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Higher-order regulatable rogue wave and hybrid interaction patterns for a new discrete complex coupled mKdV equation associated with the fourth-order linear spectral problem

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Abstract

A new discrete complex coupled mKdV equation related to the fourth-order linear spectral problem is proposed. Firstly, we construct a new discrete integrable hierarchy for this equation by Tu-scheme method, from which its higher-order versions can be expressed. Secondly, using the continuous limit, we map this discrete complex coupled equation to a continuous coupled mKdV equation. Thirdly, we discuss the modulation instability to analyze the generation mechanism of different localized waves. Then, the discrete generalized \((z, N-z)\)-fold Darboux transformation in matrix form is constructed for the first time, from which three types of position regulatable rogue waves, periodic waves and their hybrid interaction patterns are discussed graphically. Finally, the large-parameter asymptotic is applied to analyze the asymptotic states of second-order rogue wave at infinity, and the numerical simulations of some rogue wave solutions are performed to analyze their dynamical behaviors. These results might be helpful for understanding the wave motion and soliton propagation in shallow water and ultrashort pulsed media.

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Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

This work has been supported by National Natural Science Foundation of China Under Grant No. 12071042 and Beijing Natural Science Foundation Under Grant No. 1202006.

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Authors and Affiliations

  1. School of Applied Science, Beijing Information Science and Technology University, Beijing, 100192, China

    Xue-Ke Liu, Xiao-Yong Wen & Zhe Lin

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  1. Xue-Ke Liu
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Correspondence to Xiao-Yong Wen.

