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N-fold Darboux transformation of the two-component Kundu–Eckhaus equations and non-symmetric doubly localized rogue waves

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The two-component Kundu–Eckhaus (KE) equations were introduced for the first time in 1999. Very recently, the two-component KE equations considered as a model of describing the effect of quintic nonlinearity on the ultra-short optical pulse propagation in non-Kerr media have been intensively studied. In this paper, we construct an analytical and explicit representation of the Darboux transformation (DT) for the two-component KE equations. Compared with the DT constructed by researchers before, the DT here is expressed by the initial eigenfunctions, spectral parameters, and ‘seed’ solution. As applications of DT, the explicit expressions of non-symmetric rogue wave of two-component KE equations and KE equation are displayed, and the differences between the non-symmetric and symmetric rogue wave for the KE equation are discussed in detail.

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The work of D. Q. Qiu is supported by the National Natural Science Foundation of China (NNSFC) [Grant numbers 11871471,11931017], and the Fundamental Research Funds for Central Universities. The work of W. G. Cheng is supported by the Scientific Research Foundation of Educational Committee of Yunnan Province (No. 2019J0735). D. Q. Qiu acknowledges sincerely Prof. Q. P. Liu for many useful discussions.

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Correspondence to Wenguang Cheng.

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Formulae for the second-order RW solution:

