Appendix A:    The Nonlinear Schrödinger Equation

The nonlinear Schrödinger (NLS) equation

$$iU_t+U_{xx}+2U\vert U\vert^2=0\eqno (A1)$$

may be written as the system [3]

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...+2U^2V=0\ ,\cr &-iV_t+V_{xx}+2UV^2=0\ ,\cr\crcr}}\,\eqno (A2)$$

which reduces to (A1) with the identification

$$V=U^{\ast}\ .\eqno (A3)$$

The system (A2) has the Painlevé property [3] with expansions

$$U=\varphi^{-1}\sum_{j=0}^{\infty} U_j\varphi^j\ ,\qquad V=\varphi^{-1}\sum_{j=0}^{\infty} V_J\varphi^j\ ,\eqno (A4)$$

and resonances at

$$j=-1,0,3,4\ .\eqno (A5)$$

The Bäcklund transformation is

$$U=U_0/\varphi+U_1\ ,\qquad V=V_)/\varphi+V_1\eqno (A6)$$

which determines the following system of equations for $(\varphi,U_0, V_0,U_1,V_1)$:

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...=0\ ,\cr &-iV_{1t}+V_{1xx}+U_1V_1^2=0\ ,\cr\crcr}}\,\eqno (A7)$$

Taking into account the resonances at $j=0,3$, (A7) is, effectively, a system of ``six" equations for the five variables $(\varphi,U_0, V_0,U_1,V_1)$.

From (A7) it is found that

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...i_t\over \varphi_x}-\lambda^2\right\}\ ,\cr\crcr}}\,\eqno (A8)$$

and

$${\partial\over\partial t}\left({\varphi_t\over\varphi_x}\righ... ...arphi_t\over\varphi_x}+{\lambda^2\over 2}\right]=0\ ,\eqno (A9)$$

where $\lambda$ is a constant of integration. The above system of equations were studied in [3] and further applied in [6] to derive the Hirota formulation of the NLS equation from the Bäcklund transformation (A6).

In this section we will find a scalar Lax pair for the NLS equation by ``linearizing" the Miura transformation from the modified NLS to NLS equation. For this purpose it is convenient to let

$$\lambda=2i\beta\ ,\qquad W=\varphi_{xx}/\varphi_x\ ,\qquad \Omega=\varphi_t/\varphi_x-2\beta\ ,\eqno (A10)$$

which obtains from (A9) the system of modified NLS equations

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...yle{3\over 2}\Omega^2-2\beta\Omega)\ .\cr\crcr}}\, \eqno (A11)$$

By reduction of (A8),

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ..._x\over W+i\Omega}+i(\Omega+\beta)\ ,\cr\crcr}}\, \eqno (A12)$$

which is a Miura transformation from (A11) to (A2). Now let

$$G=W-i\Omega\ ,\qquad H=W+i\Omega\ ,\eqno (A13)$$

and find

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...V_1}={H_x\over H}+{H-G\over 2}+i\beta\ ,\cr\crcr}}\,\eqno (A14)$$

The substitutions

$$G=2i(U_1/\alpha)\ ,\qquad 2iV_1\alpha\eqno (A15)$$

reduce (A14) to a Ricati-type equation

$$\alpha_x+iV_1\alpha^2+i\beta\alpha-iU_1=0\ ,\eqno (A16)$$

that is linearized by

$$\alpha=-(i/V_1)(h_x/h)\eqno (A17)$$

to

$$h_{xx}+(i\beta-V_{1x})h_x+U_1V_1h=0\ .\eqno (A18)$$

Substitution of (A13), (A15), and (A17) into (A11) obtains

$$ih_t=h_{xx}+2U_1V_1h+2i\beta h_x\ .\eqno (A19)$$

By (A18)

$$ih_t=(V_{ix}/V_1+i\beta)h_x+U_1V_1h\ .\eqno (A20)$$

Here, (A18) and (A20) constitute a Lax pair for the NLS system (A2) in the sense that

$$h_{ixx}=h_{xxt}\eqno (A21)$$

requires that

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...x}\over V_1}\right)^2+2U_1V_1\right)\ ,\cr\crcr}}\,\eqno (A22)$$

which is ``equivalent" to the system (A2). With

$$A=V_{1x}/V_1\ ,\qquad B=U_1V_1\ ,\eqno (A23)$$

Eqs. (A22) are

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...&={\partial\over\partial x}(-B_x+2AB)\ ,\cr\crcr}}\,\eqno (A24)$$

and the Miura transformation from (A11) is

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...\Omega)_x/(W+i\Omega)+i(\Omega+\beta)\ .\cr\crcr}}\,\eqno (A25)$$

Now after a Galilean transformation,

$$t\rightarrow t\ ,\qquad x\rightarrow x-2\beta t\ ,\qquad \Om... ...i_t/\varphi_x\ ,\qquad W=\varphi_{xx}/\varphi_x\ ,\eqno (A26)$$

and

$$\null\,\vcenter{\openup\jot \ialign{\strut\hfil$\displaysty... ...r 2}W^2-\textstyle{3\over 2}\Omega^2)\ .\cr\crcr}}\,\eqno (A27)$$

At first inspection Eq. (A27) would seem to be nearly the modified Boussinesq equations, (2.18). However, a simple calculation determines that Eqs. (A27) have no discrete invariances, i.e., no transformations of the form

$$\left(\matrix{W\cr \Omega\cr}\right)=A\left(\matrix{W^{\prime}\cr \Omega^{\prime}\cr}\right)\eqno (A28)$$

that preserve the form of Eqs. (A27). Equations (A27) identically possess the Painlevé property with expansions

$$W=\epsilon^{-1}\sum_{j=0}^{\infty} W_j\epsilon^j\ ,\qquad \Omega=\epsilon^{-1}\sum_{j=0}^{\infty}\Omega_j\epsilon^j\eqno (A29)$$

and resonances at

$$j=-1,2,2,3\ .\eqno (A30)$$

From (A29):

(i) $\Omega_0=0\ ,\qquad W_0=-2$ ;(A31)

(ii) $\Omega_0^2=-1\ ,\qquad W_0=1$ . (A32)

As was the case for the modified Boussinesq equations a transformation

$$A_{\pm}=\left(\matrix{-\textstyle{1\over 2}&\pm\textstyle{3\o... ...extstyle{1\over 2}i&-\textstyle{1\over 2}\cr}\right)\eqno (A33)$$

interchanges the ``leading-order" vectors

$$\left(\matrix{W_0\cr \noalign{\vskip 5pt} \Omega_0\cr}\right)... ...cr}\right) \ ,\ \left(\matrix{1\cr -i\cr}\right)\ .\eqno (A34)$$

However, the substitution (A28) and (A33) is not invariant for (A27). Therefore, the method of analysis that was developed for the Boussinesq sequence is hot directly applicable to the NLS sequence.