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Matter rogue waves and management by external potentials for coupled Gross–Pitaevskii equation

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Abstract

The analytical matter rogue wave solutions are reported for the coupled Gross–Pitaevskii equation by using the similarity transformation and Darboux transformation. We study the effect of the time-dependent linear and quadratic potentials (flying-bird potential) on the profiles and dynamics of non-autonomous rogue wave solution. A non-autonomous rogue wave and bright-dark rogue wave solutions are constructed and exhibited. The managements of external potential and the dynamic behaviors of the rogue wave solutions are investigated analytically. We present the general approach and use it to calculate non-autonomous rogue wave solutions and consider the potential applications for the rogue wave phenomena.

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Acknowledgments

This work was supported by the Natural Science Foundation of Liaoning Province, China (Grant No. 2013020056) and Project supported by the National Natural Science Foundation of China (Grant No.11301349).

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  1. School of Mathematics and Systematic Sciences, Shenyang Normal University, Shenyang, 110034, China

    Fajun Yu

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Appendix: The elements of the matrix \(\Psi \)

Appendix: The elements of the matrix \(\Psi \)

$$\begin{aligned}&\psi _{{11}}= \psi _{{12}}=\psi _{{13}}\\&=-\frac{1}{24} {\frac{1}{{\left( \sqrt{3}{\alpha }^{3}-9\,\sqrt{3}{a}^{2}\alpha -9\,i{\alpha }^{2}a+9\,i{a}^{3}\right) ^{2}}}}\\&\quad \times \left( 1944\,\alpha _{{1}}{a}^{6}-72\,\alpha _{{1}}{\alpha }^{6}-9\,i\eta {a}^{3}\alpha _{{3}}-4320\,i{\alpha }^{3}\sqrt{3}{a}^{3}\alpha _{{1}}\right. \\&\quad \left. +\,432\,i{\alpha }^{5}\sqrt{3}a\alpha _{{1}}+27\,{a}^{4}\alpha _{{3}}{\eta }^{2}-1080\,t\alpha _{{3}}\eta {a}^{3}{\alpha }^{2}\right. \\&\quad \left. +\,3888\,i\alpha \,\sqrt{3}\alpha _{{1}}{a}^{5}+1620\,\alpha _{{3}}{\tau }^{2}{\alpha }^{4}{a}^{2}-4860\,\alpha _{{3}}{\tau }^{2}{a}^{4}{\alpha }^{2}\right. \\&\quad \left. -\,1080\,\eta {a}^{3}\alpha _{{2}}{\alpha }^{2}+180\,\eta {\alpha }^{4}a\alpha _{{2}}+540\,i\tau \alpha _{{3}}\eta \sqrt{3}{a}^{4}\alpha \right. \\&\quad \left. +\,12\,i\tau \alpha _{{3}}\eta {\alpha }^{5}\sqrt{3}-360\,it\alpha _{{3}}\eta \sqrt{3}{a}^{2}{\alpha }^{3}+432\,i\tau \alpha _{{2}}\sqrt{3}{\alpha }^{5}a\right. \\&\quad \left. +\,1944\,\tau \alpha _{{2}}{a}^{6}{-}72\,\tau \alpha _{{2}}{\alpha }^{6}-54\,{\alpha }^{2}{a}^{2}\alpha _{{3}}{\eta }^{2}{-}9720\,t\alpha _{{2}}{a}^{4}{\alpha }^{2}\right. \\&\quad \left. -\,4320\,i\tau \alpha _{{2}}\sqrt{3}{\alpha }^{3}{a}^{3}\!