Integrable Variants of the Toda Lattice

Abstract

By introducing bilinear operators of trigonometric type, we propose several novel integrable variants of the famous Toda lattice, two of which can be regarded as integrable discretizations of the Kadomtsev–Petviashvili equation—a universal model describing weakly nonlinear waves in media with dispersion of velocity. We also demonstrate that two one-dimensional reductions of these variants can approximate the nonlinear Schrödinger equation and a generalized nonlinear Schrödinger equation well. It turns out that these equations admit meaningful solutions including solitons, breathers, lumps and rogue waves, which are expressed in terms of explicit and closed forms. In particular, it seems to be the first time that rogue wave solutions have been obtained for Toda-type equations. Furthermore, g-periodic wave solutions are also produced in terms of Riemann theta function. An approximation solution of the three-periodic wave is successfully carried out by using a deep neural network. The introduction of trigonometric-type bilinear operators is also efficient in generating new variants together with rich properties for some other integrable equations.

Appendices

Appendix 1

Appendix 2: From Bilinear Equations to Nonlinear Equations

According to the definition of bilinear operators, the bilinear equation (1.4) can be expressed as

$$\begin{aligned} \bigl ( \frac{\partial ^2}{\partial x^2} f_n \bigr ) f_n - \bigl ( \frac{\partial }{\partial x} f_n \bigr )^2 - \bigl ( \frac{\partial ^2}{\partial t^2} f_n \bigr ) f_n + \bigl ( \frac{\partial }{\partial t} f_n \bigr )^2 - 2\sin ^2\bigl (\frac{1}{2} D_n\bigr ) f_n \cdot f_n = 0, \end{aligned}$$

which leads to

$$\begin{aligned} v_n = \frac{2\sin ^2\bigl (\frac{1}{2} D_n\bigr ) f_n \cdot f_n}{f_n^2} = \frac{(1-\cos {(D_n)})f_n \cdot f_n}{f_n^2}. \end{aligned}$$

Then we have

$$\begin{aligned} \ln {(1-v_n)} = \ln {\frac{\cos {(D_n)} f_n \cdot f_n}{f_n^2}} = \ln {(\cos {(D_n)} f_n \cdot f_n)} - 2 \ln {f_n}. \end{aligned}$$

It follows from the formula (1.188) in (Hirota 2004), $2 \cosh {\bigl (\delta \frac{\partial }{\partial z} \bigr )} \ln {f(z)} = \ln {(\cosh {(\delta D_z)} f(z) \cdot f(z))}$, that

$$\begin{aligned} \ln {(\cos {(\delta D_n)} f_n \cdot f_n)} = 2 \cos {\bigl (\delta \frac{\partial }{\partial n} \bigr )} \ln {f_n}. \end{aligned}$$

In fact, by expanding these two equations with respect to $\delta$ and comparing the terms in powers of $\delta$, it can be observed that they are completely identical. It turns out that

$$\begin{aligned} \ln {(1-v_n)} = \bigl ( 2 \cos {\bigl (\frac{\partial }{\partial n} \bigr )} - 2 \bigr ) \ln {f_n} = -4 \sin ^2\bigl (\frac{1}{2} \frac{\partial }{\partial n} \bigr ) \ln {f_n}, \end{aligned}$$

which then yields

$$\begin{aligned} \bigl ( \frac{\partial ^2}{\partial x^2} - \frac{\partial ^2}{\partial t^2} \bigr ) \ln {(1-v_n)} = -4 \bigl ( \frac{\partial ^2}{\partial x^2} - \frac{\partial ^2}{\partial t^2} \bigr ) \sin ^2\bigl (\frac{1}{2} \frac{\partial }{\partial n} \bigr ) \ln {f_n} = -4 \sin ^2\bigl (\frac{1}{2} \frac{\partial }{\partial n} \bigr ) v_n. \end{aligned}$$

Therefore, we derive the nonlinear equation (1.9) from the bilinear equation (1.4), and the other two nonlinear equations (1.10) and (1.11) can be obtained similarly.

