\partial_{t}v_{n}\partial_{x}v_{n}\left(\frac{1}{v_{n}-v_{n-1}}-\frac{1}{v_{n+1}-v_{n}}\right),

(1)

which we refer to as Ferapontov-Shabat-Yamilov (FSY) lattice, was introduced independently in [1] and [2]. In the work of Shabat and Yamilov, equation (1) was derived as one of the examples in the list of two-dimensional generalisations of the Toda-type lattices. There is a close connection between the latter and the nonlinear Shrödinger-type (NLS) systems: the lattices serve as auto-Bäcklund transformations for the NLS-type systems. Respectively, the two-dimensional generalisations of the Toda-type lattices play a role of auto-Bäcklund transformations of Davey-Stewartson-type systems. It turns out that (1) is compatible with the well-known Ishimori equation which is a two-dimensional generalization of the Heisenberg spin model [3, 4].

In the work of Ferapontov, the lattice (1) emerged in the context of two-component hydrodynamic-type systems in Riemann invariants. It was shown that the characteristic velocities of such systems, related by the Laplace transformation, satisfy lattice (1). Ferapontov also investigated periodic reductions of the corresponding chain of the Laplace transformation and lattice (1).

In regards to the integrability of reductions of lattice equations, normally the periodic boundary conditions, i.e. $v_{n+N}=v_{n}$ , lead to solitonic equations, whereas finite (non-periodic) reductions, often referred to as cutoff constraints, yield linearisable equations. For hyperbolic lattices like (1), the finite reductions turn out to be Darboux integrable. This type of integrability is characterised by existence of complete sets of integrals along each characteristic direction.

A finite reduction of the lattice (1) corresponding to the boundary condition

v_{0}=0,\ \ v_{N}=\infty

(2)

was introduced by the author [5] in connection with the determinantal equation

\left|\begin{array}[]{cccc}u&u_{t}&\ldots&u_{t\ldots t}\\ u_{x}&u_{tx}&\ldots&u_{t\ldots tx}\\ \vdots&\vdots&\ddots&\vdots\\ u_{x\ldots x}&u_{tx\ldots x}&\ldots&u_{t\ldots tx\ldots x}\end{array}\right|=0,

(3)

where

u_{t\ldots t}=\partial_{t}^{N-1}u,\,u_{x\ldots x}=\partial_{x}^{N-1}u,\,u_{t\ldots tx\ldots x}=\partial_{x}^{N-1}\partial_{t}^{N-1}u,\,\mbox{etc.}

It was pointed out that the quantities below (see also formula (25))

v_{1}=\frac{1}{u},\,v_{2}=\frac{u_{tx}}{\left|\begin{array}[]{ll}u&u_{t}\\ u_{x}&u_{tx}\end{array}\right|},\,v_{3}=\frac{\left|\begin{array}[]{ll}u_{tx}&u_{ttx}\\ u_{txx}&u_{ttxt}\end{array}\right|}{\left|\begin{array}[]{cccc}u&u_{t}&u_{tt}\\ u_{x}&u_{tx}&u_{ttx}\\ u_{xx}&u_{txx}&u_{ttxx}\end{array}\right|},\dots

(4)

satisfy lattice (1).

On the other hand, equation (3) is known [6, 7, 8] to be closely related to the two-dimesional Toda lattice in the algebraic form

w_{0}=1,\ \ \displaystyle\partial^{2}_{tx}\ln w_{n}=\frac{w_{n+1}w_{n-1}}{w_{n}^{2}},\ \ w_{N}=0,\ \ 1<n<N.

(5)

Specifically, the relationship between the scalar equation (3) and the lattice (5) is realised if we introduce the lattice variables $w_{n}$ according to the formula

\begin{array}[]{l}w_{1}=u,\,w_{2}=\left|\begin{array}[]{ll}u&u_{t}\\ u_{x}&u_{tx}\end{array}\right|,\,w_{3}=\left|\begin{array}[]{cccc}u&u_{t}&u_{tt}\\ u_{x}&u_{tx}&u_{ttx}\\ u_{xx}&u_{txx}&u_{ttxx}\end{array}\right|,\dots\end{array}

(6)

i.e. quantities $w_{n}$ represent the principal minors of the $n\times n$ matrix corresponding to the determinant in the left-hand side of formula (3).

