An Alternative Approach to Integrable Discrete Nonlinear Schrödinger Equations

Abstract

In this article we develop the direct and inverse scattering theory of a discrete matrix Zakharov-Shabat system with solutions U _n and W _n . Contrary to the discretization scheme enacted by Ablowitz and Ladik, a central difference scheme is applied to the positional derivative term in the matrix Zakharov-Shabat system to arrive at a different discrete linear system. The major effect of the new discretization is that we no longer need the following two conditions in theories based on the Ablowitz-Ladik discretization: (a) invertibility of I _N −U _n W _n and I _M −W _n U _n , and (b) I _N −U _n W _n and I _M −W _n U _n being nonzero multiples of the respective identity matrices I _N and I _M .

Appendices

Appendix A: Summation by Parts Formula

Let $\{b_{k}\}_{k=n}^{\infty}$ be a sequence of nonnegative numbers. Then in [6, Lemma A.2] the following fundamental equality has been established:

$$ \sum_{k=n}^\infty b_k \Biggl(\sum_{j=k+1}^\infty b_j \Biggr)^m =\frac{1}{m+1} \Biggl(\sum_{j=n}^\infty b_j \Biggr)^{m+1}-B_n^{(m)}, $$

(A.1)

where $B_{n}^{(m)}\ge B_{n+1}^{(m)}\ge B_{n+2}^{(m)}\ge\cdots\ge0$.

Although the general form of the discrete Gronwall inequality is well-known [21, Corollary 1.6.2], here we apply (A.1) to prove the version needed.

Proposition A.1

(Gronwall’s inequality)

Suppose $\{p_{k}\}_{k=n}^{\infty}$ and $\{q_{k}\}_{k=n}^{\infty}$ are sequences of nonnegative numbers such that the series $\sum_{k=n+1}^{\infty}q_{k}$ converges and

$$ p_n\le1+\sum_{k=n+1}^\infty q_kp_k. $$

(A.2)

Then

$$ p_n\le\exp \Biggl(\sum_{k=n+1}^\infty q_k \Biggr). $$

(A.3)

Proof

Iterating (A.2) we get

where (A.1) has been used repeatedly. □

Appendix B: Marchenko Equations in More Detail

In this appendix, we decouple (5.6a)–(5.6d) by using that the Marchenko kernels vanish for even values of their arguments. Equations (B.1a)–(B.3d) below have been used in [11] to derive explicit matrix IDNLS solutions by using matrix triplets to parametrize the Marchenko kernels.

Let us decouple (5.6a) and (5.6b), using that the Marchenko kernels F _r (s) and $\overline {\boldsymbol {F}}_{r}(s)$ vanish for even s. We get

(B.1a)

(B.1b)

(B.1c)

(B.1d)

Equations (B.1a) and (B.1d) are valid for σ≥1, whereas (B.1b) and (B.1c) are valid for σ≥0. This distinction in the ranges of the summation index σ is to bear in mind when deriving exact solutions to (1.5). The potentials are then computed as follows:

$$ \boldsymbol{u}_n=\frac{i}{2h}\boldsymbol {K}^{\text {up}}(n-1,n), \qquad\boldsymbol {w}_n=\frac{-i}{2h}\overline {\boldsymbol {K}}^{\text {dn}}(n-1,n). $$

(B.2)

Let us decouple (5.6c) and (5.6d), using that the Marchenko kernels F _l (s) and $\overline {\boldsymbol {F}}_{l}(s)$ vanish for even s. We get

(B.3a)

(B.3b)

(B.3c)

(B.3d)

Equations (B.3a) and (B.3d) are valid for σ≥1, whereas (B.3b) and (B.3c) are valid for σ≥0. This distinction in the ranges of the summation index σ is to bear in mind when deriving exact solutions of (1.5). The potentials are then computed as follows:

$$ \boldsymbol{u}_n=\frac{i}{2h}\overline {\boldsymbol {L}}^{\text {up}}(n+1,n), \qquad\boldsymbol {w}_n=\frac{-i}{2h}\boldsymbol {L}^{\text {dn}}(n+1,n). $$

(B.4)