+\!3888 i\tau \alpha _{{2}}\sqrt{3}{a}^{5}\alpha \!-\!360 i{\alpha }^{3}\sqrt{3}\eta {a}^{2}\alpha _{{2}}\right. \\&\quad \left. -\,12\,i{\alpha }^{3}\sqrt{3}a\alpha _{{3}}{\eta }^{2}+12\,i{\alpha }^{5}\sqrt{3}\eta \alpha _{{2}}+540\,i\alpha \,\sqrt{3}{a}^{4}\alpha _{{2}}\eta \right. \\&\quad \left. +\,36\,i\alpha \,\sqrt{3}{a}^{3}\alpha _{{3}}{\eta }^{2}+9\,i\eta {\alpha }^{2}a\alpha _{{3}}+216\,i\alpha _{{3}}{\tau }^{2}\sqrt{3}{\alpha }^{5}a\right. \\&\quad \left. -\,2160\,i\alpha _{{3}}{\tau }^{2}\sqrt{3}{\alpha }^{3}{a}^{3}+1944\,i\alpha _{{3}}{\tau }^{2}\sqrt{3}{a}^{5}\alpha -9720\,{a}^{4}{\alpha }^{2}\alpha _{{1}}\right. \\&\quad \left. +\,3240\,{\alpha }^{4}{a}^{2}\alpha _{{1}}-36\,\alpha _{{3}}{\tau }^{2}{\alpha }^{6}+9\,\alpha \,\sqrt{3}\eta {a}^{2}\alpha _{{3}}-{\alpha }^{3}\sqrt{3}\eta \alpha _{{3}}\right. \\&\quad \left. +\,3\,{\alpha }^{4}\alpha _{{3}}{\eta }^{2}{+}180\,t\alpha _{{3}}\eta a{\alpha }^{4}{+}324\,t\alpha _{{3}}\eta {a}^{5}{+}3240\,\tau \alpha _{{2}}{\alpha }^{4}{a}^{2}\right. \\&\quad \left. +\,972\,\alpha _{{3}}{\tau }^{2}{a}^{6}+324\,{a}^{5}\alpha _{{2}}\eta \right) , \end{aligned}$$
$$\begin{aligned}&\psi _{{21}}=\psi _{{22}}=\psi _{{23}}\\&=-{\frac{1}{96}}\,{\frac{1}{{\alpha }^{2} \left( \sqrt{3}{\alpha }^{3}-9\,\sqrt{3}{a}^{2}\alpha -9\,i{\alpha }^{2}a+9\,i{a}^{3} \right) ^{2}}}\\&\quad \times \left( -1080\,{\alpha }^{4}{a}^{2}\alpha _{{3}}\tau -90\,{\alpha }^{4}\alpha _{{3}}a\eta -1080\,{a}^{3}{\alpha }^{3}\alpha _{{3}}\tau \right. \\&\quad \left. -\,162\,{a}^{2}{\alpha }^{3}\alpha _{{3}}\eta +1620\,{a}^{4}{\alpha }^{2}\alpha _{{3}}\tau +234\,{a}^{3}{\alpha }^{2}\alpha _{{3}}\eta \right. \\&\quad \left. +\,324\,{a}^{5}\alpha \,\alpha _{{3}}\tau +54\,{a}^{4}\alpha \,\alpha _{{3}}\eta +36\,{\alpha }^{6}\alpha _{{3}}\tau +12\,{\alpha }^{5}\alpha _{{3}}\eta \right. \\&\quad \left. -\,19440\,i{\alpha }^{4}{a}^{4}\alpha _{{1}}-23328\,i{a}^{5}{\alpha }^{3}\alpha _{{1}}+9\,ia{\alpha }^{3}\alpha _{{3}}\right. \\&\quad \left. +\, 3888\,i{a}^{6}{\alpha }^{2}\alpha _{{1}}-135\,i{a}^{2}{\alpha }^{2}\alpha _{{3}}-9\,i{a}^{3}\alpha \,\alpha _{{3}}+12\,i{\alpha }^{6}\sqrt{3}\alpha _{{2}}\right. \\&\quad \left. -\, 72\,i{\alpha }^{8}\alpha _{{3}}{\tau }^{2}+180\,{\alpha }^{5}a\alpha _{{2}}-81\,\alpha \,\sqrt{3}{a}^{3}\alpha _{{3}}+72\,{\alpha }^{8}\sqrt{3}\alpha _{{3}}{\tau }^{2}\right. \\&\quad \left. +\,144\,{\alpha }^{8}\sqrt{3}\tau \alpha _{{2}}-864\,{\alpha }^{7}\sqrt{3}a\alpha _{{1}}-24\,{\alpha }^{7}\sqrt{3}\eta \alpha _{{2}}\right. \\&\quad \left. -\,6480\,{\alpha }^{6}\sqrt{3}{a}^{2}\alpha _{{1}}-6\,{\alpha }^{6}\sqrt{3}\alpha _{{3}}{\eta }^{2}+8640\,{\alpha }^{5}\sqrt{3}{a}^{3}\alpha _{{1}}\right. \\&\quad \left. -\,7776\,{\alpha }^{3}\sqrt{3}{a}^{5}\alpha _{{1}}+33\,{\alpha }^{3}\sqrt{3}a\alpha _{{3}}+2160\,{\alpha }^{4}\sqrt{3}\eta {a}^{3}\alpha _{{3}}\tau \right. \\&\quad \left. -\,1080\,{\alpha }^{3}\sqrt{3}{a}^{4}\eta \alpha _{{3}}\tau +36\,{\alpha }^{6}\alpha _{{2}}-864\,{\alpha }^{7}\sqrt{3}at\alpha _{{2}}\right. \\&\quad \left. -\,432\,{\alpha }^{7}\sqrt{3}a\alpha _{{3}}{\tau }^{2}-24\,{\alpha }^{7}\sqrt{3}\eta \alpha _{{3}}\tau -3240\,{\alpha }^{6}\sqrt{3}{a}^{2}\alpha _{{3}}{\tau }^{2}\right. \\&\quad \left. -\,6480\,{\alpha }^{6}\sqrt{3}{a}^{2}\tau \alpha _{{2}}-360\,{\alpha }^{6}\sqrt{3}\eta a\alpha _{{2}}+4320\,{\alpha }^{5}\sqrt{3}{a}^{3}\alpha _{{3}}{\tau }^{2}\right. \\&\quad \left. +\,8640\,{\alpha }^{5}\sqrt{3}{a}^{3}\tau \alpha _{{2}}+720\,{\alpha }^{5}\sqrt{3}\eta {a}^{2}\alpha _{{2}}+24\,{\alpha }^{5}\sqrt{3}a\alpha _{{3}}{\eta }^{2}\right. \\&\quad \left. -\,1296\,i{\alpha }^{7}a\alpha _{{3}}{\tau }^{2}-2592\,i{\alpha }^{7}a\tau \alpha _{{2}}-72\,i{\alpha }^{7}\eta \alpha _{{3}}\tau \right. \\&\quad \left. +\,3240\,i{\alpha }^{6}{a}^{2}\alpha _{{3}}{\tau }^{2}+6480\,i{\alpha }^{6}{a}^{2}\tau \alpha _{{2}}\right. \\&\quad \left. +\,360\,i{\alpha }^{6}a\eta \alpha _{{2}}+360\,i{\alpha }^{6}a\eta \alpha _{{3}}\tau -360\,i{\alpha }^{4}\sqrt{3}{a}^{2}\alpha _{{3}}\tau \right. \\&\quad \left. -\,42\,i{\alpha }^{4}\sqrt{3}\alpha _{{3}}a\eta +1080\,i{\alpha }^{3}\sqrt{3}{a}^{3}\alpha _{{3}}\tau +126\,i{\alpha }^{3}\sqrt{3}{a}^{2}\alpha _{{3}}\eta \right. \\&\quad \left. +\,90\,i{\alpha }^{2}\sqrt{3}{a}^{3}\alpha _{{3}}\eta -324\,i\alpha \,\sqrt{3}{a}^{5}\alpha _{{3}}\tau -1080\,{a}^{3}{\alpha }^{3}\alpha _{{2}}\right. \\&\quad \left. +\,1620\,{a}^{4}{\alpha }^{2}\alpha _{{2}}+324\,{a}^{5}\alpha \,\alpha _{{2}}+144\,{\alpha }^{8}\sqrt{3}\alpha _{{1}}-{\alpha }^{4}\sqrt{3}\alpha _{{3}}\right. \\&\quad \left. -144\,i{\alpha }^{8}\alpha _{{1}}+9\,i{\alpha }^{4}\alpha _{{3}}+54\,i{a}^{4}\alpha _{{3}}+180\,{\alpha }^{5}a\alpha _{{3}}\tau \right. \\&\quad \left. -\,7776\,{\alpha }^{3}\sqrt{3}{a}^{5}\tau \alpha _{{2}}-1080\,{\alpha }^{3}\sqrt{3}{a}^{4}\eta \alpha _{{2}}-72\,{\alpha }^{3}\sqrt{3}{a}^{3}\alpha _{{3}}{\eta }^{2}\right. \\&\quad \left. -1944\,{\alpha }^{2}\sqrt{3}{a}^{6}\alpha _{{3}}{\tau }^{2}-3888\,{\alpha }^{2}\sqrt{3}{a}^{6}\tau \alpha _{{2}}\right. \\&\quad \left. -\,648\,{\alpha }^{2}\sqrt{3}{a}^{5}\eta \alpha _{{2}}-54\,{\alpha }^{2}\sqrt{3}{a}^{4}\alpha _{{3}}{\eta }^{2}-3888\,{\alpha }^{2}\sqrt{3}{a}^{6}\alpha _{{1}}\right. \\&\quad \left. +\,9\,{\alpha }^{2}\sqrt{3}{a}^{2}\alpha _{{3}}+19440\,{\alpha }^{4}\sqrt{3}{a}^{4}\alpha _{{1}}-1080\,{\alpha }^{4}{a}^{2}\alpha _{{2}}\right. \\&\quad \left. -\,54\,i\alpha \,\sqrt{3}{a}^{4}\alpha _{{3}}\eta -180\,i{\alpha }^{5}\sqrt{3}a\alpha _{{3}}\tau -360\,i{\alpha }^{4}\sqrt{3}{a}^{2}\alpha _{{2}}\right. \\&\quad \left. +\,1080\,i{\alpha }^{3}\sqrt{3}{a}^{3}\alpha _{{2}}+540\,i{\alpha }^{2}\sqrt{3}{a}^{4}\alpha _{{2}}+540\,i{\alpha }^{2}\sqrt{3}{a}^{4}\alpha _{{3}}\tau \right. \\&\quad \left. -\,324\,i\alpha \,\sqrt{3}{a}^{5}\alpha _{{2}}+12\,i{\alpha }^{6}\sqrt{3}\alpha _{{3}}\tau -180\,i{\alpha }^{5}\sqrt{3}a\alpha _{{2}}\right. \\&\quad \left. -\, 8\,i{\alpha }^{5}\sqrt{3}\alpha _{{3}}\eta -144\,i{\alpha }^{8}\tau \alpha _{{2}}-2592\,i{\alpha }^{7}a\alpha _{{1}}\right. \\&\quad \left. -\, 72\,i{\alpha }^{7}\eta \alpha _{{2}}+6480\,i{\alpha }^{6}{a}^{2}\alpha _{{1}}+6\,i{\alpha }^{6}\alpha _{{3}}{\eta }^{2}\right. \\&\quad \left. +\, 25920\,i{\alpha }^{5}{a}^{3}\alpha _{{1}}-648\,{\alpha }^{2}\sqrt{3}{a}^{5}\eta \alpha _{{3}}\tau \right. \\&\quad \left. -\, 360\,{\alpha }^{6}\sqrt{3}a\eta \alpha _{{3}}\tau +720\,{\alpha }^{5}\sqrt{3}\eta {a}^{2}\alpha _{{3}}\tau \right. \\&\left. +9720\,{\alpha }^{4}\sqrt{3}{a}^{4}\alpha _{{3}}{\tau }^{2}+19440\,{\alpha }^{4}\sqrt{3}{a}^{4}\tau \alpha _{{2}}\right. \\&\quad \left. +\, 2160\,{\alpha }^{4}\sqrt{3}\eta {a}^{3}\alpha _{{2}}+108\,{\alpha }^{4}\sqrt{3}{a}^{2}\alpha _{{3}}{\eta }^{2}\right. \\&\quad \left. -\, 3888\,{\alpha }^{3}\sqrt{3}{a}^{5}\alpha _{{3}}{\tau }^{2}+25920\,i{\alpha }^{5}{a}^{3}\tau \alpha _{{2}}\right. \\&\quad \left. +\, 12960\,i{\alpha }^{5}{a}^{3}\alpha _{{3}}{\tau }^{2}+2160\,i{\alpha }^{5}\eta {a}^{2}\alpha _{{2}}\right. \end{aligned}$$