Appendix 3: Proof of the Theorem 2.1

For computational simplicity, we express the Grammian determinant solution (2.1) by means of a Pfaffian

$$\begin{aligned} f_n&= (1,2,\ldots ,N,N',\ldots ,2',1')_n,\\ (j,k')_n&= \delta _{jk} + \frac{1}{p_j + q_j + p_k^*+ q_k^*} e^{\eta _j + \eta _k^*},\\ (j,k)_n&= (j',k')_n=0. \end{aligned}$$

Introduce the linear differential operators $L_\pm$ defined by

$$\begin{aligned} L_\pm =\frac{\partial }{\partial x} \pm \frac{\partial }{\partial t}, \end{aligned}$$

(B.1)

from which, the bilinear equation (1.4) can be written as

$$\begin{aligned} (L_+L_-f_n)f_n - (L_+f_n)(L_-f_n) + f_{n+\textrm{i}}f_{n-\textrm{i}} - f_n^2=0. \end{aligned}$$

(B.2)

The derivatives with respect to x and t and shifts in n of the Pfaffian entry $(j,k')_n$ are as follows

$$\begin{aligned} L_+ (j,k')_n&= e^{\eta _j + \eta _k^*} = (d_n,d_n',j,k')_n, \\ L_- (j,k')_n&= e^{\eta _j + \eta _k^*+ \textrm{i}r_j - \textrm{i}r_k^*} = (d_{n-\textrm{i}},d_{n+\textrm{i}}',j,k')_n, \\ L_+L_- (j,k')_n&=( -\textrm{i}e^{\textrm{i}r_j} + \textrm{i}e^{-\textrm{i}r_k^*} ) e^{\eta _j + \eta _k^*} = -\textrm{i}(d_n,d_{n+\textrm{i}}',j,k')_n + \textrm{i}(d_{n-\textrm{i}},d_{n}',j,k')_n, \\ (j,k')_{n+\textrm{i}}&= \delta _{jk} + \frac{1}{p_j + q_j + p_k^*+ q_k^*} e^{\eta _j + \eta _k^*} + \textrm{i}e^{\eta _j + \eta _k^*+\textrm{i}r_j} \\&= (j,k')_n + \textrm{i}(d_n,d_{n+\textrm{i}}',j,k')_n, \\ (j,k')_{n-\textrm{i}}&= \delta _{jk} + \frac{1}{p_j + q_j + p_k^*+ q_k^*} e^{\eta _j + \eta _k^*} - \textrm{i}e^{\eta _j + \eta _k^*-\textrm{i}r_k^*} \\&= (j,k')_n - \textrm{i}(d_{n-\textrm{i}},d_{n}',j,k')_n, \end{aligned}$$

where

$$\begin{aligned} (d_m',j)_n= & e^{p_j x + q_j t +r_j m + \eta _j^0}, \quad (d_m,k')_n=e^{p_k^* x + q_k^* t +r_k^* m + (\eta _k^0)^*}, \\ (d_l,d_m)_n= & (d_l',d_m')_n=(d_l,d_m')_n=0, \quad l,m \in \{ z + \epsilon \textrm{i} | z \in \mathbb {Z}, \, \epsilon = 0, \pm 1 \}. \end{aligned}$$

It should be noted that $d_{n-\textrm{i}}$ and $d_{n+\textrm{i}}'$ are merely two notations defined for the sake of convenience in expression, and they can be replaced with other notations that do not involve the imaginary unit “i". Therefore, rewriting $f_n=(\bullet )_n$, we obtain

$$\begin{aligned} L_+ f_n&= (d_n,d_n',\bullet )_n, \\ L_- f_n&= (d_{n-\textrm{i}},d_{n+\textrm{i}}',\bullet )_n, \\ L_+L_- f_n&= -\textrm{i}(d_n,d_{n+\textrm{i}}',\bullet )_n + \textrm{i}(d_{n-\textrm{i}},d_{n}',\bullet )_n+(d_n,d_n',d_{n-\textrm{i}},d_{n+\textrm{i}}',\bullet )_n, \\ f_{n+\textrm{i}}&=(\bullet )_n + \textrm{i}(d_n,d_{n+\textrm{i}}',\bullet )_n, \\ f_{n-\textrm{i}}&=(\bullet )_n - \textrm{i}(d_{n-\textrm{i}},d_{n}',\bullet )_n. \end{aligned}$$

