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Darboux transformation, exact solutions and conservation laws for the reverse space-time Fokas–Lenells equation

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Abstract

In this paper, we firstly deduce a reverse space-time Fokas–Lenells equation which can be derived from a rather simple but extremely important symmetry reduction of corresponding local equation. Next, the determinant representations of one-fold Darboux transformation and N-fold Darboux transformation are expressed in detail by special eigenfunctions of spectral problem. Depending on zero seed solution and nonzero seed solution, exact solutions, including bright soliton solutions, kink solutions, periodic solutions, breather solutions, rogue wave solutions and several types of mixed soliton solutions, can be presented. Furthermore, the dynamical behaviors are discussed through some figures. It should be mentioned that the solutions of nonlocal Fokas–Lenells equation possess new characteristics different from the ones of local case. Besides, we also demonstrate the integrability by providing infinitely many conservation laws. The above results provide an alternative possibility to understand physical phenomena in the field of nonlinear optics and related fields.

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Acknowledgements

This work has been supported by the National Natural Science Foundation of China under Grant No. 11771111.

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  1. School of Mathematics, Harbin Institute of Technology, Harbin, 150001, People’s Republic of China

    Jiang-Yan Song, Yu Xiao & Chi-Ping Zhang

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  1. Jiang-Yan Song
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  2. Yu Xiao
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Correspondence to Yu Xiao.

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This work has been supported by the National Natural Science Foundation of China (NNSFC) under Grant No. 11771111.

Appendices

Appendix A

From Eq. (2.5a), comparing the coefficients of \(\lambda ^{j}, j=3, 2, 1, 0, -1\), we can get

$$\begin{aligned}&\lambda ^{3}: b_{1}=0, c_{1}=0,\\&\quad \lambda ^{2}: 2\,i\,b_{0}+a_{1}\,q_{x}-d_{1}\\&\quad q^{[1]}_{x}=0, -2\,i\,c_{0}+d_{1}\\&\quad q_{x}(-x,-t)-a_{1}\,q^{[1]}_{x}(-x,-t)=0,\\&\quad \lambda ^{1}: a_{1x}+b_{0}\,q_{x}(-x,-t)-c_{0}\\&\quad q^{[1]}_{x}=0, 2\,i\,b_{-1}+a_{0}\,q_{x}-d_{0}\,q^{[1]}_{x}=0,\\&\quad -2\,i\,c_{-1}+d_{0}\,q_{x}(-x,-t)-a_{0}\\&\quad q^{[1]}_{x}(-x,-t)=0, d_{1x}+c_{0}\,q_{x}\\&\quad -b_{0}\,q^{[1]}_{x}(-x,-t)=0,\\&\quad \lambda ^{0}: a_{0x}+b_{-1}\,q_{x}(-x,-t)-c_{-1}\,q^{[1]}_{x}=0, \\&\quad b_{0x}+a_{-1}\,q_{x}-d_{-1}\,q^{[1]}_{x}=0,\\&\quad c_{0x}+d_{-1}\,q_{x}(-x,-t)-a_{-1}\,q^{[1]}_{x}(-x,-t)=0, \\&\quad d_{0x}+c_{-1}\,q_{x}-b_{-1}\,q^{[1]}_{x}(-x,-t)=0,\\&\quad \lambda ^{-1}: a_{-1x}=0, b_{-1x}=0, c_{-1x}=0, d_{-1x}=0. \end{aligned}$$

Similarly, from Eq. (2.5b), comparing the coefficients of \(\lambda ^{j}, j=2, 1, 0, -1, -2, -3\) under the condition \(b_{1}=0, c_{1}=0\), we can get