$$\begin{aligned}&\quad \left. +\,2160\,i{\alpha }^{5}\eta {a}^{2}\alpha _{{3}}\tau +72\,i{\alpha }^{5}a\alpha _{{3}}{\eta }^{2}\right. \\&\quad \left. -\,19440\,i{\alpha }^{4}{a}^{4}\tau \alpha _{{2}}-9720\,i{\alpha }^{4}{a}^{4}\alpha _{{3}}{\tau }^{2}-2160\,i{\alpha }^{4}\eta {a}^{3}\alpha _{{2}}\right. \\&\quad \left. -\,2160\,i{\alpha }^{4}\eta {a}^{3}\alpha _{{3}}\tau -108\,i{\alpha }^{4}{a}^{2}\alpha _{{3}}{\eta }^{2}-23328\,i{a}^{5}{\alpha }^{3}\tau \alpha _{{2}}\right. \\&\quad \left. -\,11664\,i{a}^{5}{\alpha }^{3}\alpha _{{3}}{\tau }^{2}-3240\,i{a}^{4}{\alpha }^{3}\eta \alpha _{{2}}-3240\,i{a}^{4}{\alpha }^{3}\eta \alpha _{{3}}t\right. \\&\quad \left. -\,216\,i{a}^{3}{\alpha }^{3}\alpha _{{3}}{\eta }^{2}+3888\,i{a}^{6}{\alpha }^{2}\tau \alpha _{{2}}+1944\,i{a}^{6}{\alpha }^{2}\alpha _{{3}}{\tau }^{2}\right. \\&\quad \left. +\,648\,i{a}^{5}{\alpha }^{2}\eta \alpha _{{2}}+648\,i{a}^{5}{\alpha }^{2}\eta \alpha _{{3}}t+54\,i{a}^{4}{\alpha }^{2}\alpha _{{3}}{\eta }^{2} \right) ,\\ \end{aligned}$$
$$\begin{aligned}&\psi _{{31}}=\psi _{{32}}=\psi _{{33}}\\&={\frac{1}{96}}\,{\frac{1}{{\alpha }^{2} \left( \sqrt{3}{\alpha }^{3}-9\,\sqrt{3}{a}^{2}\alpha -9\,i{\alpha }^{2}a+9\,i{a}^{3} \right) ^{2}}}\\&\quad \times \,\left( -2160\,{\alpha }^{4}\sqrt{3}\eta {a}^{3}\alpha _{{3}}\tau -108\,{\alpha }^{4}\sqrt{3}{a}^{2}\alpha _{{3}}{\eta }^{2}\right. \\&\quad \left. -\, 3888\,{\alpha }^{3}\sqrt{3}{a}^{5}\alpha _{{3}}{\tau }^{2}-7776\,{\alpha }^{3}\sqrt{3}{a}^{5}\tau \alpha _{{2}}\right. \\&\quad \left. -\, 1080\,{\alpha }^{3}\sqrt{3}{a}^{4}\eta \alpha _{{2}}-72\,{\alpha }^{3}\sqrt{3}{a}^{3}\alpha _{{3}}{\eta }^{2}\right. \\&\quad \left. +\, 720\,{\alpha }^{5}\sqrt{3}\eta {a}^{2}\alpha _{{3}}\tau -1080\,{\alpha }^{3}\sqrt{3}{a}^{4}\eta \alpha _{{3}}\tau \right. \\&\quad \left. +\, 1944\,{\alpha }^{2}\sqrt{3}{a}^{6}\alpha _{{3}}{\tau }^{2}+3888\,{\alpha }^{2}\sqrt{3}{a}^{6}\tau \alpha _{{2}}\right. \\&\quad \left. -\, 864\,{\alpha }^{7}\sqrt{3}a\tau \alpha _{{2}}+3240\,{\alpha }^{6}\sqrt{3}{a}^{2}\alpha _{{3}}{\tau }^{2}\right. \\&\quad \left. +\,6480\,{\alpha }^{6}\sqrt{3}{a}^{2}\tau \alpha _{{2}}+360\,{\alpha }^{6}\sqrt{3}a\eta \alpha _{{2}}\right. \\&\quad \left. +\, 648\,{\alpha }^{2}\sqrt{3}{a}^{5}\eta \alpha _{{3}}\tau -324\,i\alpha \,\sqrt{3}{a}^{5}\alpha _{{2}}\right. \\&\quad \left. -\, 540\,i{\alpha }^{2}\sqrt{3}{a}^{4}\alpha _{{3}}\tau +23328\,i{a}^{5}{\alpha }^{3}\tau \alpha _{{2}}\right. \\&\quad \left. +\, 11664\,i{a}^{5}{\alpha }^{3}\alpha _{{3}}{\tau }^{2}+3240\,i{a}^{4}{\alpha }^{3}\eta \alpha _{{2}}\right. \\&\quad \left. +\, 3240\,i{a}^{4}{\alpha }^{3}\eta \alpha _{{3}}\tau +216\,i{a}^{3}{\alpha }^{3}\alpha _{{3}}{\eta }^{2}\right. \\&\quad \left. +\, 3888\,i{a}^{6}{\alpha }^{2}\tau \alpha _{{2}}+1944\,i{a}^{6}{\alpha }^{2}\alpha _{{3}}{\tau }^{2}\right. \\&\quad \left. +\, 648\,i{a}^{5}{\alpha }^{2}\eta \alpha _{{2}}+648\,i{a}^{5}{\alpha }^{2}\eta \alpha _{{3}}\tau \right. \\&\quad \left. +\, 1080\,{a}^{3}{\alpha }^{3}\alpha _{{2}}+1620\,{a}^{4}{\alpha }^{2}\alpha _{{2}}\right. \\&\quad \left. -\, 324\,{a}^{5}\alpha \,\alpha _{{2}}-144\,{\alpha }^{8}\sqrt{3}\alpha _{{1}}+{\alpha }^{4}\sqrt{3}\alpha _{{3}}\right. \\&\quad \left. +\, 54\,i{a}^{4}{\alpha }^{2}\alpha _{{3}}{\eta }^{2}+2592\,i{\alpha }^{7}a\tau \alpha _{{2}}\right. \\&\quad \left. +\, 72\,i{\alpha }^{7}\eta \alpha _{{3}}\tau +6480\,i{\alpha }^{6}{a}^{2}\tau \alpha _{{2}}\right. \\&\quad \left. +\, 360\,i{\alpha }^{6}a\eta \alpha _{{2}}+360\,i{\alpha }^{6}a\eta \alpha _{{3}}\tau \right. \\&\quad \left. -\, 25920\,i{\alpha }^{5}{a}^{3}\tau \alpha _{{2}}-12960\,i{\alpha }^{5}{a}^{3}\alpha _{{3}}{\tau }^{2}\right. \\&\quad \left. -\, 2160\,i{\alpha }^{5}\eta {a}^{2}\alpha _{{2}}-2160\,i{\alpha }^{5}\eta {a}^{2}\alpha _{{3}}t\right. \\&\quad \left. -\, 72\,i{\alpha }^{5}a\alpha _{{3}}{\eta }^{2}-19440\,i{\alpha }^{4}{a}^{4}\tau \alpha _{{2}}\right. \\&\quad \left. -\, 9720\,i{\alpha }^{4}{a}^{4}\alpha _{{3}}{\tau }^{2}-2160\,i{\alpha }^{4}\eta {a}^{3}\alpha _{{2}}\right. \\&\quad \left. -\, 2160\,i{\alpha }^{4}\eta {a}^{3}\alpha _{{3}}\tau +648\,{\alpha }^{2}\sqrt{3}{a}^{5}\eta \alpha _{{2}}\right. \\&\quad \left. +\, 54\,{\alpha }^{2}\sqrt{3}{a}^{4}\alpha _{{3}}{\eta }^{2}-12\,i{\alpha }^{6}\sqrt{3}\alpha _{{2}}\right. \\&\quad \left. -\, 72\,i{\alpha }^{8}\alpha _{{3}}{\tau }^{2}-144\,i{\alpha }^{8}\tau \alpha _{{2}}\right. \\&\quad \left. +\, 9\,i{a}^{3}\alpha \,\alpha _{{3}}+2592\,i{\alpha }^{7}a\alpha _{{1}}+72\,i{\alpha }^{7}\eta \alpha _{{2}}\right. \\&\quad \left. +\, 6480\,i{\alpha }^{6}{a}^{2}\alpha _{{1}}+6\,i{\alpha }^{6}\alpha _{{3}}{\eta }^{2}-25920\,i{\alpha }^{5}{a}^{3}\alpha _{{1}}\right. \\&\quad \left. -\,19440\,i{\alpha }^{4}{a}^{4}\alpha _{{1}}+23328\,i{a}^{5}{\alpha }^{3}\alpha _{{1}}-9\,ia{\alpha }^{3}\alpha _{{3}}\right. \\&\quad \left. +\,3888\,i{a}^{6}{\alpha }^{2}\alpha _{{1}}\!-\!135i{a}^{2}{\alpha }^{2}\alpha _{{3}}\!+\!9\,i{\alpha }^{4}\alpha _{{3}}\!