By employing the above equations, (B.2) is expressed as

$$\begin{aligned}&(d_n,d_n',d_{n-\textrm{i}},d_{n+\textrm{i}}',\bullet )_n (\bullet )_n - (d_n,d_n',\bullet )_n (d_{n-\textrm{i}},d_{n+\textrm{i}}',\bullet )_n \\&+ (d_n,d_{n+\textrm{i}}',\bullet )_n(d_{n-\textrm{i}},d_{n}',\bullet )_n = 0, \end{aligned}$$

which is nothing but the Pfaffian identity, thus the conclusion follows. We remark that based on the relationship between Pfaffians and determinants, each term in this Pfaffian identity can be expressed by a (bordered) determinant, and thus this Pfaffian identity essentially corresponds to the Jacobi identity for determinants (Hirota 2004).

Appendix 4: From the Variant 1DTL to the (generalized) NLS Equation

Let us derive the NLS equation and a generalized NLS equation from the equation (2.9). Hereafter, we write $q_n(t)$ as $q_n$ for simplicity. Firstly, rewrite Eq. (2.9) as

$$\begin{aligned} \left[ \left( \frac{\textrm{d}^2}{\textrm{d}t^2} q_n \right) q_n - \left( \frac{\textrm{d}}{\textrm{d}t} q_n\right) ^2 \right] q_{n-\textrm{i}} = q_{n+\textrm{i}} q_{n} q_{n-\textrm{i}} - q_{n}^3. \end{aligned}$$

(C.1)

It is noted that $q_n=\frac{f_{n+\textrm{i}}}{f_n}$ and $f_n$ is a real-valued function of the real variables t and n, which gives $q_n^*=\frac{1}{q_{n-\textrm{i}}}$. Multiplying both sides of Eq. (C.1) by $\frac{1}{q_{n-\textrm{i}}}$ yields

$$\begin{aligned} \left( \frac{\textrm{d}^2}{\textrm{d}t^2} q_n \right) q_n - \left( \frac{\textrm{d}}{\textrm{d}t} q_n\right) ^2 = q_{n+\textrm{i}} q_{n} - q_{n}^3 q_n^*. \end{aligned}$$

(C.2)

Introduce a positive parameter $h \, (0<h\le 1)$ and consider the variable transformations

$$\begin{aligned} t=\frac{\tau }{h}, \quad n=\frac{\xi }{h^2}, \quad q_n=h \, q(\tau ,\xi ). \end{aligned}$$

(C.3)

Then (C.2) is transformed into

$$\begin{aligned} h^4 q_{\tau \tau }q - h^4 q_\tau ^2 = h^2 (q + \textrm{i} h^2 q_\xi + \mathcal {O}(h^4) )q - h^4q^3q^*, \end{aligned}$$

which is equivalent to

$$\begin{aligned} \textrm{i} h^2 q_\xi - h^2 q_{\tau \tau } - h^2q|q|^2 + q + h^2 \frac{q_\tau ^2}{q} + \mathcal {O}(h^4) = 0. \end{aligned}$$

Leaving the $\mathcal {O}(h^2)$ terms as higher-order ones, this leads to

$$\begin{aligned} \textrm{i} h^2 q_\xi - h^2 q_{\tau \tau } - h^2q|q|^2 + q + h^2 \frac{q_\tau ^2}{q} = 0 \end{aligned}$$

or, with a suitable rescaling of the dependent and independent variables ($\xi \rightarrow h^2\xi , \tau \rightarrow h\tau , q \rightarrow h^{-1}q$),

$$\begin{aligned} \textrm{i} q_\xi - q_{\tau \tau } - q|q|^2 + q + \frac{q_\tau ^2}{q} = 0 \end{aligned}$$

which is a generalized NLS equation that describes wave propagation in fluids and plasmas with sharp boundaries and dissipation (see Stenflo 1988) and Kivshar and Malomed 1989, page 769).