$$\begin{aligned}&\lambda ^{2}: 2\,i\,b_{0}+a_{1}\,q_{x}-d_{1}\,q^{[1]}_{x}=0, \\&\quad -2\,i\,c_{0}+d_{1}\,q_{x}(-x,-t)\\&\quad -a_{1}\,q^{[1]}_{x}(-x,-t)=0,\\&\quad \lambda ^{1}: d_{0}\,q_{x}(-x,-t)-a_{0}\,q^{[1]}_{x}(-x,-t)=0, \\&\quad a_{0}\,q_{x}-d_{0}\,q^{[1]}_{x}=0,\\&\quad -\frac{1}{2}\,i\,a_{1}\,q(-x,-t)\,q+\frac{1}{2}\,i\,a_{1}\\&\quad q^{[1]}(-x,-t)\,q^{[1]}+a_{1t}+b_{0}\,q_{x}(-x,-t)\\&\quad -c_{0}\,q^{[1]}_{x}=0,\\&\quad \frac{1}{2}\,i\,d_{1}\,q(-x,-t)\,q-\frac{1}{2}\,i\,d_{1}\\&\quad q^{[1]}(-x,-t)\,q^{[1]}+d_{1t}+c_{0}\,q_{x}\\&\quad -b_{0}\,q^{[1]}_{x}(-x,-t)=0,\\&\quad \lambda ^{0}: -\frac{1}{2}\,i\,a_{0}\,q(-x,-t)\,q+\frac{1}{2}\,i\,a_{0}\\&\quad q^{[1]}(-x,-t)\,q^{[1]}+a_{0t}=0,\\&\quad \frac{1}{2}\,i\,d_{0}\,q(-x,-t)\,q-\frac{1}{2}\,i\,d_{0}\\&\quad q^{[1]}(-x,-t)\,q^{[1]}+d_{0t}=0,\\&\quad -2ib_{0}+\frac{1}{2}ia_{1}\,q+\frac{1}{2}ib_{0}q(-x,-t)q\\&\quad -\frac{1}{2}id_{1}q^{[1]}+\frac{1}{2}ib_{0}q^{[1]}(-x,-t)q^{[1]}+b_{0t}\\&\quad +a_{-1}q_{x} -d_{-1}q^{[1]}_{x}=0,\\&\quad 2\,i\,c_{0}-\frac{1}{2}\,i\,d_{1}\,q(-x,-t)\\&\quad -\frac{1}{2}\,i\,c_{0}\,q(-x,-t)\\&\quad q+\frac{1}{2}\,i\,a_{1}\,q^{[1]}(-x,-t)\\&\quad -\frac{1}{2}\,i\,c_{0}\,q^{[1]}(-x,-t)\,q^{[1]}\\&\quad +c_{0t}+d_{-1}\,q_{x}(-x,-t)-a_{-1}\\&\quad q^{[1]}_{x}(-x,-t)=0,\\&\quad \lambda ^{-1}: a_{0}\,q-d_{0}\,q^{[1]}=0, \\&\quad -d_{0}\,q(-x,-t)+a_{0}\,q^{[1]}(-x,-t)=0,\\&\quad -\frac{1}{2}\,i\,b_{0}\,q(-x,-t)-\frac{1}{2}\,i\,a_{-1}\\&\quad q(-x,-t)\,q-\frac{1}{2}\,i\,c_{0}\,q^{[1]}\\&\quad +\frac{1}{2}\,i\,a_{-1}\,q^{[1]}(-x,-t)\,q^{[1]}+a_{-1t}=0,\\&\quad \frac{1}{2}\,i\,c_{0}\,q+\frac{1}{2}\,i\,d_{-1}\,q(-x,-t)\,q\\&\quad +\frac{1}{2}\,i\,b_{0}\,q^{[1]}(-x,-t)-\frac{1}{2}\,i\,d_{-1}\,q^{[1]}(-x,-t)\,q^{[1]}\\&\quad +d_{-1t}=0,\\&\quad \lambda ^{-2}: b_{0}+a_{-1}\,q-d_{-1}\,q^{[1]}=0, \\&\quad c_{0}+d_{-1}\,q(-x,-t)-a_{-1}\,q^{[1]}(-x,-t)=0,\\&\quad \lambda ^{-3}: b_{-1}=0, c_{-1}=0. \end{aligned}$$

