The ($1+1$) Sine-Gordon Equation

An interesting discussion of the long history of the ($1+1$) sine-Gordon equation

$$u_{xt}=\sin u\eqno (2.1)$$

can be found in Chap. 1 of [14]. Suffice it to say that the original Bäcklund transformation [15] was defined for this equation, while the Lax pair is contained in the inverse scattering transforms of Zakharov and Shabat [16] and Ablowitz et al. [17].

In [1] the sine-Gordon equation was shown to possess the Painlevé property. For reference, we present part of the analysis here.

Since the nonlinearity of (2.1) is nonalgebraic it is convenient to transform Eq. (2.1) into a different form. That is, let

$$V=e^{iu} ,\eqno (2.2)$$

and find

$$VV_{xt}-V_xV_t=\textstyle{1\over 2}(V^3-V) .\eqno (2.3)$$

By a leading order and resonance analysis this equation has an expansion

$$V=\varphi^{-2}\sum_{j=0}^{\infty} V_j\varphi_j ,\eqno (2.4)$$

where the ``resonances" occur at

$$j=-1,2 ,\eqno (2.5)$$

and

$$V_0=4\varphi_x\varphi_t ,\qquad V_1=-4\varphi_{xt} .\eqno (2.6)$$

The compatibility condition at $j=2$ is satisfied identically ($u_2$ is arbitrary) and (2.3) and (2.1) possesses the Painlevé property [1].

To proceed further, we now define the transform

$$V=\varphi^{-2}V_0+v\varphi^{-1}V_1+V_2 ,\eqno (2.7)$$

or, using (2.6),

$$V=-4{\partial^2\over\partial x\partial t}\ln\varphi+V_2 .\eqno (2.8)$$

Substitution of (2.7) and (2.8) into Eq. (2.3) obtains an overdetermined system of equations for $(\varphi_0,V_2)$. This system arises from the recursion relations for the $V_j$ and the requirement that

$$V_3=V_4=V_5=V_6=0 ,\eqno (2.9)$$

where $V_0$ and $V_1$ are defined by (2.6) and the condition $V_6=0$ requires $V_2$ to satisfy Eq. (2.2). There is no condition when $j=2$ since this is a resonance of the recursion relations.

To effect the reduction of the system (2.9) of four equations in two unknowns to the Lax pair for Eq. (2.2) involves extensive calculation. To simplify the calculation it is convenient to let

$$Y_2=W+\varphi^2_{xt}/\varphi_x\varphi_t .\eqno (2.10)$$

The reason for this is as follows. Under the inversion,

$$\varphi=1/\psi ,\eqno (2.11)$$

$$V_2=-4{\partial^2\over\partial x\partial t}\ln\psi+V ,\eqno (2.12)$$

and the form

$$W=V_2-\varphi^2_{xt}/\varphi_x\varphi_t\eqno (2.13)$$

becomes

$$W=V-\psi^2_{xt}/\psi_x\psi_t .\eqno (2.14)$$

This invariance of $W$ under (2.11) is a useful check on the calculation.

We then recast the overdetermined (2.9) in the variables $(W,\varphi)$ into a form that is, insofar as possible, invariant under the transformation (2.11). The resulting equations involve $W$, $W_x$, K$W_t$, etc. and the expressions

$$\varphi_x{\partial\over\partial x}\Omega_1+\varphi_t{\partial\over\partial x}\Omega_2 ,\eqno (2.15)$$

and

$$\Omega_1\Omega_2=-\textstyle{1\over 4} ,\eqno (2.16)$$

where

$$\Omega_1={\varphi_{xtt}\over\varphi_x}-{\varphi_{tt}\varphi_{... ..._t}-{1\over 2}{\varphi^2_{xt}\over\varphi^2_x} , \eqno (2.17)$$

$$\Omega_2={\varphi_{xxt}\over\varphi_t}-{\varphi_{xx}\varphi_{... ..._t}-{1\over 2}{\varphi^2_{xt}\over\varphi^2_t} . \eqno (2.18)$$

The forms $\Omega_1$ and $\Omega_2$ are similar to the Schwarzian derivative (1.4) in that they are invariant under the Moebius group (1.5).