Using the relationship between $q_n$ and $f_n$ again, we obtain $q_n^{-1}=(q_{n-\textrm{i}})^*$. Multiplying both sides of Eq. (C.2) by $q_n^{-1}$ gives

$$\begin{aligned} \frac{\textrm{d}^2}{\textrm{d}t^2} q_n - \left( \frac{\textrm{d}}{\textrm{d}t} q_n\right) ^2 \left( q_{n-\textrm{i}}\right) ^*= q_{n+\textrm{i}} - q_{n}^2 q_n^*. \end{aligned}$$

Then considering the variable transformations given by (C.3) yields

$$\begin{aligned} \textrm{i} h^2 q_\xi - h^2 q_{\tau \tau } - h^2q|q|^2 + q + \mathcal {O}(h^4) = 0. \end{aligned}$$

Retain terms up to $\mathcal {O}(h^2)$, and then we have

$$\begin{aligned} \textrm{i} h^2 q_\xi - h^2 q_{\tau \tau } - h^2q|q|^2 + q = 0. \end{aligned}$$

(C.4)

Furthermore, applying the transformations

$$\begin{aligned} \xi \rightarrow - h^2\xi , \quad \tau \rightarrow h\tau , \quad q \rightarrow \sqrt{2} h^{-1}q e^{-\textrm{i}\xi }, \end{aligned}$$

then the above Eq. (C.4) reduces to

$$\begin{aligned} \textrm{i} q_\xi + q_{\tau \tau } + 2|q|^2q=0, \end{aligned}$$

which is nothing but the focusing NLS equation in standard dimensionless form.

Finally, let us give a remark. Recall that the corresponding bilinear equation for Eq. (2.9) is

$$\begin{aligned} \bigl ( D_t^2 + 4 \sin ^2\bigl (\frac{1}{2} D_n\bigr ) \bigr ) f_n \cdot f_n =0. \end{aligned}$$

By contrast, the nonlinear equation that corresponds to another 1D variant

$$\begin{aligned} \bigl ( D_t^2 - 4 \sin ^2\bigl (\frac{1}{2} D_n\bigr ) \bigr ) f_n \cdot f_n =0, \end{aligned}$$

can approximate another form of the generalized NLS equation, as well as the defocusing NLS equation in a similar way.

Appendix 5: Proof of the Theorem 2.5

Firstly, we demonstrate that the determinant $f_n=|\tilde{F}|$ satisfies the bilinear equation (1.4). According to the derivative rule of determinants, we have

$$\begin{aligned} L_+ f_n = \sum _{j,k=1}^N (L_+ \tilde{f}_{jk}) \Delta _{jk} = \sum _{j,k=1}^N \lambda _j^{-1}\delta _{jk} \Delta _{jk} = \sum _{k=1}^N \lambda _k^{-1} \Delta _{kk}, \end{aligned}$$

where $L_+$ is defined by (B.1) and $\Delta _{jk}$ denotes the cofactor of $\tilde{f}_{jk}$ in $|\tilde{F}|$. From the expansion formulae for determinant $|\tilde{F}|$, $\sum _{j=1}^N \tilde{f}_{jk} \Delta _{jk} = |\tilde{F}| \, (k=1,2,\ldots ,N)$ and $\sum _{k=1}^N \tilde{f}_{jk} \Delta _{jk} = |\tilde{F}| \, (j=1,2,\ldots ,N)$, the following equations are derived immediately

