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Soliton molecules, rational positons and rogue waves for the extended complex modified KdV equation

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Abstract

In this paper, we consider the integrable extended complex modified Korteweg–de Vries equation. Based on Darboux transformation, we obtain soliton molecules, positon solutions, rational positon solutions and rogue waves for integrable extended complex modified Korteweg–de Vries equation. Further, under the standard decomposition, we divide the rogue waves into three patterns: fundamental pattern, triangular pattern and ring pattern. On the basis of fundamental pattern, we define the length and width of rogue waves and discuss the effect of different parameters on rogue waves.

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Data Availability Statement

All data generated or analyzed during this study are included in this article.

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Acknowledgements

The authors thank the two referees for valuable comments. The research was supported by the National Natural Science Foundation of China, Grant No. 11901141.

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  1. School of Science, Hangzhou Dianzi University, 310018, Zhejiang, China

    Lin Huang & Nannan Lv

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  1. Lin Huang
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Correspondence to Nannan Lv.

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Appendices

Appendix A

$$\begin{aligned} L_{2}&=34560000\,{\beta }^{4}{t}^{4}{c}^{21}-27648000\,\alpha \,{\beta }^{3}{t}^ {4}{c}^{19}\\&\quad +4608000\,{\beta }^{3}{t}^{3}x{c}^{17}+8294400\,{\alpha }^{2} {\beta }^{2}{t}^{4}{c}^{17}\\&\quad -1105920\,{\alpha }^{3}\beta \,{t}^{4}{c}^{15} -2764800\,\alpha \,{\beta }^{2}{t}^{3}x{c}^{15}\\&\quad +55296\,{\alpha }^{4}{t}^{ 4}{c}^{13}+230400\,{\beta }^{2}{t}^{2}{x}^{2}{c}^{13}\\&\quad +552960\,{\alpha }^ {2}\beta \,{t}^{3}x{c}^{13}\\&\quad -36864\,{\alpha }^{3}{t}^{3}x{c}^{11}-92160\, \alpha \,\beta \,{t}^{2}{x}^{2}{c}^{11}\\&\quad +364800\,{\beta }^{2}{t}^{2}{c}^{ 11}+9216\,{\alpha }^{2}{t}^{2}{x}^{2}{c}^{9}\\&\quad +5120\,\beta \,t{x}^{3}{c}^{ 9}-8\,c\\&\quad -115200\,\alpha \,\beta \,{t}^{2}{c}^{9}-1024\,\alpha \,t{x}^{3}{c}^{7}\\&\quad +8448\,{\alpha }^{2}{t}^{2}{c}^{7}+14080\,\beta \,tx{c}^{7}\\&\quad +{\frac{128 \,{x}^{4}{c}^{5}}{3}}-1792\,\alpha \,tx{c}^{5}+64\,{x}^{2}{c}^{3}, \end{aligned}$$
