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Rogue waves on the periodic background in the high-order discrete mKdV equation

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Abstract

We construct new exact solutions for the high-order discrete mKdV (dmKdV) equation. The exact solutions are obtained by using the nonlinearization of spectral problem, one-, and two-fold Darboux transformations. Two families of traveling periodic waves of the high-order dmKdV equation are expressed by the Jacobi dnoidal and cnoidal elliptic functions. Since the dnoidal traveling wave is modulationally stable and the traveling cnoidal wave is modulationally unstable, the rogue waves only generated on the cnoidal wave background while the algebraic solitons propagating on the dnoidal wave background.

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  1. School of Mathematics, Southeast University, Nanjing, 210096, Jiangsu, China

    Yanpei Zhen & Jinbing Chen

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  1. Yanpei Zhen
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Correspondence to Yanpei Zhen.

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Supplementary file 1 (nb 72 KB)

Appendix A.

Appendix A.

Using the formula for addition of Jacobi elliptic function

$$\begin{aligned}{} & {} dn(\xi \pm \alpha ,k)\\{} & {} \quad =\frac{dn(\xi ,k)dn(\alpha ,k)\mp k^2sn(\xi ,k)cn(\xi ,k)sn(\alpha ,k)cn(\alpha ,k)}{1-k^2sn^2(\xi ,k)sn^2(\alpha ,k)}, \end{aligned}$$

substituting (27) into (24), we reach

$$\begin{aligned} \begin{aligned}&-ck^2sn(\xi ,k)cn(\xi ,k)=\omega ^2dn(\xi ,k)-2\omega \\&\quad \times \frac{dn(\xi ,k)dn(\alpha ,k)+k^2sn(\xi ,k)cn(\xi ,k)sn(\alpha ,k)cn(\alpha ,k)}{1-k^2sn^2(\xi ,k)sn^2(\alpha ,k)}\\&\quad -2\omega dn(\xi ,k)F_1+2[1+\frac{sn^2(\alpha ,k)}{cn^2(\alpha ,k)}dn^2(\xi ,k)]\\&\quad \frac{2dn(\xi ,k)dn(\alpha ,k)}{1-k^2sn^2(\xi ,k)sn^2(\alpha ,k)}F_1\\&\quad -2\frac{sn^2(\alpha ,k)}{cn^2(\alpha ,k)}[1+\frac{sn^2(\alpha ,k)}{cn^2(\alpha ,k)}dn^2(\xi ,k)]\\&\quad \times \frac{dn(\xi ,k)dn(\alpha ,k)+k^2sn(\xi ,k)cn(\xi ,k)sn(\alpha ,k)cn(\alpha ,k)}{1-k^2sn^2(\xi ,k)sn^2(\alpha ,k)}\\ {}&\qquad dn(\xi ,k)\\&\quad \times \frac{2dn(\xi ,k)dn(\alpha ,k)}{1-k^2sn^2(\xi ,k)sn^2(\alpha ,k)}. \end{aligned} \end{aligned}$$

Then, we can obtain

$$\begin{aligned}{} & {} -ck^2sn(\xi ,k)cn(\xi ,k)\bigg (cn^4(\alpha ,k)[1\\{} & {} \qquad -k^2sn^2(\xi ,k)sn^2(\alpha ,k)]^2\bigg )\\{} & {} \quad =\omega ^2dn(\xi ,k)\bigg (cn^4(\alpha ,k)[1-k^2sn^2(\xi ,k)sn^2(\alpha ,k)]^2\bigg )\\{} & {} \qquad -2\omega (dn(\xi ,k)dn(\alpha ,k)\\{} & {} \qquad +k^2sn(\xi ,k)cn(\xi ,k)sn(\alpha ,k)cn(\alpha ,k))\\{} & {} \qquad \times \bigg (cn^4(\alpha ,k)[1-k^2sn^2(\xi ,k)sn^2(\alpha ,k)]\bigg )\\{} & {} \qquad -2\omega dn(\xi ,k)F_1\\{} & {} \qquad \times \bigg (cn^4(\alpha ,k)[1-k^2sn^2(\xi ,k)sn^2(\alpha ,k)]^2\bigg )\\{} & {} \qquad +2[cn^2(\alpha ,k)+sn^2(\alpha ,k)dn^2(\xi ,k)]\\{} & {} \qquad 2dn(\xi ,k)dn(\alpha ,k)F_1\\{} & {} \qquad \times \bigg (cn^2(\alpha ,k)[1-k^2sn^2(\xi ,k)sn^2(\alpha ,k)]\bigg )\\{} & {} \qquad -4sn^2(\alpha ,k)cn^2(\alpha ,k)dn^2(\alpha ,k)dn^3(\xi ,k)\\{} & {} \qquad -4k^2sn^3(\alpha ,k)cn^3(\alpha ,k)\\{} & {} \qquad \times dn(\alpha ,k)sn(\xi ,k)cn(\xi ,k)dn^2(\xi ,k)\\{} & {} \qquad -4sn^4(\alpha ,k)dn^2(\alpha ,k)dn^5(\xi ,k)\\{} & {} \qquad -4k^2sn^5(\alpha ,k)cn(\alpha ,k)dn(\alpha ,k)sn(\xi ,k)\\ {}{} & {} \qquad cn(\xi ,k)dn^4(\xi ,k). \end{aligned}$$