Now, from the system (2.9) we find the ``reduced" system of equations

$$W=0\qquad\hbox{or}\qquad V_2={\varphi^2_{xt}\over\varphi_x\varphi_t} ,\eqno (2.19)$$

$$\varphi_x{\partial\over\partial x}\Omega_1+\varphi_t{\partial\over\partial x}\Omega_2=0 ,\eqno (2.20)$$

and

$$\Omega_1\Omega_2=\textstyle{1\over 4} .\eqno (2.21)$$

The system of two equations [(2.20) and (2.21)] in one unknown $(\varphi)$ can be reduced further by using the identity

$$\varphi_x{\partial\over\partial x}\Omega_1=\varphi_t{\partial\over\partial x}\Omega_2 .\eqno (2.22)$$

Thus, there results

$$\Omega_1=\alpha ,\qquad\Omega_2=\beta ,\eqno (2.23)$$

where

$$\alpha\beta=\textstyle{1\over 4} .\eqno (2.24)$$

We now let

$$Z^2=\varphi_x/\varphi_t ,\eqno (2.25)$$

$$W^2=\varphi_t/\varphi_x ,\eqno (2.26)$$

and find

$$\Omega_1=\{\varphi;t\}+2Z_{tt}/Z=\alpha ,\eqno (2.27)$$

$$\Omega_2=\{\varphi;x\}+2W_{xx}/W=\beta ,\eqno (2.28)$$

where $\alpha\beta={1\over 4}$ are Schwarzian derivatives.

To find the Lax pair we now assume that

$$\varphi=Y_1/Y_2 ,\eqno (2.29)$$

where $Y_1$ and $Y_2$ satisfy

$$Y_{xx}=aY ,\qquad\hbox{and}\qquad Y_t=bY_x+cY .\eqno (2.30)$$

By the condition

$$Y_{xxt}=Y_{txx} ,\eqno (2.31)$$

it is found that

$$2c_x+b_{xx}=0 ,\eqno (2.32)$$

$$a_t=-b_{xxx}/2+2ab_x+ba_x .\eqno (2.33)$$

By the Wronskian relation for (2.30),

$$W^2=Z^{-2}=b ,\eqno (2.34)$$

and

$$\{\varphi;x\}=-2a .\eqno (2.35)$$

Evaluating Eq. (2.28), we find

$$a={1\over 2}\left({b_{xx}\over b}-{1\over 2}{b^2_x\over b^2}\right)- {\beta\over 2} ,\eqno (2.36)$$

and substitution into Eq. (2.33) obtains

$$a_t=-\beta b_x .\eqno (2.37)$$

On the other hand, evaluationüon of (2.27) obtains

$$b_{xt}+bb_{xx}-b_tb_x/b-\textstyle{1\over 2}b^2_x-2b^2a=\alpha .\eqno (2.38)$$

Using Eq. (2.36),

$$b_{xt}-b_tb_x/b=\alpha-\beta b^2 .\eqno (2.39)$$

We now let

$$\alpha=-\lambda^{-1}/4 ,\qquad\beta=-\lambda ,\qquad b=(\lambda^{-1} /2)\Theta ,\eqno (2.40)$$

and find that $\Theta$ satisfies the equation

$$\Theta_{xt}/\Theta-\Theta_x\Theta_t/\Theta^2=\textstyle{1\over 2}(\Theta-\Theta^{-1}) ,\eqno(2.41)$$

which is Eq. (2.2).

Now substitution of (2.36) into (2.37) produces

$${\partial\over\partial x}\left({b_{xt}\over b}-{b_xb_t\over b... ...xt}\over b}-b_x{b_t\over b^2}\right)=-2\beta b_x ,\eqno (2.42)$$

or, by (2.39),

$${\partial\over\partial x}\left({b_{xt}\over b}-{b_xb_t\over b^2}-\alpha b^{-1}+\beta b \right)=0 .\eqno (2.43)$$

Thus, Eqs. (2.39) and (237) are consistent, and (2.30), (2.36), and (2.40) define the Lax pair for Eq. (2.41) or (2.2).

Having reduced (2.9) to the Lax pair for Eq. (2.2) and, thus, effectively defining the Bäcklund transform (2.8), we next consider some consequence for this reduction.