$$\begin{aligned} L_{3}&\quad =-20736000000\,{\beta }^{6}{t}^{6}{c}^{30}+24883200000\,\alpha \,{\beta }^ {5}{t}^{6}{c}^{28}\\&\quad -12441600000\,{\alpha }^{2}{\beta }^{4}{t}^{6}{c}^{26} -4147200000\,{\beta }^{5}{t}^{5}x{c}^{26}\\&\quad +4147200000\,\alpha \,{\beta }^{ 4}{t}^{5}x{c}^{24}+3317760000\,{\alpha }^{3}{\beta }^{3}{t}^{6}{c}^{24}\\&\quad - 1658880000\,{\alpha }^{2}{\beta }^{3}{t}^{5}x{c}^{22}-345600000\,{\beta } ^{4}{t}^{4}{x}^{2}{c}^{22}\\&\quad -497664000\,{\alpha }^{4}{\beta }^{2}{t}^{6}{c }^{22}+167040000\,{\beta }^{4}{t}^{4}{c}^{20}\\&\quad +276480000\,\alpha \,{\beta }^{3}{t}^{4}{x}^{2}{c}^{20}+331776000\,{\alpha }^{3}{\beta }^{2}{t}^{5}x {c}^{20}\\&\quad +39813120\,{\alpha }^{5}\beta \,{t}^{6}{c}^{20}-115200000\, \alpha \,{\beta }^{3}{t}^{4}{c}^{18}\\&\quad -1327104\,{\alpha }^{6}{t}^{6}{c}^{18 }-82944000\,{\alpha }^{2}{\beta }^{2}{t}^{4}{x}^{2}{c}^{18}\\&\quad -33177600\,{ \alpha }^{4}\beta \,{t}^{5}x{c}^{18}-15360000\,{\beta }^{3}{t}^{3}{x}^{3} {c}^{18}\\&\quad +16128000\,{\beta }^{3}{t}^{3}x{c}^{16}+29030400\,{\alpha }^{2}{ \beta }^{2}{t}^{4}{c}^{16}\\&\quad +9216000\,\alpha \,{\beta }^{2}{t}^{3}{x}^{3}{c }^{16}+11059200\,{\alpha }^{3}\beta \,{t}^{4}{x}^{2}{c}^{16}\\&\quad +1327104\,{ \alpha }^{5}{t}^{5}x{c}^{16}-3133440\,{\alpha }^{3}\beta \,{t}^{4}{c}^{14 }\\&\quad -7833600\,\alpha \,{\beta }^{2}{t}^{3}x{c}^{14}-1843200\,{\alpha }^{2} \beta \,{t}^{3}{x}^{3}{c}^{14}\\&\quad -384000\,{\beta }^{2}{t}^{2}{x}^{4}{c}^{14 }-552960\,{\alpha }^{4}{t}^{4}{x}^{2}{c}^{14}\\&\quad +119808\,{\alpha }^{4}{t}^{ 4}{c}^{12} +499200\,{\beta }^{2}{t}^{2}{x}^{2}{c}^{12}\\&\quad +1198080\,{\alpha } ^{2}\beta \,{t}^{3}x{c}^{12}\\&\quad +153600\,\alpha \,\beta \,{t}^{2}{x}^{4}{c}^{ 12}+122880\,{\alpha }^{3}{t}^{3}{x}^{3}{c}^{12}\\&\quad -606400\,{\beta }^{2}{t}^ {2}{c}^{10}\\&\quad -55296\,{\alpha }^{3}{t}^{3}x{c}^{10}-138240\,\alpha \,\beta \,{t}^{2}{x}^{2}{c}^{10}\\&\quad -15360\,{\alpha }^{2}{t}^{2}{x}^{4}{c}^{10}- 5120\,\beta \,t{x}^{5}{c}^{10}\\&\quad +145280\,\alpha \,\beta \,{t}^{2}{c}^{8}\\&\quad +7680\,{\alpha }^{2}{t}^{2}{x}^{2}{c}^{8}+{\frac{12800\,\beta \,t{x}^{3} {c}^{8}}{3}}\\&\quad +1024\,\alpha \,t{x}^{5}{c}^{8}-{\frac{256\,{x}^{6}{c}^{6} }{9}}-8896\,{\alpha }^{2}{t}^{2}{c}^{6}\\&\quad -8000\,\beta \,tx{c}^{6}-{\frac{ 512\,\alpha \,t{x}^{3}{c}^{6}}{3}}\\&\quad +1088\,\alpha \,tx{c}^{4}-{\frac{64\, {x}^{4}{c}^{4}}{3}}-48\,{x}^{2}{c}^{2}-4. \end{aligned}$$