The boundary condition

v_{0}=a,\,v_{N}=b,

(7)

where $a,b=\mbox{const},$ is equivalent to (2) due to invariance of the lattice (1) under the Möbius transformation

\varphi_{n}=\frac{\alpha v_{n}+\beta}{\gamma v_{n}+\delta}.

(8)

Thus, by choosing the parameters such that (8) becomes

\varphi_{n}=\frac{(a-b)v_{n}}{a-b-v_{n}}+a,\ \ a\neq b,

(9)

we observe that

\varphi_{0}=a,\ \ \varphi_{N}=b,

(10)

provided that $v_{n}$ satisfies the condition (2). Furthermore, the degenerate case $b=a$ can be recovered in two steps:

1.

Apply the reduction $v_{N-1}=0$ in the system obtained from the condition (2). This implies the homogeneous boundary condition $v_{0}=0,\,v_{N-1}=0$ ;
2.

Shift the variables: $\varphi_{n}=v_{n}+a.$

The Darboux integrability of the Toda lattice (5), defined as the existence of complete sets of characteristic integrals, is well-established [9]. Consequently, the lattice (1) with the boundary condition (2), or equivalently (7), is also Darboux integrable. Furthermore, an explicit formula for the characteristic integrals was provided in [5] (see formula (10) therein), where the integrals of (1) were expressed in terms of a single variable, $u=1/v_{1}$ , and its derivatives. This formula yields characteristic integrals for all lattices derived from the scalar equation (3) by introducing lattice variables.

However, a more natural representation of the integrals should involve all lattice variables and avoid mixed derivatives. While such expressions can be derived from the aforementioned formula, the process becomes increasingly computationally expensive as the number of lattice variables grows. The related approaches in [10] and [11], which rely on the system’s linear representation (Lax pair), allow the construction of integrals as coefficients of a factorised differential operator. The drawback of this approach, however, is its labour-intensive nature, as it requires expanding the differential operator to determine its coefficients.

Thus, it would be advantageous to find a direct formula for the integrals that avoids the need to compute additional objects. In the following section, equation (1) will be derived from the determinantal equation (3) ab initio. Subsequently, recurrent formulae for the integrals and the general solutions will be calculated in the variables of the hyperbolic systems obtained by constraining lattice (1) with boundary conditions (2) and (10). In conclusion, we will discuss discrete and semi-discrete analogues of (1) that are related to the determinantal equation (3).

2 Determinantal structure of the FSY lattice

First, we introduce some notation to prepare the ground for a derivation of the lattice (1). Consider a function $w_{n}$ defined by the $n$ -th order determinant

w_{n}(u)=\mbox{det}\left(a_{ij}\right),

(11)

where

\quad a_{i+1,j+1}=\frac{\partial^{i+j}u}{\partial x^{i}\partial t^{j}},\ \ i,j=1\dots n-1.

Hence, the argument of $w_{n}$ is the entry in the top left corner of the determinant. Unless otherwise stated, it will be suppressed, i.e. $w_{n}=w_{n}(u).$ In this notation, equation (3) can be re-stated as follows

w_{N}(u)=0.

(12)

Without any further restrictions, $w_{n}$ satisfies the lattice equation

w_{0}=1,\ \ \displaystyle\partial^{2}_{tx}\ln w_{n}=\frac{w_{n+1}w_{n-1}}{w_{n}^{2}}

(13)

which is infinite in the positive direction. Formula (11) is essentially a definition of new variables $w_{n}$ which transform the scalar equation (3) into a chain of hyperbolic equations (13). Imposing the condition $w_{N}(u)=\mbox{const}$ turns (13) into a Darboux integrable system of PDEs. However, we note that (13) then implies $w_{N+1}(u)=0,$ thus, without loss of generality we may assume that equation (12) is satisfied.

Jacobi determinantal identity.

We denote the minor of the entry in the $p$ -th row and $q$ -th column of the matrix $(a_{ij})$ as $m_{pq}$ and the minor obtained from $(a_{ij})$ by removing its $p$ - and $q$ -th rows as well as $r$ - and $s$ -th columns as $m_{pqrs}$ . In such notation, the Jacobi determinant identity can be written as follows [12]

m_{pqrs}w_{n}=m_{pr}m_{qs}-m_{ps}m_{qr}.

