(pp. 13-22)

The initial-value problem (1.1) is solvable for arbitrary$\hbar$because the focusing nonlinear Schrödinger equation can be represented as the compatibility condition for two systems of linear ordinary differential equations:

$\hbar\partial_{x}\begin{bmatrix} u_{1} \\ u_{2} \end{bmatrix}=\begin{bmatrix}-i\lambda & \psi\\ -\psi^{\ast} & i\lambda\end{bmatrix}\begin{bmatrix}u_{1} \\ u_{2} \end{bmatrix},\quad\caption{(2.1)}$

$i\hbar\partial_{t}\begin{bmatrix} u_{1} \\ u_{2} \end{bmatrix}=\begin{bmatrix}\lambda^{2}-|\psi|^{2}/2 & i\lambda\psi-\hbar\partial_{x}\psi/2\\ -i\lambda\psi^{\ast}-\hbar\partial_{x}\psi^{\ast}/2 & -\lambda^{2}+|\psi|^{2}/2\end{bmatrix}\begin{bmatrix}u_{1} \\ u_{2} \end{bmatrix},\quad\caption{(2.2)}$

where*λ*is an arbitrary complex parameter. The compatibility condition for (2.1) and (2.2) does not depend on the value of*λ*and is equivalent to the nonlinear Schrödinger equation.

The*N*-soliton solutions of the nonlinear Schrödinger equation are those complex functions*ψ*(*x*,*t*) for which there exist simultaneous column vector solutions of (2.1) and (2.2) of the particularly simple form

$\begin{matrix} \textrm{u}^{+}(x,t,\lambda)=\begin{bmatrix}\sum_{p=0}^{N-1}A_{p}(x,t)\lambda^{p} \\ \lambda^{N}+\sum_{p=0}^{N-1}B_{p}(x,t)\lambda^{p}\end{bmatrix}\exp(i(\lambda x+\lambda^{2}t)/\hbar), \\ \textrm{u}^{-}(x,t,\lambda)=\begin{bmatrix}\lambda^{N}+\sum_{p=0}^{N-1}C_{p}(x,t)\lambda^{p} \\ \sum_{p=0}^{N-1}D_{p}(x,t)\lambda^{p}\end{bmatrix}\exp(-i(\lambda x+\lambda^{2}t)/\hbar), \end{matrix}\quad \caption{(2.3)}$

satisfying the relations

$\begin{matrix}\textrm{u}^{+}(x,t,\lambda_{k})=\gamma_{k}\textrm{u}^{-}(x,t,\lambda_{k}), \\ -\gamma_{k}^{\ast}\textrm{u}^{+} (x,t,\lambda_{k}^{\ast})=\textrm{u}^{-}(x,t,\lambda_{k}^{\ast}),\quad k=1,\ldots,N, \\ \end{matrix}\quad\caption{(2.4)}$

for...