Substituting (28) into the above equation and using \(cn^2(\alpha ,k)=1-sn^2(\alpha ,k),dn^2(\alpha ,k)=1-k^2sn^2(\alpha ,k)\), we have

$$\begin{aligned}{} & {} -4k^2sn(\alpha ,k)cn(\alpha ,k)dn(\alpha ,k)sn(\xi ,k)cn(\xi ,k)\\{} & {} \qquad +8k^4sn^3(\alpha ,k)cn(\alpha ,k)\\{} & {} \qquad \times dn(\alpha ,k)sn^3(\xi ,k)cn(\xi ,k)\\{} & {} \qquad -4k^6sn^5(\alpha ,k)cn(\alpha ,k)dn(\alpha ,k)sn^5(\xi ,k)cn(\xi ,k)\\{} & {} \quad =\omega ^2cn^4(\alpha ,k)dn(\xi ,k)-2k^2\omega ^2\\{} & {} \qquad sn^2(\alpha ,k)cn^4(\alpha ,k)sn^2(\xi ,k)dn(\xi ,k)\\{} & {} \qquad +k^4\omega ^2sn^4(\alpha ,k)cn^4(\alpha ,k)sn^4(\xi ,k)dn(\xi ,k)\\{} & {} \qquad -2\omega cn^4(\alpha ,k)dn(\alpha ,k)dn(\xi ,k)\\{} & {} \qquad +2k^2\omega sn^2(\alpha ,k)cn^4(\alpha ,k)dn(\alpha ,k)\\{} & {} \qquad sn^2(\xi ,k)dn(\xi ,k)-2k^2\omega sn(\alpha ,k)\\{} & {} \qquad \times cn^5(\alpha ,k)sn(\xi ,k)cn(\xi ,k)+2k^4\omega \\{} & {} \qquad sn^3(\alpha ,k)cn^5(\alpha ,k)sn^3(\xi ,k)cn(\xi ,k)\\{} & {} \qquad -2\omega cn^4(\alpha ,k)F_1dn(\xi ,k)+4k^2\omega \\{} & {} \qquad sn^2(\alpha ,k)cn^4(\alpha ,k)F_1sn^2(\xi ,k)dn(\xi ,k)\\{} & {} \qquad -2k^4\omega sn^4(\alpha ,k)cn^4(\alpha ,k)F_1sn^4(\xi ,k)dn(\xi ,k)\\{} & {} \qquad +4cn^2(\alpha ,k)dn(\alpha ,k)F_1dn(\xi ,k)\\{} & {} \qquad -8k^2sn^2(\alpha ,k)cn^2(\alpha ,k)dn(\alpha ,k)F_1sn^2(\xi ,k)dn(\xi ,k)\\{} & {} \qquad +4k^4sn^4(\alpha ,k)cn^2(\alpha ,k)\\{} & {} \qquad \times dn(\alpha ,k)F_1sn^4(\xi ,k)dn(\xi ,k)-4sn^2(\alpha ,k)dn(\xi ,k)\\{} & {} \qquad +4k^2sn^4(\alpha ,k)dn(\xi ,k)+4sn^4(\alpha ,k)dn(\xi ,k)\\{} & {} \qquad -4k^2sn^6(\alpha ,k)dn(\xi ,k)\\{} & {} \qquad +4k^2sn^2(\alpha ,k)sn^2(\xi ,k)dn(\xi ,k)\\{} & {} \qquad -4k^4sn^4(\alpha ,k)sn^2(\xi ,k)dn(\xi ,k)\\{} & {} \qquad -4k^2sn^4(\alpha ,k)sn^2(\xi ,k)dn(\xi ,k)\\{} & {} \qquad +4k^4sn^6(\alpha ,k)sn^2(\xi ,k)dn(\xi ,k)\\{} & {} \qquad -4k^2sn^3(\alpha ,k)cn(\alpha ,k)dn(\alpha ,k)sn(\xi ,k)cn(\xi ,k)\\{} & {} \qquad +4k^4sn^3(\alpha ,k)cn(\alpha ,k)dn(\alpha ,k)sn^3(\xi ,k)cn(\xi ,k)\\{} & {} \qquad +4k^2sn^5(\alpha ,k)cn(\alpha ,k)\times dn(\alpha ,k)sn(\xi ,k)cn(\xi ,k)\\{} & {} \qquad -4k^4sn^5(\alpha ,k)cn(\alpha ,k)dn(\alpha ,k)sn^3(\xi ,k)cn(\xi ,k)\\{} & {} \qquad -4sn^4(\alpha ,k)dn(\xi ,k)+8k^2sn^4(\alpha ,k)sn^2(\xi ,k)dn(\xi ,k)\\{} & {} \qquad -4k^4sn^4(\alpha ,k)sn^4(\xi ,k)dn(\xi ,k)+4k^2sn^6(\alpha ,k)dn(\xi ,k)\\{} & {} \qquad -8k^4sn^6(\alpha ,k)sn^2(\xi ,k)dn(\xi ,k)\\{} & {} \qquad +4k^6sn^6(\alpha ,k)sn^4(\xi ,k)dn(\xi ,k)\\{} & {} \qquad -4k^2sn^5(\alpha ,k)cn(\alpha ,k)dn(\alpha ,k)sn(\xi ,k)cn(\xi ,k)\\{} & {} \qquad +8k^4sn^5(\alpha ,k)cn(\alpha ,k)\times dn(\alpha ,k)sn^3(\xi ,k)cn(\xi ,k)\\{} & {} \qquad -4k^6sn^5(\alpha ,k)cn(\alpha ,k)dn(\alpha ,k)sn^5(\xi ,k)cn(\xi ,k). \end{aligned}$$