$$\begin{aligned} \sum _{j,k=1}^N \lambda _k^{-1} \tilde{f}_{jk} \Delta _{jk}&= \sum _{k=1}^N \lambda _k^{-1} |\tilde{F}|, \\ \sum _{j,k=1}^N \lambda _j^{-1} \tilde{f}_{jk} \Delta _{jk}&= \sum _{j=1}^N \lambda _j^{-1} |\tilde{F}|, \end{aligned}$$

which lead to

$$\begin{aligned} \sum _{j,k=1}^N (\lambda _j^{-1} - \lambda _k^{-1}) \tilde{f}_{jk} \Delta _{jk} =0 \end{aligned}$$

or equivalently

$$\begin{aligned} \sum _{j \ne k} \lambda _k^{-1} \Delta _{jk} =0. \end{aligned}$$

Then $L_+ f_n$ can be written as

$$\begin{aligned} L_+ f_n = \sum _{j,k=1}^N \lambda _k^{-1} \Delta _{jk} = \begin{vmatrix} \tilde{F}&\varvec{-1} \\ \varvec{\lambda ^{-1}}&0 \\ \end{vmatrix}, \end{aligned}$$

where $\varvec{-1}=(-1,-1,\ldots ,-1)^T$ and $\varvec{\lambda ^{-1}}=(\lambda ^{-1}_1,\lambda ^{-1}_2,\ldots ,\lambda ^{-1}_N)$. Similarly, $L_- f_n$ and $L_+ L_- f_n$ can also be written as bordered determinants, that is

$$\begin{aligned} L_- f_n&= \begin{vmatrix} \tilde{F}&\varvec{\lambda } \\ \varvec{-1}&0 \\ \end{vmatrix}, \\ L_+ L_- f_n&= \begin{vmatrix} \tilde{F}&\varvec{\lambda }&\varvec{-1} \\ \varvec{-1}&0&0 \\ \varvec{\lambda ^{-1}}&0&0 \\ \end{vmatrix} - \textrm{i} \begin{vmatrix} \tilde{F}&\varvec{\lambda } \\ \varvec{\lambda ^{-1}}&0 \\ \end{vmatrix} + \textrm{i} \begin{vmatrix} \tilde{F}&\varvec{-1} \\ \varvec{-1}&0 \\ \end{vmatrix}, \end{aligned}$$

where $\varvec{\lambda }=(\lambda _1,\lambda _2,\ldots ,\lambda _N)^T$, and $\varvec{-1}$ represents an N-dimensional row vector or column vector with all its elements equal to $-1$.

Furthermore, based on the row–column transformations of determinants, we have

$$\begin{aligned} f_{n+\textrm{i}}&= \left| (\theta _j + \textrm{i}) \delta _{jk} - \frac{\lambda _j \textrm{i} }{\lambda _j - \lambda _k} (1-\delta _{jk}) \right| \\&= \begin{vmatrix} (\theta _j + \textrm{i}) \delta _{jk} - \frac{\lambda _k \textrm{i} }{\lambda _j - \lambda _k} (1-\delta _{jk})&\varvec{\textrm{i}} \\ \varvec{0}&1 \\ \end{vmatrix} \\&= \begin{vmatrix} \tilde{F}&\varvec{\textrm{i}} \\ \varvec{-1}&1 \\ \end{vmatrix} = |\tilde{F}| - \textrm{i} \begin{vmatrix} \tilde{F}&\varvec{-1} \\ \varvec{-1}&0 \\ \end{vmatrix}, \\ f_{n-\textrm{i}}&= \begin{vmatrix} (\theta _j - \textrm{i}) \delta _{jk} - \frac{\lambda _j \textrm{i} }{\lambda _j - \lambda _k} (1-\delta _{jk})&\varvec{\textrm{i}} \\ \varvec{0}&1 \\ \end{vmatrix} \\&= \begin{vmatrix} \theta _j \delta _{jk} - \frac{\lambda _k \textrm{i} }{\lambda _j - \lambda _k} (1-\delta _{jk})&\varvec{\textrm{i}} \\ \varvec{1}&1 \\ \end{vmatrix} \\&= \begin{vmatrix} \tilde{F}&\varvec{\lambda \textrm{i}} \\ \varvec{\lambda ^{-1}}&1 \\ \end{vmatrix} = |\tilde{F}| + \textrm{i} \begin{vmatrix} \tilde{F}&\varvec{\lambda } \\ \varvec{\lambda ^{-1}}&0 \\ \end{vmatrix}, \end{aligned}$$

