Sine-Gordon, Boussinesq and KP Equations

A. The sine-Gordon equation

$$u_{xt}=\sin u\ .\eqno (A1)$$

The substitution

$$V=e^{iu}\eqno (A2)$$

leads to the equation

$$2VV_{xt}-2V_xV_t=V^3-V\ .\eqno (A3)$$

We find that

$$V=\phi^{-2}\sum_{j=0j}^{\infty} V_j\phi_j\ ,\eqno (A4)$$

where

$$V_0=4\phi_x\phi_t\qquad\hbox{and}\qquad V_1=-4\phi_{xt}\ .$$

The resonances occur at

$$j=-1,2\ .\eqno (A5)$$

Again, the compatability condition at $j=2$ is satisfied identically and the equation possesses the Painlevé property. The recursion relations are found to be

$$2\sum_{m=0}^j V_{j-m}A_m-2\sum_{m=0}^j B_{j-m}C_m=\sum_{m=0}^j \sum_{l=0}^m V_{j-m}V_{m-l}V_l-V_{j-4}\ ,\eqno (A6)$$

where

$$\begin{array}{lcl} A_j&=&V_{j-2,xt}+(j-3)(\phi_t V_{j-1,x}+\... ...n{\vskip 5pt} B_j&=&V_{j-1,x}+(j-2)\phi_x V_j\ , \end{array}$$

and

$$C_j=V_{j-1,t}+(j-2)\phi_t V_j\ .$$

From the recursion relations or more directly from (A3), it follows that any truncation of (A4) must satisfy

$$V_j=0\ ,\qquad j\ge 3\ .\eqno (A7)$$

In the search for a possible Bäcklund transformation, we are thus led to examine the truncation

$$V=\phi^{-2} V_0+\phi^{-1} V_1+V_2=-4{\partial^2\over\partial x \partial t}\ln\phi+V_2\ .\eqno (A8)$$

It then follows that the resonance function $V_2$ must satisfy Eq. (A3). In addition, for (A7) to be verified $(V_2,\phi)$ must satisfy an overdetermined system of three nonlinear pde's obtained from the conditions

$$V_3=V_4=V_5=0\ .\eqno (A9)$$

[From $V_6=0$, $V_2$ must satisfy (A31). Simplification of the conditions defining the BT (A8) is currently in progress.]

B. The Boussinesq equation

$$u_{tt}+2uu_{xx}+2u_x^2+\textstyle{1\over 3}u_{xxxx}=0\ .\eqno (A10)$$

It is found that

$$u=\phi^{-2}\sum_{j=0}^{\infty} u_j\phi^j\ .\eqno (A11)$$

The resonances occur at

$$j=-1,4,5,6\ .\eqno (A12)$$

From the recursion relations we find that

truecm $j=0\ ,\qquad u_0=-2\phi_x^2\ ,$ (A13)

truecm $j=1\ ,\qquad u_1=2\phi_{xx}\ ,$(A14)

truecm $j=2\ ,\qquad\phi_t^2-\phi_{xx}^2+ \textstyle{4\over 3}\phi_x\phi_{xxx}+2u_2\phi_x^2=0\ ,$(A15)

truecm $j=3\ ,\qquad\phi_{tt}+\textstyle{1\over 3} \phi_{xxxx}+2\phi_{xx} u_2-2\phi_x^2 u_3=0\ ,$(A16)

truecm $j=4$ (resonance),    $u_4$ is arbitrary if the compatability

condition

$${\partial^2\over\partial x^2}(\phi_t^2-\phi_{xx}^2+\textstyle{4\over 3} \phi_x\phi_{xx}+2u_2\phi_x^2)=0\eqno (A17)$$

is satisfied. By (Al5) this is so.

$j=5$ (resonance), $u_5$ is arbitrary if

$${\partial^2\over\partial x^2}(\phi_{tt}+\textstyle{1\over 3}\phi_{xxxx}+ 2\phi_{xx} u_2-2\phi_x^2 u_3)=0\ .\eqno (A18)$$

By (A16) this is always satisfied.