Appendix B

The \(a_{1}, d_{1}, b_{0}, c_{0}\) for one-fold Darboux matrix \(T^{[1]}\) yield

$$\begin{aligned} a_{1}= & {} \frac{\left| \begin{array}{cc} -\lambda _{1}^{-1}\,\varphi _{1}(x,t) &{}\varphi _{1}(-x,-t)\\ -\lambda _{2}^{-1}\,\varphi _{2}(x,t) &{}\varphi _{2}(-x,-t)\\ \end{array} \right| }{\left| \begin{array}{cc} \lambda _{1}\,\varphi _{1}(x,t) &{}\varphi _{1}(-x,-t)\\ \lambda _{2}\,\varphi _{2}(x,t) &{}\varphi _{2}(-x,-t)\\ \end{array} \right| },\\ d_{1}= & {} \frac{\left| \begin{array}{cc} -\lambda _{1}^{-1}\,\varphi _{1}(-x,-t) &{}\varphi _{1}(x,t)\\ -\lambda _{2}^{-1}\,\varphi _{2}(-x,-t) &{}\varphi _{2}(x,t)\\ \end{array} \right| }{\left| \begin{array}{cc} \lambda _{1}\,\varphi _{1}(-x,-t) &{}\varphi _{1}(x,t)\\ \lambda _{2}\,\varphi _{2}(-x,-t) &{}\varphi _{2}(x,t)\\ \end{array} \right| },\\ b_{0}= & {} \frac{\left| \begin{array}{cc} \lambda _{1}\,\varphi _{1}(x,t) &{}-\lambda _{1}^{-1}\,\varphi _{1}(x,t)\\ \lambda _{2}\,\varphi _{2}(x,t) &{}-\lambda _{2}^{-1}\,\varphi _{2}(x,t)\\ \end{array} \right| }{\left| \begin{array}{cc} \lambda _{1}\,\varphi _{1}(x,t) &{}\varphi _{1}(-x,-t)\\ \lambda _{2}\,\varphi _{2}(x,t) &{}\varphi _{2}(-x,-t)\\ \end{array} \right| },\\ c_{0}= & {} \frac{\left| \begin{array}{cc} \lambda _{1}\,\varphi _{1}(-x,-t) &{}-\lambda _{1}^{-1}\,\varphi _{1}(-x,-t)\\ \lambda _{2}\,\varphi _{2}(-x,-t) &{}-\lambda _{2}^{-1}\,\varphi _{2}(-x,-t)\\ \end{array} \right| }{\left| \begin{array}{cc} \lambda _{1}\,\varphi _{1}(-x,-t) &{}\varphi _{1}(x,t)\\ \lambda _{2}\,\varphi _{2}(-x,-t) &{}\varphi _{2}(x,t)\\ \end{array} \right| }. \end{aligned}$$