(14)

When it is necessary to indicate the order of minors $m_{pq}$ and $m_{pqrs}$ explicitly, we write $m_{n;pq}$ and $m_{n;pqrs}$ respectively. This notation means that the minors are obtained by deleting rows and columns from the $n\times n$ matrix $(a_{ij}).$ We will refer to instances of the Jacobi identity by indicating the list of indices $(p,q,r,s)$ .

Consider the ratios of the determinants:

J_{n;p}=\frac{m_{pn}}{w_{n-1}},\ \ I_{n;p}=\frac{m_{np}}{w_{n-1}},\ \ p=1,\ldots,n

(15)

with the values for $p=n$ and $p=0$ :

J_{n;n}=I_{n;n}=1,\ \ J_{n;0}=I_{n;0}=0.

The quantities $J_{n;p}$ and $I_{n;p}$ are connected by the involution $t\leftrightarrow x$ . Broadly speaking, they represent characteristic $t$ - and $x$ -integrals, respectively. A more precise description is provided below.

Proposition.

[13] The quantities $J_{n;p}$ and $I_{n;p}$ satisfy the equations

\partial_{t}J_{n;p}=\frac{w_{n-2}w_{n}}{w_{n-1}^{2}}J_{n-1;p},

(16)

J_{n;p}=J_{n-1;p-1}-\partial_{x}J_{n-1;p}+J_{n-1;p}\partial_{x}\ln\frac{w_{n-1}}{w_{n-2}},

(17)

and

\partial_{x}I_{n;p}=\frac{w_{n-2}w_{n}}{w_{n-1}^{2}}I_{n-1;p},

(18)

I_{n;p}=I_{n-1;p-1}-\partial_{t}I_{n-1;p}+I_{n-1;p}\partial_{t}\ln\frac{w_{n-1}}{w_{n-2}}.

(19)

Now the connection between the quantities $J_{n;p}$ and $I_{n;p}$ on one hand, and the characteristic integrals of the lattice (5) on the other, is obvious. Indeed, if $w_{n}=0$ for some $n=N,$ then, as follows from (16) and (18), formula (15) gives expressions for the characteristic integrals of equation (3) as well as of the lattice (5), depending on whether $w_{n},$ present in the formulae are interpreted as determinants or the lattice variables as in (5), i.e.

\partial_{t}J_{N;p}=0,\ \ \partial_{x}I_{N;p}=0\ \ \mbox{mod}\,\,w_{N}=0.

Thus, we have $2N-2$ integrals which equals the order of equation (3). Additionally, the independence of these integrals is evident from their structure in (15). Indeed, all these integrals have the same order, $2N-3,$ yet by construction each integral depends on the unique set of variables. \IfBlankTF1.1

Example 1.

Consider the case $N=3.$ The scalar equation (3) takes the form

\left|\begin{array}[]{cccc}u&u_{t}&u_{tt}\\ u_{x}&u_{tx}&u_{ttx}\\ u_{xx}&u_{txx}&u_{ttxx}\end{array}\right|=0.

(20)

The respective lattice (5) becomes a system for variables $w_{1}$ and $w_{2}:$

w_{0}=1,\ \ \displaystyle\partial^{2}_{tx}\ln w_{1}=\frac{w_{2}}{w_{1}^{2}},\ \ \partial^{2}_{tx}\ln w_{2}=0,\ \ w_{3}=0.

(21)

Formula (15) gives the following expressions for the $t$ -integrals

J_{3;1}=\frac{\left|\begin{array}[]{ll}u_{x}&u_{tx}\\ u_{xx}&u_{txx}\end{array}\right|}{\left|\begin{array}[]{ll}u&u_{t}\\ u_{x}&u_{tx}\end{array}\right|},\ \ J_{3;2}=\frac{\left|\begin{array}[]{ll}u&u_{t}\\ u_{xx}&u_{txx}\end{array}\right|}{\left|\begin{array}[]{ll}u&u_{t}\\ u_{x}&u_{tx}\end{array}\right|}.

