Minus-one functionals and the two-dimensional Toda lattice

To begin, consider the sine-Gordon equation

$$U_{xt}=\sin(U)\ ,\eqno (2.1)$$

which is equivalent to the equation

$$VV_{xt}-V_xV_t=\textstyle{1\over 2}(V^3-V)\ ,\eqno (2.2)$$

where

$$V=e^{iu}\ .\eqno (2.3)$$

Equation (2.2) has the Painlevé property [1] and a Bäcklund transformation [5]

$$V=-4{\partial^2\over\partial x\partial t}\ln\phi+V_2\ ,\eqno (2.4)$$

where

$$V_2=\phi_{xt}^2/\phi_x\phi_t\ ,\eqno (2.5)$$

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ... \Omega_2&=\{\phi;x\}+2W_{xx}/W=\beta\ ,\cr\crcr}}\,\eqno (2.6)$$

$$\alpha\beta=\textstyle{1\over 4}\ .$$

On the other hand, the Bullough-Dodd equation [13]-[15]

$$U_{xt}=ae^u-be^{-2u}\eqno (2.7)$$

is equivalent to the equation

$$VV_{xt}-V_xV_t=-aV+bV^4\ ,\eqno (2.8)$$

where $V=e^{-u}$. Equation (2.8) has the Painlevé property with singularities of the form

$$V=\phi^{-1}\sum_{j=0}^{\infty} V_j\phi^j\ ,\eqno (2.9)$$

and resonances at $j=-1,2$. The Bäcklund transformation for Eq. (2.8)

$$V=(V_0/\phi)+V\eqno (2.10)$$

obtains, with $b=1$,

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...2}{\phi_{xt}\over\phi_x \phi_t} V_0\ ,\cr\crcr}}\,\eqno (2.11)$$

and the following overdetermined system of equations for $\phi$:

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...)^{1/2} \ ,\cr &\Omega_1\Omega_2=0\ ,\cr\crcr}}\,\eqno (2.12)$$

where $\Omega_1$ and $\Omega_2$ are defined by (2.6). From the identity

$$\phi_x{\partial\over\partial x}\Omega_1=\phi_t{\partial\over\partial x}\Omega_2\ ,\eqno (2.13)$$

Eqs. (2.12) have only the trivial solution and, as a consequence, the Bäcklund transformation (2.10) does not exist. This corresponds to the general result that Eq. (2.7) is known not to have a Bäcklund transformation [14], [16], although it does have a Lax pair [14,15] and is completely integrable.

To proceed further we note the following direct formulation of Eq. (2.2) (sine-Gordon equation) in terms of the Schwarzian derivative. With

$$V=\phi_x\ ,\eqno (2.14)$$

we find that

$${\partial\over\partial t}\{\phi;x\}=-{\partial\over\partial x}\left({1\over\phi_x}\right)\ .\eqno (2.15)$$

From the symmetry $V\rightarrow 1/V$ of (2.2) we also have

$$V=1/\psi_x=\phi_x\ ,\eqno (2.16)$$

$${\partial\over\partial t}\{\psi;x\}=-{\partial\over\partial x}\left({1\over\psi_x}\right)\ .\eqno (2.17)$$

The symmetry (2.16) is identical to (1.9) for the Schwarzian formulation (1.5) of the KdV sequence. With reference to the KdV sequence (1.1) Eq. (2.15 ) is identically

$$U_t+{\partial\over\partial x}b^{-1}(U)=0\ ,\eqno (2.18)$$

where

$$U=\{\phi;x\}\ ,\qquad b^{-1}=1/\phi_x\ .\eqno (2.19)$$

From Eqs. (1.11)-(1.13), the minus-one functional $b^{-1}(U)$, with $U=\{\phi;x\}$ satisfies the condition

$$\bigl(D-(\phi_{xx}/\phi_x)\bigr)D\bigl(+(\phi_{xx}/\phi_x)\bigr) b^{-1}(U)=0\ ,\eqno (2.20)$$

or

$$b^{-1}={a\over\phi_x}+b{\phi\over\phi_x}+c{\phi^2\over\phi_x}\ ,\eqno (2.21)$$

which obtains Eqs. (2.18) and (2.19) with

$$a=l\ ,\qquad b=c=0\ .$$

In other words, the sine-Gordon equation is a specialization of the minus-one KdV equation. A comparison of the respective formulations (2.5), (2.6) and (2.14), (2.15), where the variable $\phi$ is not identified as the same in each, yields

$$\left({V_{2x}\over V_2}+{V_x\over V}\right)=2\sigma\left({V^{... ...\over V_2^{1/2}}-{V_2^{1/2}\over V^{1/2}}\right)\ ,\eqno (2.22)$$

where, in Eq. (2.6),

$$\beta=-2\sigma^2\ .$$

Equation (2.22) is the classical BT for the sine-Gordon equation [5].