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Appendix

Appendix

$$\begin{aligned} \zeta&=\sqrt{(-1+\text {i})(\sqrt{6}+3)},\\ \alpha _1&=(-320\sqrt{6}+784)n^4+(-1408\sqrt{6}\aleph _0+3456\aleph _0\\&\quad -1424\sqrt{6}+3488)n^3+[-768\sqrt{6}\hbar _0^2+1920\hbar _0^2\\&\quad -2304\sqrt{6}\aleph _0^2\\&\quad +5760\aleph _0^2+(-15360\sqrt{6}+37632)t^2-864\text {i}\sqrt{6}\hbar _0\\&\quad +2112\text {i}\hbar _0-4704\sqrt{6}\aleph _0+11520\aleph _0\\&\quad +(6912\sqrt{6}\hbar _0-16896\hbar _0\\&\quad +3840\text {i}\sqrt{6}-9408\text {i})t-196\sqrt{6}\zeta +196\text {i}\sqrt{3}\zeta \\&\quad -240\text {i}\sqrt{2}\zeta -240\sqrt{2}\zeta +196\sqrt{3}\zeta +480\zeta \\&\quad -1888\sqrt{6}+4624]n^2\\&\quad +[1656+4608\text {i}\hbar _0\aleph _0+1056\text {i}\hbar _0\zeta +10176\aleph _0\\&\quad -480\hbar _0-676\sqrt{6}+4608\hbar _0^2\aleph _0+1056\aleph _0\zeta \\&\quad -828\sqrt{2}\zeta +676\sqrt{3}\zeta \\&\quad +12672\aleph _0^2+4224\hbar _0^2+4608\aleph _0^3-1920\text {i}\sqrt{6}\hbar _0\\&\quad -432\text {i}\sqrt{3}\hbar _0\zeta +528\text {i}\sqrt{2}\hbar _0\zeta -528\text {i}\sqrt{2}\aleph _0\zeta \\&\quad +432\text {i}\sqrt{3}\aleph _0\zeta -828\text {i}\sqrt{2}\zeta \\&\quad +676\text {i}\sqrt{3}\zeta +108\text {i}\sqrt{6}-1920\text {i}\sqrt{6}\hbar _0\aleph _0\\&\quad -432\sqrt{6}\aleph _0\zeta -1536\sqrt{6}\hbar _0^2\aleph _0-264\text {i}\\&\quad +(15360\sqrt{6}\hbar _0\aleph _0-36864\hbar _0\aleph _0\\&\quad +15360\sqrt{6}\hbar _0-37632\hbar _0+8448\text {i}\sqrt{6}\aleph _0\\&\quad -20736\text {i}\aleph _0+1920\text {i}\sqrt{6}\zeta \\&\quad -2352\text {i}\sqrt{2}\zeta +1920\text {i}\sqrt{3}\zeta +2352\sqrt{2}\zeta -1920\sqrt{3}\zeta \\&\quad -4704\text {i}\zeta +8544\text {i}\sqrt{6}-864\sqrt{6}+2112-20928\text {i})t\\&\quad -196\sqrt{6}\zeta +192\sqrt{6}\hbar _0-1536\sqrt{6}\aleph _0^3\\&\quad +432\sqrt{3}\hbar _0\zeta -5184\sqrt{6}\aleph _0^2\\&\quad -1728\sqrt{6}\hbar _0^2-4160\sqrt{6}\aleph _0-528\sqrt{2}\aleph _0\zeta \\&\quad +432\sqrt{3}\aleph _0\zeta -528\sqrt{2}\hbar _0\zeta +480\zeta +(-33792\sqrt{6}\aleph _0\\&\quad +82944\aleph _0-34176\sqrt{6}\\&\quad +83712)t^2\\&\quad -432\text {i}\sqrt{6}\hbar _0\zeta +4704\text {i}\hbar _0]n\\&\quad +(-184320 \sqrt{6}+451584) t^4+(165888\sqrt{6}\hbar _0\\&\quad -405504\hbar _0+92160\text {i}\sqrt{6}\\&\quad -225792\text {i})t^3+(-55296\sqrt{6}\hbar _0^2+138240\hbar _0^2\\ \end{aligned}$$