(22)

In order to express $J_{3;1}$ and $J_{3;2}$ in terms of $w_{1}$ and $w_{2}$ one can substitute $u=w_{1}$ and eliminate mixed derivatives using (21). However, a simpler way to calculate them is to use formula (17) which gives

\begin{array}[]{ll}J_{2;1}&\displaystyle=J_{1;0}-\partial_{x}J_{1;1}+J_{1;1}\partial_{x}\ln\frac{w_{1}}{w_{0}}\\ &=\partial_{x}\ln w_{1},\end{array}

(23)

Then the integrals are calculated as follows

\begin{array}[]{ll}J_{3;1}&\displaystyle=J_{2;0}-\partial_{x}J_{2;1}+J_{2;1}\partial_{x}\ln\frac{w_{2}}{w_{1}}\\[11.38109pt] &\displaystyle=-\partial_{x}^{2}\ln w_{1}+\left(\ln w_{1}\right)_{x}\partial_{x}\ln\frac{w_{2}}{w_{1}}\\[11.38109pt] &\displaystyle=-\frac{w_{1,xx}}{w_{1}}+\frac{w_{1,x}w_{2,x}}{w_{1}w_{2}},\\[11.38109pt] J_{3;2}&\displaystyle=J_{2;1}-\partial_{x}J_{2;2}+J_{2;2}\partial_{x}\ln\frac{w_{2}}{w_{1}}\\[5.69054pt] &\displaystyle=\partial_{x}\ln w_{1}+\partial_{x}\ln\frac{w_{2}}{w_{1}}\\[5.69054pt] &\displaystyle=\partial_{x}\ln w_{2}.\end{array}

(24)

New variables.

Let us introduce a new variable $v_{n}$ as the ratio of the determinants

v_{n}=\frac{w_{n-1}\left(u_{tx}\right)}{w_{n}},\ \ n\geq 1,

(25)

or the same $v_{n}=\partial_{u}\ln w_{n}.$ It follows from this definition that

v_{1}=\frac{1}{u}

(26)

since $w_{1}=u.$

Lemma.

Quantity $v_{n}$ satisfies the following equation

\partial_{t}v_{n}=-\frac{w_{n-1}\left(u_{x}\right)w_{n}\left(u_{t}\right)}{w_{n}^{2}}

(27)

Proof.

Identity (14) with $(p,q,r,s)=(1,n+1,1,n+1)$ can be cast into the form

w_{n-1}(u_{tx})w_{n+1}=w_{n}(u_{tx})w_{n}-w_{n}(u_{x})w_{n}(u_{t}).

(28)

Dividing it by $w_{n}w_{n+1}$ and employing (25) we obtain

v_{n+1}=v_{n}+\frac{w_{n}\left(u_{x}\right)}{w_{n}}\frac{w_{n}\left(u_{t}\right)}{w_{n+1}}.

(29)

Setting $p=1$ in $(\ref{recJ})$ we can rewrite it as follows

\partial_{t}\frac{w_{n-1}\left(u_{x}\right)}{w_{n-1}}=\frac{w_{n-2}\left(u_{x}\right)w_{n}}{w_{n-1}^{2}}.

(30)

Additionally, setting $p=1$ in $(\ref{recI2})$ and noting that $I_{n-1,0}=0$ we get the following equation

\partial_{t}\frac{w_{n-1}\left(u_{t}\right)}{w_{n}}=-\frac{w_{n-1}w_{n}\left(u_{t}\right)}{w_{n}^{2}}.

(31)

Further, we employ induction on $n$ . For $n=1$ equation (27) takes the form $\partial_{t}(1/u)=-u_{t}/u^{2}$ hence satisfied. Differentiating (29) and employing (27), (30) and (31), we get

\partial_{t}v_{n+1}=-\frac{w_{n}\left(u_{x}\right)w_{n+1}\left(u_{t}\right)}{w_{n+1}^{2}}

which is nothing but the upshifted formula (27). ∎

Corollary.

i.

Due to symmetry $t\leftrightarrow x$ in the definition of $v,$ formula (27) implies the following equation

$\partial_{x}v_{n}=-\frac{w_{n-1}\left(u_{t}\right)w_{n}\left(u_{x}\right)}{w_{n}^{2}}.$ (32)

ii.

Additionally, let us state the $x$ -counterparts of formulae (30) and (31), which will be useful later:

\partial_{x}\frac{w_{n-1}\left(u_{t}\right)}{w_{n-1}}=\frac{w_{n-2}\left(u_{t}\right)w_{n}}{w_{n-1}^{2}},\ \ \partial_{x}\frac{w_{n-1}\left(u_{x}\right)}{w_{n}}=-\frac{w_{n-1}w_{n}\left(u_{x}\right)}{w_{n}^{2}}.

(33)

iii.