Taking into account the various scalings,

$$a={1\over 2}\left({\Theta_{xx}\over\Theta}-{1\over 2}{\Theta^2_x \over\Theta^2}\right)+{\lambda\over 2} .\eqno (2.44)$$

In the scattering problem (2.30) $\lambda$ is the spectral parameter and

$$d={1\over 2}\left({\Theta_{xx}\over\Theta}-{1\over 2}{\Theta^2_x \over\Theta^2}\right) ,\eqno (2.45)$$

where $\lim_{\vert x\vert\rightarrow\infty} d=0$, is the (in general, complex) ``potential".

From (2.45),

$$2\Theta\Theta_{xx}-\Theta^2_x-4d\Theta^2=0 ,\eqno (2.46)$$

and differentiating with respect to $x$,

$$\Theta_{xxx}-4d\Theta_x-2e_x\Theta=0 .\eqno (2.47)$$

Now formally, the Lenard recursion relations [18] are

$$\psi_{n+1,x}=-\psi_{n,xxx}+4d\psi_{n,x}+2d_x\psi_n ,\eqno (2.48)$$

where

$$\psi_0=1 ,\qquad\psi_1=d ,\qquad\psi_2=-d_{xx}+3d^2\eqno (2.49)$$

are obtained from the generating function $\psi$, where

$$2\psi\psi_{xx}-\psi^2_x-4d\psi^2+2\lambda\psi^2-2\lambda=0 ,\eqno (2.50)$$

and

$$\psi=\sum_{n=0}^{\infty} {\psi_n\over\lambda_n} .\eqno (2.51)$$

From (2.48) and (2.47),

$$\Theta=\psi_{-1} ,\eqno (2.52)$$

and the sine-Gordon equation is, with the scaling employed,

$$d_t={\partial\over\partial x}\psi_{-1} .\eqno (2.53)$$

The sequence of higher-order KdV equations are

$$d_t={\partial\over\partial x}\psi_n ,\eqno (2.54)$$

for $n=0,1,2,\ldots$ .

It seems appropriate that

$$d_t={\partial\over\partial x}\psi_{-n} ,\eqno (2.55)$$

for $n=1,2,3,4,\ldots$ be termed the higher-order sine-Gordon equations. The results of [19] demonstrate that the flows of (2.54) and (2.55) ``commute" in the sense of Hamiltonian systems. This result is essentially equivalent to tiret round in [20].

Next, we note that Eqs. (2.27) and (2.28) are, in effect, the ``classical" Bäcklund transformation for the sine-Gordon equation. Let

$$H^2=\varphi^2_{xt}/\varphi_x\varphi_t ,\eqno (2.56)$$

then

$$\null \vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...Z_x-\textstyle{1\over 2}Z^2H^2=\beta .\cr\crcr}} \eqno (2.57)$$

With

$$\null \vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...W^2&=\varphi_t/\varphi_x=e^{i\omega} ,\cr\crcr}} \eqno (2.58)$$

the Eqs. (2.57) become

$$\left({u-\omega-\omega_0\over 2}\right)_t=e^{-i\omega_0/2}\sin\left( {u+\omega+\omega_0\over 2}\right) ,\eqno (2.59)$$

and

$$\left({u+\omega+\omega_0\over 2}\right)_x=e^{i\omega_0/2}\sin\left( {u-\omega-\omega_0\over 2}\right) ,\eqno (2.60)$$

where

$$u_{xt}=\sin u ,\qquad (\omega+\omega_0)_{xt}=\sin(\omega+\omega_0) .\eqno (2.61)$$

Now, Eqs. (2.57) may be reduced by the substitution

$$\Theta=-{H\over W}=-{\varphi_{xt}\over\varphi_t} ,\qquad\Theta=- {H\over Z}=-{\varphi_{xt}\over\varphi_x} ,\eqno (2.62)$$

to the form

$$\null \vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...da^{-1}\over 2}{\Phi\over\Theta}=0 ,\cr\crcr}} \eqno (2.63)$$

where

$$V=e^{iu}=\Theta\Phi ,\qquad\alpha=\lambda/2 ,\qquad\beta= \lambda^{-1}/2 .\eqno (2.64)$$

We term Eqs. (2.63), the ``modified" sine-Gordon equations (See Appen-
dix B).