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

Herein, we consider the $N$ space-one rime ($N+1$) dimension sine-Gordon equation (SGE). For the ($2+1$) SGE explicit soliton type solutions were obtained by Hirota [5], while a Bäcklund transform was found by Leibbrandt [7]. Basically, the $n$-soliton solution found by these authors consists of a superposition of $n$ plane, traveling waves [8]. The parameter (directions) of these waves (soliton) are required to satisfy a certain set of compatibility conditions for the solutions to exist [5,6,8,9]. For the ($1+1$) SGE these conditions are trivial. For the two-soliton solution of the ($2+1$) SGE, Gibbon and Zambotti [6] have shown the compatibility conditions to be trivial; while, for the three-soliton solution, the area of the triangle formed by the three plane waves is time invariant. All the known exact solutions of the ($N+1$) have an infinite energy since they are constructed from plane waves. It is not known if there exist exact solutions with finite energy.

In what follows we apply the Painlevé analysis to the ($N+1$) SGE and find that (for $N>1$) this equation is not identically Painlevé. In addition, it can be shown that the directions of the $n$-plane waves must lie in the same plane if the compatibility conditions are to be satisfied for solutions of this type. Hence, these solutions can be obtained by a Lorenz transformation of the solutions of the ($1+1$) SGE.

Without loss of generality and for notational convenience, we consider the ($N+1$) elliptic SGE

$$-\Box u=\sin u ,\eqno (4.1)$$

where

$$\Box=\partial^2_{xj}=\nabla^{\prime}\nabla ,\eqno (4.2)$$

and

$$\nabla_j={\partial\over\partial x_j} .$$

By the substitution

$$V=e^{iu} ,\eqno (4.3)$$

we find

$$-V\Box V+\nabla V\cdot\nabla V=\textstyle{1\over 2}(V^3-V) .\eqno (4.4)$$

The Painlevé representation

$$V=\varphi^{-2}\sum_{j=0}^{\infty} V_j\varphi^j ,\eqno (4.5)$$

with resonances at

$$j=-1,2 ,\eqno (4.6)$$

will be valid if $\varphi=\varphi(x_1,\ldots,x_{n+1})$ satisfies a compatibility condition. Using the expressions

$$V_0=-4\nabla\varphi\cdot\nabla\varphi ,\qquad V_1=4\Box\varphi , \eqno (4.7)$$

the compatibility condition is found to be

$$\nabla\varphi\cdot D\nabla\varphi=0 ,\eqno (4.8)$$

where

$$D_{ii}=\sum_{l=1\atop {l\ne i\atop m\ne i}}^{^{N+1}}\sum_{m=1... ...{N+1}} (\varphi^2_{im}-\varphi_{ii}\varphi_{mm}) ,\eqno (4.9)$$

and

$$d_{ij}=\sum_{k=1}^{^{N+1}} (\varphi_{ij}\varphi_{kk}-\varphi_{ik} \varphi_{jk} .\eqno (4.10)$$

We note the following observations:

(1) The matrix $D$ is symmetric $(D_{ij}=D_{ji})$ and Eq. (4.8) is trivial when $N=1$ [($1+1$) SGE].

(2) Equation (4.8) is invariant under the change of variables, $x_j \rightarrow ix_j$ (hyperbolic SGE).

(3) Equation (4.8) is translation invariant, i.e., $x_j\rightarrow x_j+c_j$.

(4) Equation (4.8) is invariant under orthogonal changes of independent variables,

$$\nabla=B\nabla^{\prime} ,\eqno (4.11)$$

where

$$B^t=B^{-1} .$$

Observation (4) follows from the orthogonal invariance of (4.4) and (4.7).

Therefore, consider the hypersurface $M$ defined by

$$M=\bigl\{\widehat{x}\colon\varphi(\widehat{x})=\varphi_0\bigr\} ,\eqno (4.12)$$

where $\widehat{x}=(x_1,x_2,\ldots,x_{n+1})$ and $\nabla\varphi\mid M$.

By translation and rotation we may locate the origin of the coordinate system at a point $\widehat{x}_0\in M$ so that

$${\partial\over\partial x_2},\ldots,{\partial\over\partial x_{n+1}}$$

provide an orthogonal basis for the tangent space of $M$ at $\widehat{x}_0$. Since $M$ is a hypersurface there is a unique normal to $M$ at $wt{x}_0$:

$$\widehat{N}=\left(\matrix{\varphi_{x_1}\cr 0\cr \vdots\cr 0\cr}\right) . \eqno (4.13)$$

By observations (3) and (4) and (4.13), Eq. (4.8) reduces to

$$\varphi^2_{x_1}\sum_{l=2}^{^{N+1}}\sum_{m=2}^{^{N+1}} (\varphi^2_{lm} -\varphi_{ll}\varphi_{mm})=0 ,\eqno (4.14)$$

at the ``arbitrary point" $\widehat{x}_0$.