$$\begin{aligned} N_{12}&=-\,6\,\sqrt{2} \left( -8\,{\rho _1}^{4}+64\,{\rho _1}^{3}-256\,{\rho _1}^{2}+ 512\,\rho _1-512 \right) {t}^{5}+ \left( -6\,\sqrt{2} \left( -32\,{\rho 1}^{3}+192\,{\rho _1}^{2}-512\,\rho _1\right. \right. \\&\quad \left. \left. +\,512\right) x-6\,\sqrt{2} \left( 2 \,{\mathrm{i}}{\rho _1}^{4}-16\,{\mathrm{i}}{\rho _1}^{3}+96\,{\mathrm{i}}{\rho _1}^{2}+16\,{\rho _1}^{3}-256\, {\mathrm{i}}\rho _1-96\,{\rho _1}^{2}-256+384\,{\mathrm{i}}+256\,\rho _1 \right) \right) {t}^{4}\\&\quad +\left( -6\,\sqrt{2} \left( -48\,{\rho _1}^{2}+192\,\rho _1-256 \right) { x}^{2}-6\,\sqrt{2} \left( 8\,{\mathrm{i}}{\rho _1}^{3}-48\,{\mathrm{i}}{\rho _1}^{2}+192\,\mathrm{i}\rho _ 1+48\,{\rho _1}^{2}+256-256\,{\mathrm{i}}\right. \right. \\&\quad \left. \left. -\,192\,\rho _1 \right) x-6\,\sqrt{2} \left( -4\,{\mathrm{i}}{\rho _1}^{3}+24\,{\mathrm{i}}{\rho _1}^{2}-96\,{\mathrm{i}}\rho _1-32+128\,{\mathrm{i}} \right) \right) {t}^{3}+ \left( -6\,\sqrt{2} \left( -32\,\rho _1+64 \right) {x }^{3}\right. \\&\quad \left. -\,6\,\sqrt{2} \left( 12\,{\mathrm{i}}{\rho _1}^{2}-48\,{\mathrm{i}}\rho _1-96+96\,{\mathrm{i}}+48\, \rho _1 \right) {x}^{2}-6\,\sqrt{2} \left( -12\,{\mathrm{i}}{\rho _1}^{2}+48\,{\mathrm{i}}\rho _1 -96\,{\mathrm{i}} \right) x-6\,\sqrt{2} \left( 6\,{\mathrm{i}}{\rho _1}^{2}\right. \right. \\&\quad \left. \left. -\,24\,{\mathrm{i}}\rho _1+24+72 \,{\mathrm{i}}-12\,\rho _1 \right) \right) {t}^{2}+ \left( 48\,\sqrt{2}{x}^{4}-6 \,\sqrt{2} \left( 8\,{\mathrm{i}}\rho _1+16-16\,{\mathrm{i}} \right) {x}^{3}-6\,\sqrt{2} \left( -12\,{\mathrm{i}}\rho _1+24\,{\mathrm{i}} \right) {x}^{2}\right. \\&\quad \left. -\,6\,\sqrt{2} \left( 12\,{\mathrm{i}} \rho _1-12-24\,{\mathrm{i}} \right) x-6\,\sqrt{2} \left( -3\,{\mathrm{i}}\rho _1+12+6\,{\mathrm{i}} \right) \right) t-12\,{\mathrm{i}}\sqrt{2}{x}^{4}+24\,{\mathrm{i}}\sqrt{2}{x}^{3}-36\,{\mathrm{i}} \sqrt{2}{x}^{2}+18\,{\mathrm{i}}\sqrt{2}x.\\ D_{12}&= \left( 16\,{\rho _1}^{6}-192\,{\rho _1}^{5}+1152\,{\rho _1}^{4}-4096\,{\rho _ 1}^{3}+9216\,{\rho _1}^{2}-12288\,\rho _1+8192 \right) {t}^{6}+ \left( \left( 96\,{\rho _1}^{5}-960\,{\rho _1}^{4}\right. \right. \\&\quad \left. \left. +\,4608\,{\rho _1}^{3}-12288\,{ \rho _1}^{2}+18432\,\rho _1-12288 \right) x-48\,{\rho _1}^{5}+480\,{\rho _1}^{ 4}-2304\,{\rho _1}^{3}+6144\,{\rho _1}^{2}-9216\,\rho _1\right. \\&\quad \left. +\,6144 \right) {t}^{5 }+ \left( \left( 240\,{\rho _1}^{4}-1920\,{\rho _1}^{3}+6912\,{\rho _1}^{2} -12288\,\rho _1+9216 \right) {x}^{2}+ \left( -240\,{\rho _1}^{4}+1920\,{ \rho _1}^{3}-6912\,{\rho _1}^{2}\right. \right. \\&\quad \left. \left. +\,12288\,\rho _1-9216 \right) x+72\,{\rho _1}^{ 4}-576\,{\rho _1}^{3}+1728\,{\rho _1}^{2}-2304\,\rho _1+3072 \right) {t}^{4} + \left( \left( 320\,{\rho _1}^{3}-1920\,{\rho _1}^{2}\right. \right. \\&\quad \left. \left. +\,4608\,\rho _1-4096 \right) {x}^{3}+ \left( -480\,{\rho _1}^{3}+2880\,{\rho _1}^{2}-6912\, \rho _1+6144 \right) {x}^{2}+ \left( 288\,{\rho _1}^{3}-1728\,{\rho _1}^{2}+ 3456\,\rho _1\right. \right. \\&\quad \left. \left. -\,2304 \right) x-72\,{\rho _1}^{3}+432\,{\rho _1}^{2}-576\,\rho _1 \right) {t}^{3}+ \left( \left( 240\,{\rho _1}^{2}-960\,\rho _1+1152 \right) {x}^{4}+ \left( -480\,{\rho _1}^{2}+1920\,\rho _1\right. \right. \\&\quad \left. \left. -\,2304 \right) {x }^{3}+ \left( 432\,{\rho _1}^{2}-1728\,\rho _1+1728 \right) {x}^{2}+ \left( -216\,{\rho _1}^{2}+864\,\rho _1-576 \right) x+72\,{\rho _1}^{2}-288 \,\rho _1+576 \right) {t}^{2}\\&\quad +\Big ( \left( 96\,\rho _1-192 \right) {x}^ {5}+ \left( -240\,\rho _1+480 \right) {x}^{4}+ \left( 288\,\rho _1-576 \right) {x}^{3}+ \left( -216\,\rho _1+432 \right) {x}^{2}+ \left( 144\, \rho _1-288 \right) x\\&\quad -\,36\,\rho _1+72 \Big ) t+16\,{x}^{6}-48\,{x}^{5}+72 \,{x}^{4}-72\,{x}^{3}+72\,{x}^{2}-36\,x+9\\ F_{12}&=2\,{\mathrm{i}} \left( 8\,{\rho _1}^{6}-96\,{\rho _1}^{5}+576\,{\rho _1}^{4}-2048\,{ \rho _1}^{3}+4608\,{\rho _1}^{2}-6144\,\rho _1+4096 \right) \rho _1\,{t}^{6}+ \left( 2\,{\mathrm{i}} \Big ( 48\,{\rho _1}^{5}-480\,{\rho _1}^{4}\right. \\&\quad \left. +\,2304\,{\rho _1}^{3} -6144\,{\rho _1}^{2}+9216\,\rho _1-6144 \Big ) \rho _1\,x+2\,{\mathrm{i}} \Big ( -12\, {\rho _1}^{5}+120\,{\rho _1}^{4}-576\,{\rho _1}^{3}+1536\,{\rho _1}^{2}-2304\, \rho _1\right. \\&\quad \left. +\,1536 \Big ) \rho _1 \right) {t}^{5}+ \left( 2\,{\mathrm{i}} \left( 120\,{ \rho _1}^{4}-960\,{\rho _1}^{3}+3456\,{\rho _1}^{2}-6144\,\rho _1+4608 \right) \rho _1\,{x}^{2}+2\,{\mathrm{i}} \left( -60\,{\rho _1}^{4}+480\,{\rho _1}^{3}\right. \right. \\&\quad \left. \left. -\, 1728\,{\rho _1}^{2}+3072\,\rho _1-2304 \right) \rho _1\,x+2\,{\mathrm{i}} \left( 6\,{ \rho _1}^{4}-48\,{\rho _1}^{3}+384\,\rho _1+384 \right) \rho _1 \right) {t}^{4 }+ \left( 2\,{\mathrm{i}} \left( 160\,{\rho _1}^{3}-960\,{\rho _1}^{2}\right. \right. \\&\quad \left. \left. +\,2304\,\rho _1- 2048 \right) \rho _1\,{x}^{3}+2\,{\mathrm{i}} \left( -120\,{\rho _1}^{3}+720\,{\rho _1} ^{2}-1728\,\rho _1+1536 \right) \rho _1\,{x}^{2}+2\,{\mathrm{i}} \left( 24\,{\rho _1}^{ 3}-144\,{\rho _1}^{2}\right. \right. \\&\quad \left. \left. +384\right) \rho _1\,x+2\,{\mathrm{i}} \left( 144\,\rho _1-288 \right) \rho _1 \right) {t}^{3}+ \left( 2\,{\mathrm{i}} \left( 120\,{\rho _1}^{2}- 480\,\rho _1+576 \right) \rho _1\,{x}^{4}+2\,{\mathrm{i}} \left( -120\,{\rho _1}^{2}+ 480\,\rho _1-576 \right) \right. \\&\quad \left. \rho _1\,{x}^{3}+2\,{\mathrm{i}} \left( 36\,{\rho _1}^{2}-144 \,\rho _1 \right) \rho _1\,{x}^{2}+288\,{\mathrm{i}}\rho _1\,x+2\,{\mathrm{i}} \left( 9\,{\rho _1}^{ 2}-36\,\rho _1+216 \right) \rho _1 \right) {t}^{2}+ \Big ( 2\,{\mathrm{i}} \left( 48 \,\rho _1-96 \right) \rho _1\,{x}^{5}\\&\quad +\,2\,{\mathrm{i}} \left( -60\,\rho _1+120 \right) \rho _1\,{x}^{4}+2\,{\mathrm{i}} \left( 24\,\rho _1-48 \right) \rho _1\,{x}^{3}+2\,{\mathrm{i}} \left( 18\,\rho _1-36 \right) \rho _1\,x \Big ) t+16\,{\mathrm{i}}\rho _1\,{x}^{6}-24 \,{\mathrm{i}}\rho _1\,{x}^{5}+12\,{\mathrm{i}}\rho _1\,{x}^{4}\\&\quad +\,18\,{\mathrm{i}}\rho _1\,{x}^{2} \end{aligned}$$

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Qiu, D., Cheng, W. N-fold Darboux transformation of the two-component Kundu–Eckhaus equations and non-symmetric doubly localized rogue waves. Eur. Phys. J. Plus 135, 13 (2020).

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