Collecting the coefficients of \(dn(\xi ,k)\), \(sn^2(\xi ,k)dn(\xi ,k)\), \(sn^4(\xi ,k)dn(\xi ,k)\), \(sn(\xi ,k)cn(\xi ,k)\), \(sn^3(\xi ,k)cn(\xi ,k)\) and \(sn^5(\xi ,k)cn(\xi ,k)\) yield the system of equations

$$\begin{aligned}{} & {} \omega ^2cn^4(\alpha ,k)-2\omega cn^4(\alpha ,k)dn(\alpha ,k)\nonumber \\{} & {} \quad -4sn^2(\alpha ,k)+4k^2sn^4(\alpha ,k)\nonumber \\{} & {} \quad -2\omega cn^4(\alpha ,k)F_1+4cn^2(\alpha ,k)dn(\alpha ,k)F_1=0, \nonumber \\ \end{aligned}$$
(A1)
$$\begin{aligned}{} & {} -2k^2\omega ^2sn^2(\alpha ,k)cn^4(\alpha ,k)\nonumber \\{} & {} \quad +2k^2\omega sn^2(\alpha ,k)cn^4(\alpha ,k)dn(\alpha ,k)\nonumber \\{} & {} \quad +4k^2sn^2(\alpha ,k)-4k^4sn^4(\alpha ,k)+4k^2sn^4(\alpha ,k)\nonumber \\{} & {} \quad -4k^4sn^6(\alpha ,k)\nonumber \\{} & {} \quad +4k^2\omega sn^2(\alpha ,k)cn^4(\alpha ,k)F_1\nonumber \\{} & {} \quad -8k^2sn^2(\alpha ,k)cn^2(\alpha ,k)dn(\alpha ,k)F_1=0, \end{aligned}$$
(A2)
$$\begin{aligned}{} & {} k^4\omega ^2sn^4(\alpha ,k)cn^4(\alpha ,k)-4k^4sn^4(\alpha ,k)\nonumber \\{} & {} \quad +4k^6sn^6(\alpha ,k)\nonumber \\{} & {} \quad -2k^4\omega sn^4(\alpha ,k)cn^4(\alpha ,k)F_1\nonumber \\{} & {} \quad +4k^4sn^4(\alpha ,k)cn^2(\alpha ,k)dn(\alpha ,k)F_1=0, \end{aligned}$$
(A3)
$$\begin{aligned}{} & {} -2k^2\omega sn(\alpha ,k)cn^5(\alpha ,k)\nonumber \\{} & {} \quad -4k^2sn^3(\alpha ,k)cn(\alpha ,k)dn(\alpha ,k)\nonumber \\{} & {} \quad =-4k^2sn(\alpha ,k)cn(\alpha ,k)dn(\alpha ,k), \end{aligned}$$
(A4)
$$\begin{aligned}{} & {} 2k^4\omega sn^3(\alpha ,k)cn^5(\alpha ,k)\nonumber \\{} & {} \quad +4k^4sn^5(\alpha ,k)cn(\alpha ,k)dn(\alpha ,k)\nonumber \\{} & {} \quad =4k^4sn^3(\alpha ,k)cn(\alpha ,k)dn(\alpha ,k), \end{aligned}$$
(A5)
$$\begin{aligned}{} & {} -4k^6sn^5(\alpha ,k)cn(\alpha ,k)dn(\alpha ,k)=\nonumber \\{} & {} \quad -4k^6sn^5(\alpha ,k)cn(\alpha ,k)dn(\alpha ,k). \end{aligned}$$
(A6)

From (A1)-(A6), we obtain

$$\begin{aligned} \omega =\frac{2dn(\alpha ,k)}{cn^2(\alpha ,k)},\quad \alpha \in (0,K(k)). \end{aligned}$$

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Zhen, Y., Chen, J. Rogue waves on the periodic background in the high-order discrete mKdV equation. Nonlinear Dyn 111, 12511–12524 (2023). https://doi.org/10.1007/s11071-023-08481-z

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