where $\varvec{\lambda \textrm{i}}=(\lambda _1 \textrm{i},\lambda _2 \textrm{i},\ldots ,\lambda _N \textrm{i})^T$ and similarly for the other bold notations. Substituting the above equations into the bilinear equation (1.4) or equivalently (B.2), we obtain

$$\begin{aligned} \begin{vmatrix} \tilde{F}&\varvec{\lambda }&\varvec{-1} \\ \varvec{-1}&0&0 \\ \varvec{\lambda ^{-1}}&0&0 \\ \end{vmatrix} |\tilde{F}| - \begin{vmatrix} \tilde{F}&\varvec{-1} \\ \varvec{\lambda ^{-1}}&0 \\ \end{vmatrix} \begin{vmatrix} \tilde{F}&\varvec{\lambda } \\ \varvec{-1}&0 \\ \end{vmatrix} + \begin{vmatrix} \tilde{F}&\varvec{-1} \\ \varvec{-1}&0 \\ \end{vmatrix} \begin{vmatrix} \tilde{F}&\varvec{\lambda } \\ \varvec{\lambda ^{-1}}&0 \\ \end{vmatrix} = 0, \end{aligned}$$

which is nothing but the Jacobi identity, thus the conclusion follows.

Next, we prove the positivity of $f_n$. Extracting $\lambda _j ( 1 \le j \le 2\,M)$ from the j-row of the determinant $|\tilde{F}|$, we obtain

$$\begin{aligned} |\tilde{F}|=\left( \prod _{j=1}^M \lambda _j \lambda _j^*\right) |\hat{F}|, \end{aligned}$$

where $\hat{F}=(\hat{f}_{jk})$ is a $2M \times 2M$ matrix whose elements are given by

$$\begin{aligned} \hat{f}_{jk} =\lambda _j^{-1} \theta _j \delta _{jk} - \frac{\textrm{i} }{\lambda _j - \lambda _k} (1-\delta _{jk}). \end{aligned}$$

Thus, we only need to prove that $|\hat{F}|>0$. For this purpose, following the idea of that in Villarroel (1998, Proposition 3.2), we show that the following results hold.

• (i) $\hat{F}$ has the following structure

  $$\begin{aligned} \hat{F}= \begin{pmatrix} B & C \\ -C^*& B^\dag \\ \end{pmatrix}, \end{aligned}$$

  where $\dag$ denotes the conjugate transpose, and $B=(b_{jk}), C=(c_{jk})$ are $M \times M$ matrices whose elements are defined by

  $$\begin{aligned} b_{jk}&= \lambda _j^{-1} \theta _j \delta _{jk} - \frac{\textrm{i} }{\lambda _j - \lambda _k} (1-\delta _{jk}), \\ c_{jk}&= - \frac{\textrm{i} }{\lambda _j - \lambda _k^*}. \end{aligned}$$

• (ii) $|\hat{F}|$ is real.

• (iii) C is Hermitian, i.e., $C^\dag =C$, and

  $$\begin{aligned} |C| = (-\frac{1}{2})^M \left( \prod _{j=1}^M {\text {Im}}{\lambda _j} \right) ^{-1} \prod _{1\le j < k \le M} \frac{|\lambda _j-\lambda _k|^2}{|\lambda _j-\lambda _k^*|^2}. \end{aligned}$$

• (iv) If ${\text {Im}}\lambda _j >0 \, (j=1,2,\ldots ,M)$, then C is negative definite; on the contrary, if ${\text {Im}}\lambda _j<0 \, (j=1,2,\ldots ,M)$, then C is positive definite.

