Appendix B:    Rational Solutions

One consequence of a discrete symmetry group (Bäcklund transformation) for the ``modified" equations is the induced Bäcklund transformation for the ``singular manifold" equation. [See (3.76)]. This Bäcklund transformation [combined with the Moebius transformation (3.50)], determines a simple method for iteratively constructing rational and other special solutions of the equations under consideration. Therefore, discrete symmetries (of modified equations) are a sufficient condition (by construction) for the existence of sequences of ``rational" solutions. We conjecture that a necessary condition (for rational solutions) is the occurrence of a nondegenerate Bäcklund transformation for the ``modified" equations. This would imply, by the results of Appendix C, that the NLS equations (A2) have no (nontrivial) sequences of rational solutions. Effectively, the only direct (known) Bäcklund transformation for Eq. (A9) is the Moebius group (3.50), which is not sufficient for the iterative construction of solutions. In this section rational solutions are iteratively defined for the ``Boussinesq" equation

$${\partial\over\partial x}\left({\varphi_t\over\varphi_x}\righ... ...left({\varphi_t\over\varphi_x}\right)^2 \right)=0\ ,\eqno (B1)$$

where

$$\{\varphi;x\}={\partial\over\partial x}\left({\varphi_{xx}\ov... ...er 2} \left({\varphi_{xx}\over\varphi_x}\right)^2\ .\eqno (B2)$$

Equation (B1) is invariant under the Moebius group

$$\varphi=(a\psi+b)/(c\psi+d)\eqno (B3)$$

and the Bäcklund transformation

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...psi_x}- {1\over 2}{\psi_t\over\psi_x}\ .\cr\crcr}}\,\eqno (B4)$$

Now composing (B3) and (B4), where

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...varphi_j\ ,\cr \varphi&=\varphi_{j+1}\ ,\cr\crcr}}\,\eqno (B5)$$

obtains

(i) $${{\varphi_{j+1,xx}\over\varphi_{j+1,x}}= -{1\over 2}{\partial\ove... ..._{j,x}\over\varphi_j^2}\right)\mp {3\over 2}{\varphi_{j,t}\over\varphi_{j,x}}}$$ ,

(B6)

(ii) $${{\varphi_{j+1,t}\over\varphi_{j+1,x}}= \pm {1\over 2}{\partial\o... ...hi_{j,x}\over\varphi_j^2}\right)- {1\over 2}{\varphi_{j,t}\over\varphi_{j,x}}}$$ .

From (B6) with lower sign and

$$\varphi_0=x\ ,\eqno (B7)$$

it is found that after normalization

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...arphi_3&=(x^6+10tx^4+20t^2x^2+40t^3)/x\ .\cr\crcr}}\,\eqno (B8)$$

By evaluation of (B6)

(i) $\varphi_{j+1,x}=\varphi_j \varphi_{j,x}^{-1/2}\lambda_j$ ,

(B9)

(ii) $${\varphi_{j+1,t}=\pm{1\over 3} \varphi_j^4\varphi_{j,x}^{-2}{\partial\over\partial x}(\varphi_j^{-3}\varphi_{j,x}^{3/2} \lambda_j)}$$ ,

where

$$\lambda_j=\left\{\prod_{k=1}^j\left({\varphi_{j-k,x}\over \varphi_{j-k}^2}\right)^{(-1/2)^k}\right\}^{3/2}\ .\eqno (B10)$$

The identity

$$\lambda_j=(\varphi_{j-1,x}/\varphi_{j-1}^2)^{-3/4}\lambda_{j-1}^{-1/2} \eqno (B11)$$

and recursive application of (B9)(i) obtains

$$\varphi_{j+1,x}=(\varphi_j\varphi_{j-1}/\varphi_{j-2}^2) \varphi_{j-2,x}\ .\eqno (B12)$$

To simplify (B6), (B9), and (B12), let the meromorphic function

$$\varphi_j=P_j/Q_j\ ,\eqno (B13)$$

where $(P_j,Q_j)$ are entire functions of $(x,t)$. Substitutions into (B12) obtain

$$Q_{j+1}=P_{j-2}\ ,\eqno (B14)$$

$$P_{j-2}P_{j+1,x}-P_{j+1}P_{j-2,x}=P_jP_{j-1}\ ,\eqno (B15)$$

where, by (B14),

$$\varphi_j=P_j/P_{j-3}\ .\eqno (B16)$$

Substitution of (B16) into (B6)(ii) obtains

$$P_{j-2}P_{j+1,t}-P_{j+1}P_{j-2,t}=\mp(P_{j-1}P_{j,x}-P_jP_{j-1,x})\ . \eqno (B17)$$

Therefore, (B15) and (B17) define entire functions $P_j=P_j(x,t)$ and, from (B16), meromorphic $\varphi_j$. From (B8),

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...,\cr P_3&=x^6+10tx^4+20t^2x^2+40t^3\ .\cr\crcr}}\,\eqno (B18)$$

By induction, using (B15), (B17), and (B18),

$$P_j=\sum_{k=0}^j C_kt^{j-k}x^{2k}\ ,\eqno (B19)$$

where (for $j>0$) the $C_k$ are constant. By the results of Sec. II the above defines rational solutions for the Boussinesq and modified Boussinesq equations. The constructions (B15)-(B17) remain valid when, in (B6), $\varphi_0$ assumes other values than (B7). Say,

$$\varphi_0=xt$$

or

$$\varphi_0=e^{ax+bt}\ ,\eqno (B20)$$

which defines $(P_0,P_1,P_2)$ and from (B15) to (B17), $(\varphi_j, P_j)$ for $j\ge 3$.

Rational solutions of integrable partial differential equations have been studied for some time as ``pole expansions" of the solution [10]-[12]. In [3], the pole expansions are derived from the (Painlevé) expansions about the singular manifold.

Our method is similar to that of [13] and [14] in that the solution is defined in terms of a polynomial in the independent variables. However, to us, the calculation based on the (Schwarzian) modified equation seems preferable in that the Bäcklund transformations apply to ``general" forms of solutions and the ``rational" solutions are found at the last stage of the analysis as ``natural" special solutions. (See [5], Appendix B).