$j=6$ (resonance): We find that $u_6$ is arbitrary if a rather complicated compatibility condition is satisfied. After algebraic reduction, this condition is found to involve $(\phi,u_2,u_3)$, or, using (A15) and (A16), to reduce to an identity in the arbitrary function $\phi$. The verification of this identity becomes trivial when the ``reduced ansatz" [14]

$$\phi=x-\psi(t)\eqno (A19)$$

is employed. Basically, when $\phi_x\ne 0$ [see (A13)], the implicit function theorem indicates that $\phi=\phi(x,t)$ may be ``locally" (near $\phi=0$) represented in the form (A19), where

$$\phi\bigl(\psi(t),t\bigr)=0\ .\eqno (A20)$$

In this manner the compatability condition is verified, and the Boussinesq equation is found to be identically Painlevé.

To define a Bäcklund transform it is necessary to truncate (A11) at the ``constant" level term. That is,

$$u_j=0\ ,\qquad j\ge 3\ .\eqno (A21)$$

We find from (A13)-(A16) that

$$u=2{\partial^2\over\partial x^2}\ln\phi+u_2\ ,\eqno (A22)$$

where $(u,u_2)$ satisfy (A10), defines a Bäcklund transform, if

$$u_{2t}+2u_2u_{2xx}+2u_{2x}^2+\textstyle{1\over 3}u_{2xxx}=0\ ,\eqno (A23)$$

$$\phi_t^2-\phi_{xx}^2+\textstyle{4\over 3}\phi_x\phi_{xx}+2u_2\phi_x^2=0\ , \eqno (A24)$$

and

$$\phi_{tt}+\textstyle{1\over 3}\phi_{xxxx}+2\phi_{xx} u_2=0\ .\eqno (A25)$$

By a method presented in [15] the Lax pair can be found from (A24) and (A25).

C. The Kadomstev-Petvlashvili, or two-dimensional KdV, equation

The KP equation [16]

$$u_{tx}+u_x2+uu_{xx}+\delta u_{xxxx}+u_{yy}=0\eqno (A26)$$

has an expansion of the form

$$u=\phi^{-2}\sum_{j=0}^{\infty} u_j\phi^j\eqno (A27)$$

with resonances at

$$j=j-1,4,5,6\ .\eqno (A28)$$

From the recursion relations we find that

truecm $j=0\ ,\qquad u_0=-12\sigma\phi_x^2\ ,$ (A29)

truecm $j=1\ ,\qquad u_1=12\sigma\phi_{xx}\ ,$ (A30)

truecm $j=2\ ,\qquad\phi_t\phi_x+4\sigma\phi_x \phi_{xxx}-3\sigma\phi_{xx}^2+\phi_y^2+u_2\phi_x^2=0\ ,$(A31)

truecm $j=3\ ,\qquad\phi_{xt}+\sigma\phi_{xxxx}+ \phi_{yy}+\phi_{xx} u_2-\phi_x^2 u_3=0\ ,$(A32)

truecm $j=4$ (resonance),    $u_4$ is arbitrary if the compatability

condition

$${\partial^2\over\partial x^2}(\phi_x\phi_t+4\sigma\phi_x\phi_{xxx} -3\sigma\phi_{xx}^2+\phi_y^2+u_2\phi_x^2)=0\ .\eqno (A33)$$

By (A31) this is so.

$j=5$ (resonance), $u_5$ is arbitrary if

$${\partial^2\over\partial x^2}(\phi_{xt}+\sigma\phi_{xxxx}+\phi_{yy} +\phi_{xx} u_2-\phi_x^2 u_3)=0\ .\eqno (A34)$$

By (A32) this is so.

$j=6$ (resonance): Again $u_6$ will be arbitrary if a complicated compatibility condition is satisfied. By the ``reduced ansatz" (see the last section)

$$\phi=x-\psi(t,y)\eqno (A35)$$

it can be shown that this condition is satisfied identically.

By the above considerations the KP equation is found to be identically Painlevé.

The associated Bäcklund transform, defined by truncating at the ``constant" term, is

$$u=12\sigma {\partial^2\over\partial x^2}\ln\phi+u_2\ .\eqno (A36)$$

In [15] we demonstrate the consistency of this transform by finding the Lax pairs for this equation.