• (v) $\forall x,t,n \in \mathbb {R}$, $|\hat{F}|>0$.

Obviously, $\lambda _{M+j} = \lambda _j^*\, (j=1,2,\ldots ,M)$ imply that $\hat{F}$ has the structure (i) and $C^\dag =C$. We next note that matrix $\hat{F}$ has the following property

$$\begin{aligned} \hat{F}^\dag = \begin{pmatrix} B^\dag & -C^*\\ C & B \\ \end{pmatrix} = \begin{pmatrix} \varvec{0} & I \\ I & \varvec{0} \\ \end{pmatrix} \begin{pmatrix} B & C \\ -C^*& B^\dag \\ \end{pmatrix} \begin{pmatrix} \varvec{0} & I \\ I & \varvec{0} \\ \end{pmatrix}, \end{aligned}$$

which leads to $|\hat{F}|=|\hat{F}^\dag |$, i.e., (ii) holds.

To prove (iii), it is easy to show that

$$\begin{aligned} |C| = (-\textrm{i})^M |\tilde{C}|, \end{aligned}$$

where $\tilde{C}$ is the Cauchy matrix $\tilde{c}_{jk}=\frac{1}{\lambda _j - \lambda _k^*}$. The determinant of the Cauchy matrix is standard and (iii) follows.

Furthermore, to prove (iv), we know from (iii) that if ${\text {Im}}\lambda _j<0 \; (j=1,2,\ldots ,M)$ then all C’s leading principal minors are positive, that is, C is positive definite; otherwise, if ${\text {Im}}\lambda _j>0 \; (j=1,2,\ldots ,M)$, then the leading principal minors of odd order are negative and the leading principal minors of even order are positive, that is, C is negative definite.

We now prove the main claim (v). Assume that $|\hat{F}|=0$. Then there exists a nonzero vector $(e_1^T,e_2^T)^T \in \mathbb {C}^{2M}$ satisfying $\hat{F}(e_1^T,e_2^T)^T=\varvec{0}$ that is

$$\begin{aligned} B e_1 + C e_2&= \varvec{0}, \\ -C^*e_1 + B^\dag e_2&= \varvec{0}. \end{aligned}$$

Hence, we obtain

$$\begin{aligned} e_1^\dag C^*e_1 + e_2^\dag C e_2 =0. \end{aligned}$$

Since C is positive or negative definite, this yields $(e_1^T,e_2^T)^T=\varvec{0}$ and hence $|\hat{F}| \ne 0$. In addition, we have $|\hat{F}| \rightarrow +\infty$ when $x^2+t^2+n^2\rightarrow +\infty$, which in turn implies the statement.

Appendix 6: Proof of the Theorem 2.6

Let us assume the following form

$$\begin{aligned} \varvec{\eta } = x\varvec{k}+t\varvec{\omega }+n\varvec{l}+m\varvec{j}+\cdots , \end{aligned}$$