$$\begin{aligned}&\quad -18432\sqrt{6}\aleph _0^2+46080\aleph _0^2-62208\text {i}\sqrt{6}\hbar _0\\&\quad +152064\text {i}\hbar _0-37632\sqrt{6}\aleph _0\\&\quad +92160 \aleph _0-4704 \text {i}\sqrt{3}\zeta +4704\sqrt{6}\zeta +5760\text {i}\sqrt{2}\zeta \\&\quad +5760\sqrt{2}\zeta -4704\sqrt{3}\zeta -11520\zeta \\&\quad -11520\sqrt{6}+28224)t^2\\&\quad +(2352+2304\aleph _0-12672\hbar _0-41472\hbar _0\aleph _0\\&\quad -18432\hbar _0\aleph _0^2-960\sqrt{6}+5184\hbar _0\zeta +4056\sqrt{2}\zeta \\&\quad -3312\sqrt{3}\zeta -18432\hbar _0^3\\&\quad +2112\text {i}\sqrt{6}\aleph _0\zeta -23040\text {i}\aleph _0-11520\text {i}\aleph _0^2-34560\text {i}\hbar _0^2\\&\quad +960\text {i}\sqrt{6}\zeta +9216\sqrt{6}\hbar _0^3+16896\sqrt{6}\hbar _0\aleph _0\\&\quad -2112\sqrt{6}\hbar _0\zeta -14112\text {i}\\&\quad +9408\text {i}\sqrt{6}\aleph _0+5760\text {i}\sqrt{6}+5184\sqrt{6}\hbar _0\\&\quad +2112\text {i}\sqrt{3}\aleph _0\zeta -2592\text {i}\sqrt{2}\aleph _0\zeta -2592\text {i}\sqrt{2}\hbar _0\zeta \\&\quad +4608\text {i}\sqrt{6}\aleph _0^2+13824\text {i}\sqrt{6}\hbar _0^2\\&\quad +2112\text {i}\sqrt{3}\hbar _0\zeta +9216\sqrt{6}\hbar _0\aleph _0^2\\&\quad -4056\text {i}\sqrt{2}\zeta \\&\quad +2112\sqrt{3}\hbar _0\zeta -960\sqrt{6}\aleph _0+2592\sqrt{2}\aleph _0\zeta \\&\quad -2112\sqrt{3}\aleph _0\zeta -2592\sqrt{2}\hbar _0\zeta \\ \end{aligned}$$
$$\begin{aligned}&\quad +3312\text {i}\sqrt{3}\zeta -5184\text {i}\aleph _0\zeta -2352\text {i}\zeta )t\\&\quad -654\text {i}\sqrt{2}\zeta +534\text {i}\sqrt{3}\zeta +480\sqrt{3}\zeta -588\sqrt{2}\zeta \\&\quad -66\text {i}\zeta +27\text {i}\sqrt{6}\zeta +27\sqrt{6}\zeta \\&\quad -66\zeta +110\sqrt{3}+157\sqrt{6}-135\sqrt{2}-384-294\text {i}\\&\quad +120\text {i}\sqrt{6}-528\hbar _0+3168\text {i}\hbar _0-1296\text {i}\sqrt{6}\hbar _0\\&\quad +216\sqrt{6}\hbar _0+1824\aleph _0\\&\quad -288\text {i}\aleph _0+120\text {i}\sqrt{6}\aleph _0-744\sqrt{6}\aleph _0+1440\hbar _0^2\\&\quad -576\sqrt{6}\hbar _0^2-576\hbar _0\aleph _0-2112\text {i}\sqrt{6}\hbar _0\aleph _0\\&\quad +5184\text {i}\hbar _0\aleph _0+192\sqrt{6}\hbar _0\aleph _0\\&\quad -216\text {i} \sqrt{6}\hbar _0\zeta -912\sqrt{2}\hbar _0\zeta +744\sqrt{3}\hbar _0\zeta \\&\quad +528\text {i}\hbar _0\zeta -744\text {i}\sqrt{3}\hbar _0\zeta +912\text {i}\sqrt{2}\hbar _0\zeta \\&\quad +5568\aleph _0^2-2304 \sqrt{6}\aleph _0^2+744\sqrt{3}\aleph _0\zeta \\&\quad -912\sqrt{2} \aleph _0 \zeta -216\sqrt{6}\aleph _0\zeta +744 \text {i}\sqrt{3} \aleph _0\zeta \\&\quad -912\text {i}\sqrt{2} \aleph _0\zeta +528 \aleph _0 \zeta +2304\hbar _0^4\\&\quad +4608\hbar _0^2\aleph _0^2+2304\aleph _0^4+4608\hbar _0^2\aleph _0\\&\quad +4608\aleph _0^3-240\sqrt{6}\aleph _0^2\zeta -288\text {i}\sqrt{2}\aleph _0^2\zeta \\&\quad +240\text {i}\sqrt{3}\aleph _0^2\zeta -576\sqrt{2}\hbar _0\aleph _0\zeta +1152\text {i}\hbar _0\aleph _0\zeta \\&\quad +480\sqrt{3}\hbar _0\aleph _0\zeta +576\aleph _0^2\zeta \\&\quad -240\text {i}\sqrt{3}\hbar _0^2\zeta +288\text {i}\sqrt{2}\hbar _0^2\zeta -1152\text {i}\sqrt{6}\hbar _0^3\\&\quad -1920\sqrt{6}\hbar _0^2\aleph _0+240\sqrt{6}\hbar _0^2\zeta +240\sqrt{3}\aleph _0^2\zeta \\&\quad -288\sqrt{2}\aleph _0^2\zeta -1920\sqrt{6}\aleph _0^3\\&\quad -480\text {i}\sqrt{3}\hbar _0\aleph _0\zeta -480\text {i}\sqrt{6}\hbar _0\aleph _0\zeta -576\hbar _0^2\zeta \\ \end{aligned}$$
$$\begin{aligned}&\quad +2304\text {i}\hbar _0^3-1152\text {i}\sqrt{6}\hbar _0\aleph _0^2+576\text {i}\sqrt{2}\hbar _0\aleph _0\zeta \\&\quad +288\sqrt{2}\hbar _0^2\zeta +2304\text {i}\hbar _0\aleph _0^2\\&\quad -240\sqrt{3}\hbar _0^2\zeta ,\\ \beta _1&=(-48\sqrt{6}+120)n^2+(-96\sqrt{6}\aleph _0+288\aleph _0\\&\quad -24\text {i}\sqrt{6}\zeta +30\text {i}\zeta \sqrt{2}-24\text {i}\sqrt{3}\zeta \\&\quad +60\text {i}\zeta -30\zeta \sqrt{2}+24\sqrt{3}\zeta -108\sqrt{6}+264)n\\&\quad +(-1152\sqrt{6}+2880)t^2+(576\sqrt{6}\hbar _0-1152\hbar _0\\&\quad -144 \text {i}\zeta \sqrt{2}+120\text {i}\sqrt{3}\zeta \\&\quad -120\sqrt{6}\zeta -144\sqrt{2}\zeta +120\sqrt{3}\zeta +288\zeta )t+162\\&\quad -12 \sqrt{3}-66 \sqrt{6}+18 \sqrt{2}+27 \sqrt{3} \zeta +33 \text {i} \zeta \sqrt{2}\\&\quad +66 \text {i} \zeta -27 \text {i} \sqrt{6} \zeta -27 \text {i} \sqrt{3} \zeta -33 \sqrt{2} \zeta +288\hbar _0^2+288\aleph _0^2,\\ \varpi _1&=(96\sqrt{6}-240)n^2+(192\sqrt{6} \aleph _0-576 \aleph _0\\&\quad +216 \sqrt{6}-528)n+(2304 \sqrt{6}-5760)t^2\\&\quad +(-1152\sqrt{6} \hbar _0+2304 \hbar _0)t+132\sqrt{6}\\&\quad +24\sqrt{3}-36\sqrt{2}-324+240 \sqrt{6} \aleph _0\\&\quad -576 (\aleph _0+\hbar _0^2+\aleph _0^2),\\ \alpha _2&=(640\sqrt{6}-1568)n^4+(2816\sqrt{6} \aleph _0-6912 \aleph _0\\&\quad +2848 \sqrt{6}-6976)n^3+[1536\sqrt{6}\hbar _0^2\\&\quad -3840 \hbar _0^2+4608\sqrt{6} \aleph _0^2-11520 \aleph _0^2\\&\quad +(30720\sqrt{6}-75264) t^2+1728 \text {i}\sqrt{6} \hbar _0\\&\quad -4224 \text {i} \hbar _0+9408\sqrt{6} \aleph _0-23040 \aleph _0\\&\quad +(-13824\sqrt{6} \hbar _0\\&\quad +33792 \hbar _0-7680 \text {i}\sqrt{6}+18816 \text {i}) t\\&\quad +3776 \sqrt{6}-9248]n^2+[3072\sqrt{6} \hbar _0^2 \aleph _0\\&\quad -9216 \hbar _0^2 \aleph _0+3072\sqrt{6} \aleph _0^3-9216 \aleph _0^3\\&\quad +3456\sqrt{6} \hbar _0^2-8448 \hbar _0^2-9216 \text {i} \hbar _0 \aleph _0+3840 \text {i}\sqrt{6} \hbar _0 \aleph _0\\&\quad +10368\sqrt{6} \aleph _0^2-25344 \aleph _0^2+(67584\sqrt{6} \aleph _0\\&\quad -165888 \aleph _0+68352\sqrt{6}-167424) t^2-9408 \text {i} \hbar _0\\&\quad +3840 \text {i}\sqrt{6} \hbar _0-384 \sqrt{6} \hbar _0+960 \hbar _0\\ \end{aligned}$$