Now, for the Bullough-Dodd equation (2.7), we let

$$e^u=\phi_x\ ,\eqno (2.23)$$

and find the equation

$${\partial\over\partial t}\{\phi;x\}=-{3\over 2}b{\partial\over\partial x}\left({1\over\phi_x^2}\right)\ .\eqno (2.24)$$

The substitution

$$e^{-2u}=\psi_x\ ,\eqno (2.25)$$

gives us

$${\partial\over\partial t}\{\psi;x\}=-6a{\partial\over\partial x}\psi_x^{-1/2}\ .\eqno (2.26)$$

With reference to the CDG sequence (1.16), (1.17), we have, for Eqs. (2.24) and (2.26), respectively,

$$A_t+\theta_2 H_{-2}(A)=0\ ,\eqno (2.27)$$

$$U_t+\theta_1 G_{-2}(U)=0\ , (2.28)$$

where

$$A=\{\phi;x\}\ ,\qquad U=\{\psi;x\}\ .\eqno (2.29)$$

From (1.18)-(1.23) with

$$V=\phi_{xx}/\phi_x\ ,\qquad W=\psi_{xx}/\psi_x\ ,\eqno (2.30)$$

$$\theta_2 H_{-1}(A)={\partial\over\partial x}\left\{{1\over\ph... ...i^2+d^{\prime}\phi^3+e^{\prime}\phi^4)\right\}\ , \eqno (2.31)$$

$$\theta_1 G_{-2}(U)={\partial\over\partial x}\bigl\{\psi_x^{-1/2}(a^{\prime}+b^{\prime}\psi) \bigr\}\ ,\eqno (2.32)$$

where $(a^{\prime},b^{\prime},c^{\prime},d^{\prime},e^{\prime})$ are numerical constants. Thus, Eqs. (2.24) and (2.26) are specializations of the minus (two) CDG equations. From equations (1.25) and (1.26), we conclude that

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...{\psi_{xxx}^2 \over\psi_x^2}\right)\ ,\cr\crcr}}\,\eqno (2.33)$$

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...i;x\}\ ,\cr U_2&=-2(\phi_{xxx}/\phi_x)\cr\crcr}}\,\eqno (2.34)$$

are solutions of Eqs. (2.27) and (2.28), respectively, and Eqs. (2.24) and (2.26) are connected by the transformation

$$\psi_x=\phi_x^{-2}\ .\eqno (2.35)$$

However, without the invariance under the Möbius group

$$phi=(a\phi^{\prime}+b)/(c\phi^{\prime}+d)\eqno (2.36)$$

for Eqs. (2.24) and (2.26), the Bäcklund transformation (1.24) does not exist for Eqs. (2.27) and (2.28).

From the results of [7] the CDG sequence is a consistent reduction of the Boussinesq sequence. There, the modified Boussinesq sequence is found to be

$$\left(\matrix{\theta\cr Z\cr}\right)_t=L^n\Omega_2\left( \ma... ...kip 5pt} -S-\textstyle{3\over 2}Z^2\cr} \right)\ ,\eqno (2.37)$$

where

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...0&\textstyle{1\over 3}D\cr} \right)\ ,\cr\crcr}}\,\eqno (2.38)$$

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...gn{\vskip 5pt} D^{-1}&0\cr}\right)\ ,\cr\crcr}}\,\eqno (2.39)$$

$$B={1\over 3}\left(\matrix{D-\theta&3(D-Z)\cr \noalign{\vskip ... ...eta_x-\textstyle{1\over 2}\theta^2)\cr}\right) \ ,\eqno (2.40)$$

and $B^{\ast}$ is the adjoint to $B$. Letting

$$\theta=\phi_{xx}/\phi_x\ ,\qquad Z=\beta_{xx}/\beta_x\ ,\eqno (2.41)$$

we find that

$$\left(\matrix{\theta\cr Z\cr}\right)_t=L^{-2}\Omega_2\left( ... ...\cr} \right)=L^{-1}\left(\matrix{0\cr 0\cr}\right)\eqno (2.42)$$

is the equation

$${\partial\over\partial t}\left(\matrix{\phi_{xx}/\phi_x\cr \n... ...^{-3/2}+b\phi_x^{-1/2}\beta_x^{3/2}\cr}\right)\ . \eqno (2.43)$$