Multiplying (27) and (32) and then using (29) we can eliminate $w_{n-1}(u_{t})w_{n-1}(u_{x})$ and $w_{n}(u_{t})w_{n}(u_{x})$ to obtain the formula relating lattice variables $v_{n}$ and $w_{n}$ [2]

$\frac{w_{n-1}w_{n+1}}{w_{n}^{2}}=\frac{v_{n,t}v_{n,x}}{\left(v_{n}-v_{n-1}\right)\left(v_{n+1}-v_{n}\right)}.$ (34)

The following theorem was stated without a proof in [5].

Theorem.

Quantity $v_{n}$ satisfies the lattice equation (1) with the left-boundary condition $v_{0}=0.$

Proof.

The case $n=1$ can be verified directly if we consider (26) and the fact that

v_{2}=\frac{u_{tx}}{uu_{tx}-u_{t}u_{x}}.

Further, for $n>1,$ on differentiating (27) with respect to $x$ and using (33), we obtain

\partial^{2}_{tx}v_{n}=\frac{w_{n+1}w_{n-1}}{w_{n}^{2}}\left(\frac{w_{n}(u_{t})w_{n}(u_{x})}{w_{n}w_{n+1}}-\frac{w_{n-1}\left(u_{t}\right)w_{n-1}\left(u_{x}\right)}{w_{n-1}w_{n}}\right).

(35)

Then, using formulae (29) and (34) we can eliminate variables $w_{n}$ to obtain (1). ∎

3 Formulae for integrals and general solutions of the FSY lattice

Here we discuss the aspects of Darboux integrability of the obtained systems, e.g., the existence of characteristic integrals and solutions depending on arbitrary functions. Given that both systems (1) and (5) have their origin in the scalar equation (3) they inherit its Darboux integrability and the integrals. Hence the integrals (15) will be recalculated in the lattice variables $v_{n}$ given by formula (25).

3.1 Recurrent formula for characteristic integrals

Boundary condition $(0,\infty)$ .

Firstly, consider the boundary condition (2). It is convenient to calculate the $t$ -integrals using the recurrent formula (17). To recalculate them in the lattice variables $v_{n}$ , we re-write (34) and make self-substitution:

\begin{array}[]{ll}\displaystyle\frac{w_{n+1}}{w_{n}}&\displaystyle=\frac{w_{n}}{w_{n-1}}\cdot\frac{v_{n,t}v_{n,x}}{\left(v_{n+1}-v_{n}\right)\left(v_{n}-v_{n-1}\right)}\\[11.38109pt] &\displaystyle=\frac{w_{n-1}}{w_{n-2}}\cdot\frac{v_{n,t}v_{n,x}v_{n-1,t}v_{n-1,x}}{\left(v_{n+1}-v_{n}\right)\left(v_{n}-v_{n-1}\right)^{2}\left(v_{n-1}-v_{n-2}\right)}\\[11.38109pt] &\dots\\ &\displaystyle=\frac{\displaystyle\prod_{m=1}^{n}v_{m,t}v_{m,x}}{v_{1}^{2}\left(v_{n+1}-v_{n}\right)\displaystyle\prod_{m=1}^{n-1}\left(v_{m+1}-v_{m}\right)^{2}},\ \ n\geq 1.\end{array}

(36)

Then, on differentiating and eliminating $v_{m,tx}$ by means of (1), we obtain

\begin{array}[]{ll}\Lambda_{n}&\displaystyle=\partial_{x}\ln\frac{w_{n+1}}{w_{n}}\\ &\displaystyle=-\frac{v_{1,x}}{v_{1}}+\sum_{m=1}^{n}\left(\frac{v_{m,xx}}{v_{m,x}}-\frac{v_{m+1,x}}{v_{m+1}-v_{m}}\right)+\sum_{m=1}^{n-1}\frac{v_{m,x}}{v_{m+1}-v_{m}}.\end{array}

(37)

Thus, formula (17) takes the following form

J_{n;p}=J_{n-1;p-1}-\partial_{x}J_{n-1;p}+\Lambda_{n-2}J_{n-1;p},

(38)

where $\Lambda_{n}$ is given by formula (37).

\IfBlankTF

1.2

Example 2.

Consider equation (20) in variables (25) which implies the condition

v_{0}=0,\,v_{3}=\infty.