In other words, we can deduce

$$\begin{aligned} T^{[1]}_{11}= & {} \frac{\left| \begin{array}{ccc} \lambda &{}0 &{}\lambda ^{-1}\\ \lambda _{1}\,\varphi _{1}(x,t) &{}\varphi _{1}(-x,-t) &{}\lambda _{1}^{-1}\,\varphi _{1}(x,t)\\ \lambda _{2}\,\varphi _{2}(x,t) &{}\varphi _{2}(-x,-t) &{}\lambda _{2}^{-1}\,\varphi _{2}(x,t)\\ \end{array} \right| }{\left| \begin{array}{cc} \lambda _{1}\,\varphi _{1}(x,t) &{}\varphi _{1}(-x,-t)\\ \lambda _{2}\,\varphi _{2}(x,t) &{}\varphi _{2}(-x,-t)\\ \end{array} \right| },\\ T^{[1]}_{12}= & {} \frac{\left| \begin{array}{ccc} 0 &{}1 &{}0\\ \lambda _{1}\,\varphi _{1}(x,t) &{}\varphi _{1}(-x,-t) &{}\lambda _{1}^{-1}\,\varphi _{1}(x,t)\\ \lambda _{2}\,\varphi _{2}(x,t) &{}\varphi _{2}(-x,-t) &{}\lambda _{2}^{-1}\,\varphi _{2}(x,t)\\ \end{array} \right| }{\left| \begin{array}{cc} \lambda _{1}\,\varphi _{1}(x,t) &{}\varphi _{1}(-x,-t)\\ \lambda _{2}\,\varphi _{2}(x,t) &{}\varphi _{2}(-x,-t)\\ \end{array} \right| },\\ T^{[1]}_{21}= & {} \frac{\left| \begin{array}{ccc} 0 &{}1 &{}0\\ \lambda _{1}\,\varphi _{1}(-x,-t) &{}\varphi _{1}(x,t) &{}\lambda _{1}^{-1}\,\varphi _{1}(-x,-t)\\ \lambda _{2}\,\varphi _{2}(-x,-t) &{}\varphi _{2}(x,t) &{}\lambda _{2}^{-1}\,\varphi _{2}(-x,-t)\\ \end{array} \right| }{\left| \begin{array}{cc} \lambda _{1}\,\varphi _{1}(-x,-t) &{}\varphi _{1}(x,t)\\ \lambda _{2}\,\varphi _{2}(-x,-t) &{}\varphi _{2}(x,t)\\ \end{array} \right| },\\ T^{[1]}_{22}= & {} \frac{\left| \begin{array}{ccc} \lambda &{}0 &{}\lambda ^{-1}\\ \lambda _{1}\,\varphi _{1}(-x,-t) &{}\varphi _{1}(x,t) &{}\lambda _{1}^{-1}\,\varphi _{1}(-x,-t)\\ \lambda _{2}\,\varphi _{2}(-x,-t) &{}\varphi _{2}(x,t) &{}\lambda _{2}^{-1}\,\varphi _{2}(-x,-t)\\ \end{array} \right| }{\left| \begin{array}{cc} \lambda _{1}\,\varphi _{1}(-x,-t) &{}\varphi _{1}(x,t)\\ \lambda _{2}\,\varphi _{2}(-x,-t) &{}\varphi _{2}(x,t)\\ \end{array} \right| }. \end{aligned}$$

Appendix C

The \(W^{[N]}, \widetilde{T^{[N]}_{11}}, \widetilde{T^{[N]}_{12}}, \widetilde{W^{[N]}}, \widetilde{T^{[N]}_{21}}, \widetilde{T^{[N]}_{22}}\) for N-fold Darboux matrix \(T^{[N]}\) read