Let

$$c=2\ ,\qquad b=1\ ,\qquad a=-1\ ,\eqno (2.44)$$

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...qquad\phi_{1x}\phi_{2x}\phi_{3x}=1\ ,\cr\crcr}}\, \eqno (2.45)$$

and $\sum_1^3 W_i=0$. This gives us, from Eq. (2.43), the three component Toda equation

$${\partial\over\partial t}\widehat{W}=\left(\matrix{\phi_{1x}/... ...hi_{3x}/\phi_{2x}-\phi_{1x}/\phi_{3x}\cr}\right)\ .\eqno (2.46)$$

With

$$\phi_{ix}=e^{\theta i}\ ,\eqno (2.47)$$

this is

$$\theta_{ixt}=e^{\theta i-\theta i-1}-e^{\theta i+1-\theta i}\ ,\eqno (2.48)$$

for $i=1,2,3,\ldots(\hbox{mod}\ N)$, where

$$\sum^N\theta_{ix}=0\ ,\qquad N=3\ .$$

Thus, the three component Toda lattice is the minus-one equation of the Boussinesq sequence. A Bäcklund transformation is known to exist for Eq. (2.48) [14]. However, whether a Bäcklund transformation can be constructed for one of the equivalent forms of Eq. (2.48) by the Painlevé method is nearly a moot point. For instance, in Eq. (2.43), let

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...heta=\ln(\beta_x^{3/2}\phi_x^{-1/2})\ ,\cr\crcr}}\,\eqno (2.49)$$

and find the equation

$$U_{xt}=2ae^{-u}+be^{\theta}+(c/2)e^{u-\theta}\ ,\eqno (2.50)$$

$$\theta_{xt}=2ae^{-u}+2be^{\theta}-(c/2)e^{u-\theta}\ .\eqno (2.51)$$

With

$$W=e^u\ ,\qquad V=e^{-\theta}\ ,$$

we find the system

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ..._{xt}-V_xV_t)=-aV^2-2bVW+(c/2)V^3W^2\ .\cr\crcr}}\,\eqno (2.52)$$

Equations (2.52) have singularities of the form

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...^j\ ,\cr V&=\phi^{-1}\sum V_j\phi^j\ ,\cr\crcr}}\,\eqno (2.53)$$

with resonances at

$$j=-1,0,1,2\ .\eqno (2.54)$$

The Bäcklund transformation

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...^{-1}+W_1\ ,\cr V&=V_0\phi^{-1}+V_1\ ,\cr\crcr}}\,\eqno (2.55)$$

taking into account (2.54), produces a system of nine equations for five functions, $(\phi,W_0,V_0,W_1,V_1)$. An analysis of this system indicates that the BT (2.55) determines a reduction of Eq. (2.52):

$$V=\lambda^2 W\ ,\eqno (2.56)$$

where

$$\lambda^2=-b/a\ .\eqno (2.57)$$

The reduced system is Eq. (2.8), the Bullough-Dodd equation, for which a Bäcklund transformation of the form (2.10), (2.55) does not exist. This result is similar to that of [7], where the BT for the modified nonlinear Shrödinger (NLS) equations determined a reduction to Burgers equation. The BT's defined by other forms of the singularities for equations (2.52) and equivalent systems, have not been investigated, these typically being highly overdetermined and implicit systems of equations.

The situation here contrasts sharply with the analysis for the (positive) Boussinesq sequence. For the (positive) Boussinesq sequence the system of equations produced by the (Painlevé) BT is not overdetermined. This is the result of the distribution of the resonances and the linearity of the highest derivative for the positive sequence. Therefore, from the point of view of Painlevé analysis and the calculation of Bäcklund transformations, it is of interest to identify when a system is a defined by a negative functional of a sequence of equations. The results of [17] demonstrate that all the ($N$-component) two-dimensional, periodic Toda lattice equations can be identified with the minus-one functionals of equation sequences. Therefore, by suitably developing the recursion operators for these sequences, it should be possible to recursively define the (Painlevé) BT's. On the other hand, a different approach to Bäcklund transformations and the Painlevé property for the Toda lattice is presented in [18].