and consider Eq. (2.14),

$$\begin{aligned} Ff\cdot f&= \sum _{\varvec{\bar{n}}, \varvec{\tilde{n}} \in \mathbb {Z}^g } F e^{2\pi \textrm{i}\left\langle \varvec{\bar{n}},\varvec{\eta }\right\rangle - \pi \left\langle \varvec{\tau }\varvec{\bar{n}}, \varvec{\bar{n}}\right\rangle } \cdot e^{2\pi \textrm{i}\left\langle \varvec{\tilde{n}},\varvec{\eta }\right\rangle - \pi \left\langle \varvec{\tau } \varvec{\tilde{n}}, \varvec{\tilde{n}}\right\rangle }\\&= \sum _{\varvec{\bar{n}}, \varvec{\hat{n}} \in \mathbb {Z}^g} F e^{2\pi \textrm{i}\left\langle \varvec{\bar{n}},\varvec{\eta }\right\rangle - \pi \left\langle \varvec{\tau }\varvec{\bar{n}}, \varvec{\bar{n}}\right\rangle } \cdot e^{2\pi \textrm{i}\left\langle \varvec{\hat{n}}-\varvec{\bar{n}},\varvec{\eta }\right\rangle - \pi \left\langle \varvec{\tau } (\varvec{\hat{n}}-\varvec{\bar{n}}), \varvec{\hat{n}}-\varvec{\bar{n}}\right\rangle }\\&= \sum _{\varvec{\bar{n}}, \varvec{\hat{n}} \in \mathbb {Z}^g} F\left( 2\pi \textrm{i} \left\langle 2\varvec{\bar{n}}-\varvec{\hat{n}}, \varvec{k}\right\rangle , 2\pi \textrm{i} \left\langle 2\varvec{\bar{n}}-\varvec{\hat{n}},\varvec{\omega }\right\rangle , \cdots \right) \\&\qquad \times e^{2\pi \textrm{i}\left\langle \varvec{\hat{n}},\varvec{\eta }\right\rangle - \pi \left\langle \varvec{\tau }\varvec{\bar{n}}, \varvec{\bar{n}}\right\rangle - \pi \left\langle \varvec{\tau } (\varvec{\hat{n}}-\varvec{\bar{n}}), \varvec{\hat{n}}-\varvec{\bar{n}}\right\rangle }\\&= \sum _{\varvec{\hat{n}} \in \mathbb {Z}^g} \hat{F}(\varvec{\hat{n}}) e^{2\pi \textrm{i}\left\langle \varvec{\hat{n}},\varvec{\eta }\right\rangle }, \end{aligned}$$

where

$$\begin{aligned} \hat{F}(\varvec{\hat{n}}) =\sum _{\varvec{\bar{n}} \in \mathbb {Z}^g} F\left( 2\pi \textrm{i} \left\langle 2\varvec{\bar{n}}-\varvec{\hat{n}}, \varvec{k}\right\rangle , 2\pi \textrm{i} \left\langle 2\varvec{\bar{n}}-\varvec{\hat{n}},\varvec{\omega }\right\rangle ,\cdots \right) e^{-\pi \left\langle \varvec{\tau }\varvec{\bar{n}}, \varvec{\bar{n}}\right\rangle - \pi \left\langle \varvec{\tau } (\varvec{\hat{n}}-\varvec{\bar{n}}), \varvec{\hat{n}}-\varvec{\bar{n}}\right\rangle }. \end{aligned}$$

(E.1)

In Eq. (E.1), we have the following “quasi” periodic relation by shifting summation index $\varvec{\bar{n}} = \varvec{\bar{n}'} + \varvec{\delta }$ (one component is 1 and others are 0 in $\varvec{\delta }$, e.g., $\varvec{\delta }=(0,1,0,0)$)

$$\begin{aligned} \hat{F}(\varvec{\hat{n}})&=\sum _{\varvec{\bar{n}'} \in \mathbb {Z}^g} F\left( 2\pi \textrm{i} \left\langle 2\varvec{\bar{n}'}-\varvec{\hat{n}} + 2\varvec{\delta }, \varvec{k}\right\rangle , \cdots \right) \\&\quad \times e^{-\pi \left( \left\langle \varvec{\tau }(\varvec{\bar{n}'}+\varvec{\delta }), (\varvec{\bar{n}'}+\varvec{\delta })\right\rangle + \left\langle \varvec{\tau } (\varvec{\hat{n}}-\varvec{\bar{n}'}-\varvec{\delta }), \varvec{\hat{n}}-\varvec{\bar{n}'}-\varvec{\delta }\right\rangle \right) } \\&= \hat{F}(\varvec{\hat{n}} - 2\varvec{\delta }) e^{-2\pi \left\langle \varvec{\tau }(\varvec{\hat{n}}-\varvec{\delta }), \varvec{\delta } \right\rangle }. \end{aligned}$$

This implies that if $\hat{F}(\varvec{s})=0$ for all possible g-dimensional vectors $\varvec{s}$ whose components are either 0 or 1, then all other $\hat{F}(\varvec{\hat{n}})$ are also zero.