$$\begin{aligned}&W^{[N]}=\left| \begin{array}{ccccccc} \lambda _{1}^{N}\,\varphi _{1}(x,t) &{}\lambda _{1}^{N-1}\,\varphi _{1}(-x,-t) &{}\cdots &{}\lambda _{1}^{-(N-2)}\,\varphi _{1}(x,t) &{}\lambda _{1}^{-(N-1)}\,\varphi _{1}(-x,-t)\\ \lambda _{2}^{N}\,\varphi _{2}(x,t) &{}\lambda _{2}^{N-1}\,\varphi _{2}(-x,-t) &{}\cdots &{}\lambda _{2}^{-(N-2)}\,\varphi _{2}(x,t) &{}\lambda _{2}^{-(N-1)}\,\varphi _{2}(-x,-t)\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \\ \lambda _{2N-1}^{N}\,\varphi _{2N-1}(x,t) &{}\lambda _{2N-1}^{N-1}\,\varphi _{2N-1}(-x,-t) &{}\cdots &{}\lambda _{2N-1}^{-(N-2)}\,\varphi _{2N-1}(x,t) &{}\lambda _{2N-1}^{-(N-1)}\,\varphi _{2N-1}(-x,-t)\\ \lambda _{2N}^{N}\,\varphi _{2N}(x,t) &{}\lambda _{2N}^{N-1}\,\varphi _{2N}(-x,-t) &{}\cdots &{}\lambda _{2N}^{-(N-2)}\,\varphi _{2N}(x,t) &{}\lambda _{2N}^{-(N-1)}\,\varphi _{2N}(-x,-t)\\ \end{array} \right| ,\\&\widetilde{T^{[N]}_{11}}=\left| \begin{array}{cccccccc} \lambda ^{N} &{}0 &{}\cdots &{}0 &{}\lambda ^{-N}\\ \lambda _{1}^{N}\,\varphi _{1}(x,t) &{}\lambda _{1}^{N-1}\,\varphi _{1}(-x,-t) &{}\cdots &{}\lambda _{1}^{-(N-1)}\,\varphi _{1}(-x,-t) &{}\lambda _{1}^{-N}\,\varphi _{1}(x,t)\\ \lambda _{2}^{N}\,\varphi _{2}(x,t) &{}\lambda _{2}^{N-1}\,\varphi _{2}(-x,-t) &{}\cdots &{}\lambda _{2}^{-(N-1)}\,\varphi _{2}(-x,-t) &{}\lambda _{2}^{-N}\,\varphi _{2}(x,t)\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \\ \lambda _{2N-1}^{N}\,\varphi _{2N-1}(x,t) &{}\lambda _{2N-1}^{N-1}\,\varphi _{2N-1}(-x,-t) &{}\cdots &{}\lambda _{2N-1}^{-(N-1)}\,\varphi _{2N-1}(-x,-t) &{}\lambda _{2N-1}^{-N}\,\varphi _{2N-1}(x,t)\\ \lambda _{2N}^{N}\,\varphi _{2N}(x,t) &{}\lambda _{2N}^{N-1}\,\varphi _{2N}(-x,-t) &{}\cdots &{}\lambda _{2N}^{-(N-1)}\,\varphi _{2N}(-x,-t) &{}\lambda _{2N}^{-N}\,\varphi _{2N}(x,t)\\ \end{array} \right| ,\\&\widetilde{T^{[N]}_{12}}=\left| \begin{array}{cccccccc} 0 &{}\lambda ^{N-1} &{}\cdots &{}\lambda ^{-(N-1)} &{}0\\ \lambda _{1}^{N}\,\varphi _{1}(x,t) &{}\lambda _{1}^{N-1}\,\varphi _{1}(-x,-t) &{}\cdots &{}\lambda _{1}^{-(N-1)}\,\varphi _{1}(-x,-t) &{}\lambda _{1}^{-N}\,\varphi _{1}(x,t)\\ \lambda _{2}^{N}\,\varphi _{2}(x,t) &{}\lambda _{2}^{N-1}\,\varphi _{2}(-x,-t) &{}\cdots &{}\lambda _{2}^{-(N-1)}\,\varphi _{2}(-x,-t) &{}\lambda _{2}^{-N}\,\varphi _{2}(x,t)\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \\ \lambda _{2N-1}^{N}\,\varphi _{2N-1}(x,t) &{}\lambda _{2N-1}^{N-1}\,\varphi _{2N-1}(-x,-t) &{}\cdots &{}\lambda _{2N-1}^{-(N-1)}\,\varphi _{2N-1}(-x,-t) &{}\lambda _{2N-1}^{-N}\,\varphi _{2N-1}(x,t)\\ \lambda _{2N}^{N}\,\varphi _{2N}(x,t) &{}\lambda _{2N}^{N-1}\,\varphi _{2N}(-x,-t) &{}\cdots &{}\lambda _{2N}^{-(N-1)}\,\varphi _{2N}(-x,-t) &{}\lambda _{2N}^{-N}\,\varphi _{2N}(x,t)\\ \end{array} \right| ,\\&\widetilde{W^{[N]}}=\left| \begin{array}{ccccccc} \lambda _{1}^{N}\,\varphi _{1}(-x,-t) &{}\lambda _{1}^{N-1}\,\varphi _{1}(x,t) &{}\cdots &{}\lambda _{1}^{-(N-2)}\,\varphi _{1}(-x,-t) &{}\lambda _{1}^{-(N-1)}\,\varphi _{1}(x,t)\\ \lambda _{2}^{N}\,\varphi _{2}(-x,-t) &{}\lambda _{2}^{N-1}\,\varphi _{2}(x,t) &{}\cdots &{}\lambda _{2}^{-(N-2)}\,\varphi _{2}(-x,-t) &{}\lambda _{2}^{-(N-1)}\,\varphi _{2}(x,t)\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \\ \lambda _{2N-1}^{N}\,\varphi _{2N-1}(-x,-t) &{}\lambda _{2N-1}^{N-1}\,\varphi _{2N-1}(x,t) &{}\cdots &{}\lambda _{2N-1}^{-(N-2)}\,\varphi _{2N-1}(-x,-t) &{}\lambda _{2N-1}^{-(N-1)}\,\varphi _{2N-1}(x,t)\\ \lambda _{2N}^{N}\,\varphi _{2N}(-x,-t) &{}\lambda _{2N}^{N-1}\,\varphi _{2N}(x,t) &{}\cdots &{}\lambda _{2N}^{-(N-2)}\,\varphi _{2N}(-x,-t) &{}\lambda _{2N}^{-(N-1)}\,\varphi _{2N}(x,t)\\ \end{array} \right| ,\\ {}&\widetilde{T^{[N]}_{21}}=\left| \begin{array}{ccccccc} 0 &{}\lambda ^{N-1} &{}\cdots &{}\lambda ^{-(N-1)} &{}0\\ \lambda _{1}^{N}\,\varphi _{1}(-x,-t) &{}\lambda _{1}^{N-1}\,\varphi _{1}(x,t) &{}\cdots &{}\lambda _{1}^{-(N-1)}\,\varphi _{1}(x,t) &{}\lambda _{1}^{-N}\,\varphi _{1}(-x,-t)\\ \lambda _{2}^{N}\,\varphi _{2}(-x,-t) &{}\lambda _{2}^{N-1}\,\varphi _{2}(x,t) &{}\cdots &{}\lambda _{2}^{-(N-1)}\,\varphi _{2}(x,t) &{}\lambda _{2}^{-N}\,\varphi _{2}(-x,-t)\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \\ \lambda _{2N-1}^{N}\,\varphi _{2N-1}(-x,-t) &{}\lambda _{2N-1}^{N-1}\,\varphi _{2N-1}(x,t) &{}\cdots &{}\lambda _{2N-1}^{-(N-1)}\,\varphi _{2N-1}(x,t) &{}\lambda _{2N-1}^{-N}\,\varphi _{2N-1}(-x,-t)\\ \lambda _{2N}^{N}\,\varphi _{2N}(-x,-t) &{}\lambda _{2N}^{N-1}\,\varphi _{2N}(x,t) &{}\cdots &{}\lambda _{2N}^{-(N-1)}\,\varphi _{2N}(x,t) &{}\lambda _{2N}^{-N}\,\varphi _{2N}(-x,-t)\\ \end{array} \right| ,\\&\widetilde{T^{[N]}_{22}}=\left| \begin{array}{ccccccc} \lambda ^{N} &{}0 &{}\cdots &{}0 &{}\lambda ^{-N}\\ \lambda _{1}^{N}\,\varphi _{1}(-x,-t) &{}\lambda _{1}^{N-1}\,\varphi _{1}(x,t) &{}\cdots &{}\lambda _{1}^{-(N-1)}\,\varphi _{1}(x,t) &{}\lambda _{1}^{-N}\,\varphi _{1}(-x,-t)\\ \lambda _{2}^{N}\,\varphi _{2}(-x,-t) &{}\lambda _{2}^{N-1}\,\varphi _{2}(x,t) &{}\cdots &{}\lambda _{2}^{-(N-1)}\,\varphi _{2}(x,t) &{}\lambda _{2}^{-N}\,\varphi _{2}(-x,-t)\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \\ \lambda _{2N-1}^{N}\,\varphi _{2N-1}(-x,-t) &{}\lambda _{2N-1}^{N-1}\,\varphi _{2N-1}(x,t) &{}\cdots &{}\lambda _{2N-1}^{-(N-1)}\,\varphi _{2N-1}(x,t) &{}\lambda _{2N-1}^{-N}\,\varphi _{2N-1}(-x,-t)\\ \lambda _{2N}^{N}\,\varphi _{2N}(-x,-t) &{}\lambda _{2N}^{N-1}\,\varphi _{2N}(x,t) &{}\cdots &{}\lambda _{2N}^{-(N-1)}\,\varphi _{2N}(x,t) &{}\lambda _{2N}^{-N}\,\varphi _{2N}(-x,-t)\\ \end{array} \right| . \end{aligned}$$