$$\begin{aligned}&\quad +8320\sqrt{6} \aleph _0-20352 \aleph _0+(-30720\sqrt{6} \hbar _0 \aleph _0\\&\quad +73728 \hbar _0 \aleph _0-30720\sqrt{6} \hbar _0+75264 \hbar _0\\&\quad -16896 \text {i}\sqrt{6} \aleph _0+41472 \text {i} \aleph _0-17088 \text {i} \sqrt{6}\\&\quad -4224+41856 \text {i}+1728\sqrt{6}) t-216 \text {i}\sqrt{6}-3312\\&\quad +528\text {i}+1352\sqrt{6}]n+(368640\sqrt{6}-903168)t^4\\&\quad +(-331776\sqrt{6} \hbar _0+811008 \hbar _0\\&\quad -184320 \text {i}\sqrt{6}+451584 \text {i})t^3+(110592\sqrt{6} \hbar _0^2\\&\quad -276480 \hbar _0^2+36864\sqrt{6} \aleph _0^2-92160 \aleph _0^2\\&\quad +124416 \text {i}\sqrt{6} \hbar _0-304128 \text {i} \hbar _0+75264 \sqrt{6} \aleph _0\\&\quad -184320 \aleph _0+23040\sqrt{6}-56448)t^2\\&\quad +(-18432\sqrt{6} \hbar _0^3+36864 \hbar _0^3-18432\sqrt{6} \hbar _0 \aleph _0^2\\&\quad +36864 \hbar _0 \aleph _0^2-27648 \text {i}\sqrt{6} \hbar _0^2+69120 \text {i} \hbar _0^2\\&\quad -33792 \sqrt{6} \hbar _0 \aleph _0+82944 \hbar _0 \aleph _0+23040 \text {i} \aleph _0^2\\&\quad -9216 \text {i}\sqrt{6} \aleph _0^2-10368\sqrt{6} \hbar _0+25344 \hbar _0\\&\quad -18816 \text {i}\sqrt{6} \aleph _0+46080 \text {i} \aleph _0+1920\sqrt{6} \aleph _0\\ \end{aligned}$$
$$\begin{aligned}&\quad -4608 \aleph _0-11520 \text {i}\sqrt{6}-4704+28224 \text {i}+1920\sqrt{6})t\\&\quad +270 \sqrt{2}+768+588 \text {i}-240 \text {i} \sqrt{6}-314 \sqrt{6}\\&\quad -220 \sqrt{3}+1056 \hbar _0\\&\quad +2592 \text {i} \sqrt{6} \hbar _0-6336 \text {i} \hbar _0-432 \sqrt{6} \hbar _0-3648 \aleph _0\\&\quad -240 \text {i} \sqrt{6} \aleph _0+576 \text {i} \aleph _0+1488 \sqrt{6} \aleph _0-2880\hbar _0^2\\&\quad +1152\sqrt{6}\hbar _0^2+1152\hbar _0\aleph _0\\&\quad -384\sqrt{6}\hbar _0\aleph _0-10368\text {i}\hbar _0\aleph _0+4224\text {i}\sqrt{6}\hbar _0\aleph _0\\&\quad -11136\aleph _0^2+4608\sqrt{6}\aleph _0^2+2304 \text {i} \sqrt{6} \hbar _0^3-4608 \text {i} \hbar _0^3\\&\quad -9216 \hbar _0^2 \aleph _0+3840 \sqrt{6} \hbar _0^2 \aleph _0\\&\quad -4608 \text {i} \hbar _0 \aleph _0^2+2304 \text {i} \sqrt{6} \hbar _0 \aleph _0^2-9216 \aleph _0^3\\&\quad +3840 \sqrt{6} \aleph _0^3-4608\hbar _0^4-9216\hbar _0^2\aleph _0^2-4608\aleph _0^4.\\ \end{aligned}$$

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Liu, XK., Wen, XY. & Lin, Z. Higher-order regulatable rogue wave and hybrid interaction patterns for a new discrete complex coupled mKdV equation associated with the fourth-order linear spectral problem. Nonlinear Dyn 111, 15309–15333 (2023). https://doi.org/10.1007/s11071-023-08627-z

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