In terms of the hypersurface $M$, Eq. (4.14) states [23] that the elementary symmetric function of the principal curvatures of $M$ vanishes. That is,

$$K_1K_2+K_1K+3+\cdots+K_{n-1}K_n=0 ,\eqno (4.15)$$

where $K_j$, $j=1,\ldots,n$ are the principal curvatures of $M$. In effect, Eq. (4.14) is the sum of the principal minors of order 2 of the second fundamental form of $M$ [23].

Now, let $N=2$ [the ($2+1$) SGE] and find

$$K_1K_2=0\eqno (4.16)$$

or $K=K_1K_2$ (the Gaussian curvature) vanishes, defining a ``developable surface" [23]. Condition (4.8) becomes, in the variables $(t,x,y)$,

$$\null \vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...i_{yt}-\varphi_{xy} \varphi_{tt})=0 .\cr\crcr}} \eqno (4.17)$$

As noted in observation (1), Eq. (4.17) is trivial when $\varphi$ is a function of two variables, i.e., $\varphi=\varphi(t,x)$.

Now, let $\varphi$ be a product of plane, traveling waves:

$$\varphi=\prod_{j=1}^m f_j(a_jt+b_jx+c_jy-d_j) ,\eqno (4.18)$$

where the $f_j(x)$ are arbitrary.

If $m=2$ (two waves), a rotation of the coordinates can be devised so that $\varphi$ depends (effectively) on two variables, and condition (4.17) will be trivial [6].

For any $m$ a similar argument demonstrates (4.17) will satisfied identically if all of the wave directions, $\widehat{a}_j=(a_j,b_j, c_j)$ lie in the same plane. Furthermore, the necessity of this condition can be proven by direct substitution of (4.18) into (4.17) and using the requirement that the $f_j(z)$ be arbitrary.

For three waves the co-planar condition may be written

$$\left(\matrix{a_1&b_1&c_1\cr a_2&b_2&c_2\cr a_3&b_3&c_3\cr}\right)=0 .\eqno (4.19)$$

This is the condition found in [6] for the existence of the three soliton solution. It indicates that the area of the triangle formed by the plane waves is time invariant.

From the above it appears that the class of known, exact solutions for the ($2+1$) SGE is trivial in that they can be reduced to solutions of the ($1+1$) SGE. If nontrivial solutions of (4.17) (developable surfaces) correspond to exact solutions of (4.4) this class may contain solutions with nonreducible behavior.

As in Sec. III the compatibility condition (4.17) may be ``linearized" and the complete solution found by a Legendre transformation. That is,

$$\null \vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...=\varphi_y ,\qquad y=W_{\epsilon_3} ,\cr\crcr}} \eqno (4.20)$$

$$\varphi(t,x,y)+W(\epsilon_1,\epsilon_2,\epsilon_3)=t\epsilon_1+ x\epsilon_2+y\epsilon_3\eqno (4.21)$$

obtains from (4.17) the linear equation (with summation convention)

$$\epsilon_i\epsilon_j{\partial^2\over\partial\epsilon_i\partial \epsilon_j} W=0 .\eqno (4.22)$$

Letting

$${d\over ds}=\epsilon_i{\partial\over\partial\epsilon_i} ,\eqno (4.23)$$

we find

$${d^2\over ds^2}W={d\over ds}W .\eqno (4.24)$$

The complete solution of (4.24) is

$$W=W_0+W_1 ,\eqno (4.25)$$

where

$${d\over ds}W_0=0 ,\qquad {d\over ds}W_1=W_1 .\eqno (4.26)$$

Here $W_0$ and $W_1$ are ``homogeneous" functions of degree zero and one, respectively. (See Sec. III.) Again, we find

$$\varphi(t,x,y)=-W_0(\epsilon_1,\epsilon_2,\epsilon_3) ,\eqno (4.27)$$

and

$$t\epsilon_1+x\epsilon_2+y\epsilon_3=W_1(\epsilon_1,\epsilon_2, \epsilon_3) .\eqno (4.28)$$

We note that the Legendre transformation is defined when $\varphi$ depends, effectively, on three independent variables.