(39)

Condition (39) turns lattice (1) into the following two-component system

\begin{array}[]{ll}v_{1,tx}&\displaystyle=v_{1,t}v_{1,x}\left(\frac{1}{v_{1}}-\frac{1}{v_{2}-v_{1}}\right),\\[5.69054pt] v_{2,tx}&\displaystyle=\frac{v_{2,t}v_{2,x}}{v_{2}-v_{1}}.\end{array}

(40)

The most straightforward way to calculate the integrals of (40) is to use formula (22), in which we must substitute $u=1/v_{1}$ and eliminate the mixed derivatives by means of (40). However, the integrals can be calculated more efficiently by means of formula (38). Recall that $J_{n,0}=0$ and $J_{n,n}=1$ for any $n$ . Then, from formulae (37) and (38) we first obtain the auxiliary expression

J_{2,1}=\Lambda_{0}=-\frac{v_{1,x}}{v_{1}}.

(41)

Further, we have

\Lambda_{1}=\frac{v_{1,xx}}{v_{1,x}}-\frac{v_{1,x}}{v_{1}}-\frac{v_{2,x}}{v_{2}-v_{1}}

(42)

hence the first integral is given by

\begin{array}[]{ll}J_{3,1}&=-\partial_{x}J_{2,1}+\Lambda_{1}J_{2,1}\\ &\displaystyle=\frac{v_{1,xx}}{v_{1}}-\frac{v_{1,x}^{2}}{v_{1}^{2}}-\frac{v_{1,x}}{v_{1}}\left(\frac{v_{1,xx}}{v_{1,x}}-\frac{v_{1,x}}{v_{1}}-\frac{v_{2,x}}{v_{2}-v_{1}}\right)\\[11.38109pt] &\displaystyle=\frac{v_{1,x}v_{2,x}}{v_{1}\left(v_{2}-v_{1}\right)}.\end{array}

(43)

Lastly, the integral $J_{3,2}$ reads

\begin{array}[]{ll}J_{3,2}&=J_{2,1}+\Lambda_{1}J_{2,2}\\[5.69054pt] &\displaystyle=\frac{v_{1,xx}}{v_{1,x}}-\frac{2v_{1,x}}{v_{1}}-\frac{v_{2,x}}{v_{2}-v_{1}}.\end{array}

(44)

Thus, the integrals $J_{3,1}$ and $J_{3,2}$ constitute a complete set of integrals for the system (40).

Boundary condition $(a,b).$

Let us quickly discuss how integrals can be re-calculated in the case of the boundary condition (7). Obviously, one can simply make the inverse transformation

v_{n}=\frac{\varphi_{n}-a}{\varphi_{n}-b},

(45)

where the non-essential constant factor $a-b$ was suppressed.

However, instead of recalculating the integrals we would rather recalculate formula (37) in variables $\varphi_{n}$ . Calculations, similar to (36) and (37), yield the formula

\begin{array}[]{ll}\Lambda_{n}&\displaystyle=-\frac{\varphi_{1,x}}{\varphi_{1}-a}-\frac{\varphi_{n,x}}{\varphi_{n}-b}+\frac{\varphi_{n+1,x}}{\varphi_{n+1}-b}\\[5.69054pt] &\displaystyle+\sum_{m=1}^{n}\left(\frac{\varphi_{m,xx}}{\varphi_{m,x}}-\frac{\varphi_{m+1,x}}{\varphi_{m+1}-\varphi_{m}}\right)+\sum_{m=1}^{n-1}\frac{\varphi_{m,x}}{\varphi_{m+1}-\varphi_{m}}.\end{array}

(46)

\IfBlankTF

1.3

Example 3.

Consider the system obtained from (40) by applying (45)

\begin{array}[]{l}\displaystyle\varphi_{1,tx}=\varphi_{1,t}\varphi_{1,x}\left(\frac{1}{\varphi_{1}-a}-\frac{1}{\varphi_{2}-\varphi_{1}}\right),\\[8.53581pt] \displaystyle\varphi_{2,tx}=\varphi_{2,t}\varphi_{2,x}\left(\frac{1}{\varphi_{2}-\varphi_{1}}-\frac{1}{b-\varphi_{2}}\right).\end{array}

(47)

Hence, we assume that

\varphi_{0}=a,\ \ \varphi_{3}=b.

(48)

Then, formula (46) yields

J_{2,1}=\Lambda_{0}=\frac{\varphi_{1,x}}{\varphi_{1}-b}-\frac{\varphi_{1,x}}{\varphi_{1}-a}.