Appendix D

The \(\varOmega _{-(N-1)}, \widetilde{\varOmega _{-(N-1)}}\) for N-soliton solutions (\(q^{[N]}, q^{[N]}(-x, -t)\)) arrive at

$$\begin{aligned}&\varOmega _{-(N-1)}=\left| \begin{array}{ccccccc} \lambda _{1}^{N}\,\varphi _{1}(x,t) &{}\lambda _{1}^{N-1}\,\varphi _{1}(-x,-t) &{}\cdots &{}\lambda _{1}^{-(N-2)}\,\varphi _{1}(x,t) &{}-\lambda _{1}^{-N}\,\varphi _{1}(x,t)\\ \lambda _{2}^{N}\,\varphi _{2}(x,t) &{}\lambda _{2}^{N-1}\,\varphi _{2}(-x,-t) &{}\cdots &{}\lambda _{2}^{-(N-2)}\,\varphi _{2}(x,t) &{}-\lambda _{2}^{-N}\,\varphi _{2}(x,t)\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \\ \lambda _{2N-1}^{N}\,\varphi _{2N-1}(x,t) &{}\lambda _{2N-1}^{N-1}\,\varphi _{2N-1}(-x,-t) &{}\cdots &{}\lambda _{2N-1}^{-(N-2)}\,\varphi _{2N-1}(x,t) &{}-\lambda _{2N-1}^{-N}\,\varphi _{2N-1}(x,t)\\ \lambda _{2N}^{N}\,\varphi _{2N}(x,t) &{}\lambda _{2N}^{N-1}\,\varphi _{2N}(-x,-t) &{}\cdots &{}\lambda _{2N}^{-(N-2)}\,\varphi _{2N}(x,t) &{}-\lambda _{2N}^{-N}\,\varphi _{2N}(x,t)\\ \end{array} \right| ,\\&\widetilde{\varOmega _{-(N-1)}}=\left| \begin{array}{ccccccc} \lambda _{1}^{N}\,\varphi _{1}(-x,-t) &{}\lambda _{1}^{N-1}\,\varphi _{1}(x,t) &{}\cdots &{}\lambda _{1}^{-(N-2)}\,\varphi _{1}(-x,-t) &{}-\lambda _{1}^{-N}\,\varphi _{1}(-x,-t)\\ \lambda _{2}^{N}\,\varphi _{2}(-x,-t) &{}\lambda _{2}^{N-1}\,\varphi _{2}(x,t) &{}\cdots &{}\lambda _{2}^{-(N-2)}\,\varphi _{2}(-x,-t) &{}-\lambda _{2}^{-N}\,\varphi _{2}(-x,-t)\\ \vdots &{}\vdots &{}\vdots &{}\vdots &{}\vdots \\ \lambda _{2N-1}^{N}\,\varphi _{2N-1}(-x,-t) &{}\lambda _{2N-1}^{N-1}\,\varphi _{2N-1}(x,t) &{}\cdots &{}\lambda _{2N-1}^{-(N-2)}\,\varphi _{2N-1}(-x,-t) &{}-\lambda _{2N-1}^{-N}\,\varphi _{2N-1}(-x,-t)\\ \lambda _{2N}^{N}\,\varphi _{2N}(-x,-t) &{}\lambda _{2N}^{N-1}\,\varphi _{2N}(x,t) &{}\cdots &{}\lambda _{2N}^{-(N-2)}\,\varphi _{2N}(-x,-t) &{}-\lambda _{2N}^{-N}\,\varphi _{2N}(-x,-t)\\ \end{array} \right| . \end{aligned}$$

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Song, JY., Xiao, Y. & Zhang, CP. Darboux transformation, exact solutions and conservation laws for the reverse space-time Fokas–Lenells equation. Nonlinear Dyn 107, 3805–3818 (2022). https://doi.org/10.1007/s11071-021-07170-z

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