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Original system $$i \frac{d}{dt} u_n(t) = -\underbrace{(u_{n+1} - 2u_{n} + u_{n-1})}_{\Delta_2 u_{n}} - |u_{n}|^{2} \sum\limits_{r \in I_{N}\setminus\{0\}}\Big (u_{n+r} + u_{n-r} \Big )\Big ( znr(\alpha,r,N)\Big ), \; n\in\mathbb{Z}$$
where $$znr(\alpha,r,N) = \frac{1}{2(2N)^{1+\alpha}} \Big(\zeta(1+\alpha, \frac{r}{2N} + \zeta(1+\alpha, -\frac{r}{2N}) - \Big | \frac{r}{2N}\Big|^{-(1+\alpha)}\Big) \in \mathbb{C}$$
and $n \in I_{N} = \{-N, \cdots, 0, \cdots, N-1\} $

- - -

- $u_{n} = Re[u_{n}] + i Im[u_n] = a_n + b_{n} i $, and $znr(\alpha, r,N) = c+di$

LHS: \begin{align*}i\frac{d}{dt} \Big ( a_{n} + b_{n}i \Big )&= -\frac{d}{dt}b_{n} + i\frac{d}{dt} a_{n}\end{align*}

RHS:

\begin{align*}\Delta_{2} \Big ( a_{n} + b_{n}i\Big ) &= \Big (a_{n+1} -2a_{n} + a_{n-1} \Big ) + i\Big ( b_{n+1} -2b_{n} + b_{n-1}\Big )\end{align*}

\begin{align*}\Big | a_{n} + b_{n} i\Big |^{2} &= \Big ( a_{n} + b_{n}i\Big )\Big (a_{n} - b_{n}i \Big ) \\ &= a_{n}^{2} + b_{n}^{2}\end{align*}

\begin{align*}\Big (a_{n+r} + a_{n-r} + i (b_{n+r} + b_{n-r})\Big )\Big ( c + di\Big ) &=\Big (a_{n+r} + a_{n-r} \Big )c- \Big (b_{n+r} - b_{n-r} \Big )d + i \Big ( (a_{n+r} + a_{n-r})d + (b_{n+r}+b_{n-r})c \Big )\end{align*}

Real part:

\begin{align}
-\frac{d}{dt}b_{n} &= -\Big (a_{n+1} -2a_{n} + a_{n-1} \Big ) - (a_{n}^2 + b_{n}^2) \sum\limits_{r \in I_{n}\setminus \{0 \}}\Big (a_{n+r} + a_{n-r} \Big )c- \Big (b_{n+r} - b_{n-r} \Big )d\\
\frac{d}{dt}b_{n} &= \Big (a_{n+1} -2a_{n} + a_{n-1} \Big ) + (a_{n}^2 + b_{n}^2) \sum\limits_{r \in I_{n}\setminus \{0 \}}\Big (a_{n+r} + a_{n-r} \Big )c- \Big (b_{n+r} - b_{n-r} \Big )d
\end{align}

Imaginary part:

\begin{align}
\frac{d}{dt}a_{n} &= -\Big ( b_{n+1} -2b_{n} + b_{n-1}\Big ) - \Big (a_{n}^{2} + b_{n}^{2} \Big )\sum\limits_{r \in I_{n}\setminus \{0 \}} (a_{n+r} + a_{n-r})d + (b_{n+r}+b_{n-r})c
\end{align}

Define $U = \begin{pmatrix}REAL \\ IMAGINARY\end{pmatrix} = \begin{pmatrix}\vdots\\b_{n}\\\vdots \\ a_{n} \\ \vdots\end{pmatrix}\in \mathbb{R}^{4N}$

$$M = \begin{bmatrix} M_1 & M_2 \\ M_3 & M_4\end{bmatrix}$$ where $M_{1} = 0\cdot I_{2N}$, $M_{2} = diag(-2)$

Mon, 05 Feb 2024 06:33 GMT