In terms of the variables $W_i$ defined by Eqs. (2.45) for the (modified) Toda equation (2.46), the recursion operator for the Boussinesq sequence assumes a considerably more symmetric form. That is, with

$$\widehat{W}=\left(\matrix{W_1\cr \noalign{\vskip 5pt} W_2\cr \noalign{\vskip 5pt} W_3\cr}\right)\ ,\eqno (2.58)$$

the modified Boussinesq sequence is

$$\widehat{W}_t=L^n\widehat{W}_x\ ,\eqno (2.59)$$

where

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...playstyle{{\partial\over\partial x}}\ ,\cr\crcr}}\,\eqno (2.60)$$

and for $i,j=1,2,3$,

$$J=\{J_{ij}\}\ ,\eqno (2.61)$$

$$J_{ii}=-16W_iD-8W_{ix}-8W_iD^{-1}A_i-8A_iD^{-1}W_i\ ,\eqno (2.62)$$

$$A_i=(W_{i+1}-W_{i-1})_x-(W_{i+1}-W_{i-1})^2-6W_{i+1}W_{i-1}\ ,\eqno (2.63)$$

where $i=1,2,3,\ldots(\hbox{mod}\ 3)$. And

$$J_{12}=8D^2-16W_3D-8B_3-8W_1D^{-1}A_2-8A_1D^{-1}W_2\ ,\eqno (2.64)$$

$$J_{13}=-8D^2-16W_2D-8C_2-8W_1D^{-1}A_3-8A_1D^{-1}W_3\ ,\eqno (2.65)$$

$$J_{23}=-8D^2-16W_1D-8B_1-8W_2D^{-1}A_3-8A_2D^{-1}W_3\ ,\eqno (2.66)$$

$$J_{21}=-J_{21}^{\ast}\ ,\qquad J_{31}=-J_{13}^{\ast}\ ,\qquad J_{32}=-J_{23}^{\ast}\ ,\eqno (2.67)$$

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...1W_3\ ,\cr C_2&=W_{2x}-W_2^2+W_1W_3\ .\cr\crcr}}\,\eqno (2.68)$$

It is to be remarked that

$$J^{\ast}=-J\ ,\eqno (2.69)$$

and, by (2.45),

$$\sum_1^3 W_i=0\eqno (2.70)$$

implies that

$$\sum_{i=1}^3 J_{ij}=0\ ,\quad\hbox{for}\ j=1,2,3\ ,\eqno (2.71)$$

$$\sum_{j=1}^3 J_{ij}=0\ ,\quad\hbox{for}\ i=1,2,3\ .\eqno (2.72)$$

From Eqs. (2.46) and (2.60), the modified Toda lattice equations are

$$L\circ\widehat{W}_t=0\ ,\eqno (2.73)$$

or the recursion operator annihilates the right side of the modified Toda lattice equation (2.46). This suggests that explicit formulas for recursion operators of the Toda Lattice equations can be constructed from a suitable system of annihilators expressed in terms of the variables $\{W_i\}$, (2.45). It is not difficult to find formulas for the annihilators [for any $N$ in Eq. (2.48)]. Yet it is nontrivial to verify that the resulting expression is the recursion operator for a sequence. After verification of this procedure it then is necessary to find a transformation analogous to (2.45) for the Boussinesq three-component sequence in which (1) [unlike the Toda formulation (2.46)] all variables allow simultaneous singularities, and (2) a component of the transformed system is invariantly formulated (in terms of the Schwarzian derivative) thereby allowing an analysis similar to that for the Boussinesq sequence [7]. It would be most interesting to resolve the question of Schwarzian formulation through construction of Bäcklund transformations. It is our view that the Schwarzian derivative arises naturally from the essential dependence of the singularities on one preferred (spacelike) independent variable $(x)$, and not, say, from the order of the monodromy group of the associated Lax operator. In this connection, the Schwarzian derivative expresses the differential invariance of an equation when subject to the natural (unique) group of conformal transformations preserving the complex sphere $(C^1)$. When the structure of an equation's singularities depends essentially on more than one complex independent variable, various generalizations of the Schwarzian derivative are indicated.

Finally, the minus-one equation of the Hirota-Satsuma [5], [19] sequence can be shown to be equivalent to the system

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...\ ,\cr B_{xt}&=-ae^{A-B}+be^{-A-B}-e^B\cr\crcr}}\,\eqno (2.74)$$

of Toda type. Equation (2.74) is reduction of the four-component Toda lattice presented in [14]. With reference to Eq. (2.43), the minus-one Boussinesq equation tan be written as

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ... D_{xt}&=-ae^{C-D}+be^{-C-D}+cd^{2D}\ ,\cr\crcr}}\,\eqno (2.75)$$

where

$$C=\ln(\beta_x^{3/2})\ ,\qquad D=\ln(\phi_x^{1/2})\ .\eqno (2.76)$$