Further, from the same formula we obtain

\Lambda_{1}=\frac{\varphi_{2,x}}{\varphi_{2}-b}-\frac{\varphi_{1,x}}{\varphi_{1}-b}-\frac{\varphi_{1,x}}{\varphi_{1}-a}+\frac{\varphi_{1,xx}}{\varphi_{1,x}}-\frac{\varphi_{2,x}}{\varphi_{2}-\varphi_{1}}.

(49)

The integrals are given by

\begin{array}[]{ll}J_{3,1}&\displaystyle=-\partial_{x}J_{2,1}+\Lambda_{1}J_{2,1}\\[5.69054pt] &\displaystyle=\frac{(a-b)\varphi_{1,x}\varphi_{2,x}}{(\varphi_{1}-a)(\varphi_{2}-\varphi_{1})(\varphi_{2}-b)}\end{array}

(50)

and

\begin{array}[]{ll}J_{3,2}&\displaystyle=J_{2,1}+\Lambda_{1}J_{2,2}\\[5.69054pt] &\displaystyle=-\frac{2\varphi_{1,x}}{\varphi_{1}-a}+\frac{\varphi_{2,x}}{\varphi_{2}-b}-\frac{\varphi_{2,x}}{\varphi_{2}-\varphi_{1}}+\frac{\varphi_{1,xx}}{\varphi_{1,x}}.\end{array}

(51)

Remark.

System (40), given in the variables

v_{1}=-a/p,\ \ v_{2}=q

was shown to be Darboux integrable in [14], in which its integrals and higher symmetries were described. Later, the point-equivalent system (47) was used as an illustrating example in several papers (see e.g. [11, 4]).

3.2 General solutions

The general solution of (3) has a particularly simple form [6], namely

u(t,x)=\sum_{s=1}^{N-1}X_{s}(x)T_{s}(t),

(52)

where $T_{s}(t)$ and $X_{s}(x)$ are arbitrary functions of their arguments. Thus, the number of arbitrary functions in it, $2N-2$ , matches the order of equation (3).

Substituting (52) into (25) we obtain the general solution of the lattice (1) with the boundary condition (2):

v_{n}=\frac{\mbox{det}\left(a_{ij}\right)}{\mbox{det}\left(b_{kl}\right)},

(53)

where the components of the determinants are given by

\begin{array}[]{ll}a_{ij}&\displaystyle=\sum_{s=1}^{N-1}X_{s}^{(i)}T_{s}^{(j)},\ \ b_{kl}=\sum_{s=1}^{N-1}X_{s}^{(k-1)}T_{s}^{(l-1)},\\ &i,j=1,\dots,n-1,\ \ k,l=1,\dots,n.\end{array}

(54)

and the derivatives are denoted as follows:

X^{(\kappa)}_{s}=\frac{d^{\kappa}X}{dx^{\kappa}},\ \ T^{(\mu)}_{s}=\frac{d^{\mu}T}{dt^{\mu}}.

(55)

Hence, formula (53) provides us with the general solution of (1) satisfying condition (2). In order to obtain the general solution satisfying the boundary condition (7) one has to apply transformation (9).

Let us consider the degenerate boundary condition

v_{0}=v_{N}=a

(56)

in more detail. Although calculation of integrals in this case is straightforward, the general solution requires a more detailed analysis.

Firstly, we consider the homogeneous condition

v_{0}=v_{N}=0.

(57)

Comparing it with (25) we see that

w_{N-1}(u_{tx})=0.

The general solution of the latter equation is given by

u=X_{0}(x)+T_{0}(t)+\sum_{s=1}^{N-2}X_{s}T_{s}.

(58)

Substituting (58) in (25) and shifting the result, we get the general solution of (1) satisfying condition (56) in the form

v_{n}=\frac{\mbox{det}\left(\alpha_{ij}\right)}{\mbox{det}\left(\beta_{kl}\right)}+a,

(59)

where

\begin{array}[]{ll}\alpha_{ij}&\displaystyle=\sum_{s=1}^{N-2}X_{s}^{(i)}T_{s}^{(j)},\\ \beta_{kl}&\displaystyle=\delta_{k1}T_{0}^{(l-1)}+\delta_{1l}X_{0}^{(k-1)}+\sum_{s=1}^{N-2}X_{s}^{(k-1)}T_{s}^{(l-1)},\\ &i,j=1,\dots,n-1,\ \ k,l=1,\dots,n.\end{array}

(60)

Here, $\delta_{kl}$ is the Kronecker delta:

\delta_{kl}=\left\{\begin{array}[]{ll}1,&k=l\\ 0,&k\neq l\end{array}\right.