Appendix B

$$\begin{aligned} A_{1}&=100{c}^{2}{a}^{8}{\beta }^{2}{t}^{2}-800{t}^{2}{a}^{6}{\beta }^{2}{c }^{4}+6000{t}^{2}{a}^{4}{\beta }^{2}{c}^{6}\\&\quad +3600{c}^{10}{\beta }^{2} {t}^{2}+120{c}^{2}{a}^{6}\beta {t}^{2}\alpha \\&\quad -720{t}^{2}{a}^{4} \beta {c}^{4}\alpha +720{t}^{2}{a}^{2}\beta {c}^{6}\alpha \\&\quad -1440{c}^{8}\beta {t}^{2}\alpha +36{c}^{2}{a}^{4}{\alpha }^{2}{t}^{2}+144{c }^{6}{\alpha }^{2}{t}^{2}\\&\quad +40{c}^{2}{a}^{4}\beta tx+40{c}^{2}{a}^{4 }\beta ts_{{0}}-480{c}^{4}{a}^{2}\beta tx\\&\quad -480 {c}^{4}{a}^{2}\beta ts_{{0}}\\&\quad +240{c}^{6}\beta tx+240 {c}^{6}\beta ts_{{0}}+24{c}^{2}{a}^{2}\alpha tx\\&\quad +24 {c}^{2}{a}^{2}\alpha ts_{{0}}-48{c}^{4}\alpha tx-48{c}^{4}\alpha ts_{{0}}+4{c}^{2}{x}^{2}\\&\quad +8{c}^{2}xs_{{0}}+4\,{c}^{2}{s_{{0}}}^{2}.\\ A_{2}&=16\,{x}^{4}+ ( 160\,{a}^{4}t-1824\,{a}^{2}t+768\,t ) {x}^{3 }\\&\quad + ( 600\,{a}^{8}{t}^{2}-10480\,{t}^{2}{a}^{6}\\&\quad +66456\,{t}^{2}{a}^ {4}-42336\,{t}^{2}{a}^{2}+13824\,{t}^{2}\\&\quad +40 ) {x}^{2}+ ( 1000\,{a}^{12}{t}^{3}-18200\,{a}^{10}{t}^{3}+135480\,{a}^{8}{t}^{3}\\&\quad - 631464\,{a}^{6}{t}^{3}+196128\,{a}^{4}{t}^{3}\\&\quad +200\,{a}^{4}t-228096\,{a }^{2}{t}^{3}\\&\quad -2280\,{a}^{2}t+110592\,{t}^{3}+960\,t ) x\\&\quad +625\,{a}^ {16}{t}^{4}-8500\,{t}^{4}{a}^{14}\\&\quad +95350\,{t}^{4}{a}^{12}-442860\,{t}^{ 4}{a}^{10}+1733841\,{t}^{4}{a}^{8}\\&\quad +250\,{a}^{8}{t}^{2}+282600\,{t}^{4}{a}^{6}\\&\quad -8100\,{t}^{2}{a}^{6}+1563408\,{t}^{4}{a}^{4}\\&\quad +47850\,{t}^{2}{a} ^{4}+207360\,{t}^{4}{a}^{2}\\&\quad -44856\,{t}^{2}{a}^{2}+331776\,{t}^{4}\\&\quad +5760\,{t}^{2}-7. \end{aligned}$$
$$\begin{aligned} A_{3}&=-8\,{x}^{4}+ ( -80\,{a}^{4}+912\,{a}^{2}-384 ) t{x}^{3}\\&\quad + [28+ ( -300\,{a}^{8}+5240\,{a}^{6}\\&\quad -33228\,{a}^{4}+21168\,{a}^{2}-6912 ) {t}^{2} ] {x}^{2}\\&\quad + [( -500\,{a}^{12}+9100\,{a}^{10}-67740\,{a}^{8}\\&\quad +315732\,{a}^{6}-98064\,{a}^{4}\\&\quad + 114048\,{a}^{2}-55296 ) {t}^{3} +(140\,{a}^{4}-1596\,{a}^{2}\\&\quad +672 ) t ]x+ (-165888+4250\,{a}^{14}\\&\quad -{\frac{625}{2}}{a}^{16}-103680\,{a}^{2} -781704\,{a}^{4}\\&\quad -141300\,{a}^{6}-{\frac{1733841}{2}}{a}^{8}\\&\quad +221430\,{a}^{10}-47675\,{a}^{12}){t}^{4}+( 175\,{a}^{8}-7590\,{a}^{6}\\&\quad +43863\,{a}^{4}-45396\,{a}^{2}+4032) {t}^{2}-{\frac{17}{2}}. \end{aligned}$$

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Huang, L., Lv, N. Soliton molecules, rational positons and rogue waves for the extended complex modified KdV equation. Nonlinear Dyn 105, 3475–3487 (2021). https://doi.org/10.1007/s11071-021-06764-x

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