\IfBlankTF

1.4

Example 4.

For system (40), formula (53) gives a general solution in the following form

\begin{array}[]{ll}v_{1}&\displaystyle=\frac{1}{T_{1}X_{1}+T_{2}X_{2}},\\ v_{2}&\displaystyle=\frac{T_{1}^{\prime}X_{1}^{\prime}+T_{2}^{\prime}X_{2}^{\prime}}{T_{1}X_{1}T_{2}^{\prime}X_{2}^{\prime}-T_{1}X_{2}T_{2}^{\prime}X_{1}^{\prime}-T_{2}X_{1}T_{1}^{\prime}X_{2}^{\prime}+T_{2}X_{2}T_{1}^{\prime}X_{1}^{\prime}}.\end{array}

(61)

Applying (9), we get a general solution of (47) in the form

\begin{array}[]{ll}\varphi_{1}&\displaystyle=a+\frac{a-b}{(a-b)(X_{1}T_{1}+X_{2}T_{2})-1},\\[8.53581pt] \varphi_{2}&\displaystyle=a+\frac{(a-b)\big{(}T_{1}^{\prime}X_{1}^{\prime}+T_{2}^{\prime}X_{2}^{\prime}\big{)}}{(a-b)\big{(}X_{2}X_{1}^{\prime}-X_{1}X_{2}^{\prime}\big{)}\big{(}T_{2}T_{1}^{\prime}-T_{1}T_{2}^{\prime}\big{)}-T_{1}^{\prime}X_{1}^{\prime}-T_{2}^{\prime}X_{2}^{\prime}}.\end{array}

(62)

Finally, the solution to the system (47) in the degenerate case $b=a$ is obtained from the formula (59)

\begin{array}[]{ll}\varphi_{1}&\displaystyle=a+\frac{1}{T_{1}X_{1}+T_{0}+X_{0}},\\ \varphi_{2}&\displaystyle=a+\frac{T_{1}^{\prime}X_{1}^{\prime}}{T_{0}T_{1}^{\prime}X_{1}^{\prime}-T_{1}T_{0}^{\prime}X_{1}^{\prime}+X_{0}T_{1}^{\prime}X_{1}^{\prime}-X_{1}T_{1}^{\prime}X_{0}^{\prime}-T_{0}^{\prime}X_{0}^{\prime}}.\end{array}

(63)

4 Conclusion

An evident extension of the results presented in this paper concerns the discrete and semi-discrete versions of the FSY lattice (1). To derive these, one can start with the discrete scalar equation (3), where derivatives are replaced by shifts of discrete variables, and introduce new lattice variables in a similar manner, specifically using formula (25). The subsequent calculations rely on the Jacobi identity (14) and largely mirror those presented in this paper.

The resulting equations should exhibit analogous properties. In particular, when the boundary condition (2) is imposed, they become Darboux integrable. Detailed aspects of integrability, such as characteristic integrals and general solutions, will be addressed in a separate publication.

Another direction, concerning finite reductions of the FSY lattice, involves finding exact solutions to integrable models with two spatial dimensions [15, 4]. We anticipate that the formulae for the integrals and general solutions provided in this paper will facilitate progress in addressing these problems.

References

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[11] Habibullin I T and Sakieva A U 2024 On integrable reductions of two-dimensional Toda-type lattices Partial Differ. Equations Appl. Math. 11 100854
[12] Hirota R 2003 Determinants and Pfaffians: how to obtain N-soliton solutions from 2-soliton solutions RIMS Kôkyûroku 1302 220–42
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Characteristic integrals and general solution of the Ferapontov-Shabat-Yamilov lattice

Abstract

1 Introduction

2 Determinantal structure of the FSY lattice

Jacobi determinantal identity.

Proposition.

Example 1.

New variables.

Lemma.

Proof.

Corollary.

Theorem.

Proof.

3 Formulae for integrals and general solutions of the FSY lattice

3.1 Recurrent formula for characteristic integrals

Boundary condition (0,∞)(0,\infty).

Example 2.

Boundary condition (a,b).(a,b).

Example 3.

Remark.

3.2 General solutions

Example 4.

4 Conclusion

References

References

Boundary condition $(0,\infty)$ .

Boundary condition $(a,b).$