-\!144\,i{\alpha }^{8}\alpha _{{1}}\right. \\&\quad \left. -\,54\,{a}^{4}\alpha \,\alpha _{{3}}\eta +3888\,{\alpha }^{2}\sqrt{3}{a}^{6}\alpha _{{1}}-9\,{\alpha }^{2}\sqrt{3}{a}^{2}\alpha _{{3}}\right. \\&\quad \left. -\,81\,\alpha \,\sqrt{3}{a}^{3}\alpha _{{3}}-72\,{\alpha }^{8}\sqrt{3}\alpha _{{3}}{\tau }^{2}-144\,{\alpha }^{8}\sqrt{3}\tau \alpha _{{2}}\right. \\&\quad \left. -\,864\,{\alpha }^{7}\sqrt{3}a\alpha _{{1}}-24\,{\alpha }^{7}\sqrt{3}\eta \alpha _{{2}}+6480\,{\alpha }^{6}\sqrt{3}{a}^{2}\alpha _{{1}}\right. \\&\quad \left. +\,6{\alpha }^{6}\sqrt{3}\alpha _{{3}}{\eta }^{2}\!+\!8640\,{\alpha }^{5}\sqrt{3}{a}^{3}\alpha _{{1}}\!-\!19440\,{\alpha }^{4}\sqrt{3}{a}^{4}\alpha _{{1}}\right. \\&\quad \left. -\,7776\,{\alpha }^{3}\sqrt{3}{a}^{5}\alpha _{{1}}+33\,{\alpha }^{3}\sqrt{3}a\alpha _{{3}}+54\,i{a}^{4}\alpha _{{3}}\right. \\&\quad \left. -\,432\,{\alpha }^{7}\sqrt{3}a\alpha _{{3}}{t}^{2}-24\,{\alpha }^{7}\sqrt{3}\eta \alpha _{{3}}\tau -180\,{\alpha }^{5}a\alpha _{{3}}\tau \right. \end{aligned}$$
$$\begin{aligned}&\quad \left. +\,360\,i{\alpha }^{4}\sqrt{3}{a}^{2}\alpha _{{3}}\tau -108\,i{\alpha }^{4}{a}^{2}\alpha _{{3}}{\eta }^{2}\right. \\&\quad \left. -\,90\,i{\alpha }^{2}\sqrt{3}{a}^{3}\alpha _{{3}}\eta -180\,i{\alpha }^{5}\sqrt{3}a\alpha _{{2}}\right. \\&\quad \left. -\,8\,i{\alpha }^{5}\sqrt{3}\alpha _{{3}}\eta +360\,i{\alpha }^{4}\sqrt{3}{a}^{2}\alpha _{{2}}\right. \\&\quad \left. +\,1080\,i{\alpha }^{3}\sqrt{3}{a}^{3}\alpha _{{2}}-180\,i{\alpha }^{5}\sqrt{3}a\alpha _{{3}}\tau \right. \\&\quad \left. +\,42\,i{\alpha }^{4}\sqrt{3}\alpha _{{3}}a\eta +1080\,i{\alpha }^{3}\sqrt{3}{a}^{3}\alpha _{{3}}\tau \right. \\&\quad \left. +\,126\,i{\alpha }^{3}\sqrt{3}{a}^{2}\alpha _{{3}}\eta -540\,i{\alpha }^{2}\sqrt{3}{a}^{4}\alpha _{{2}}\right. \\&\quad \left. -\,324\,i\alpha \,\sqrt{3}{a}^{5}\alpha _{{3}}\tau -54\,i\alpha \,\sqrt{3}{a}^{4}\alpha _{{3}}\eta \right. \\&\quad \left. +\,3240\,i{\alpha }^{6}{a}^{2}\alpha _{{3}}{\tau }^{2}+1296\,i{\alpha }^{7}a\alpha _{{3}}{\tau }^{2}\right. \\&\quad \left. -\,180\,{\alpha }^{5}a\alpha _{{2}}-12\,{\alpha }^{5}\alpha _{{3}}\eta \right. \\&\quad \left. +\,36\,{\alpha }^{6}\alpha _{{3}}\tau +36\,{\alpha }^{6}\alpha _{{2}}+360\,{\alpha }^{6}\sqrt{3}a\eta \alpha _{{3}}\tau \right. \\&\quad \left. +\,4320\,{\alpha }^{5}\sqrt{3}{a}^{3}\alpha _{{3}}{\tau }^{2}+8640\,{\alpha }^{5}\sqrt{3}{a}^{3}\tau \alpha _{{2}}\right. \\&\quad \left. +\,720\,{\alpha }^{5}\sqrt{3}\eta {a}^{2}\alpha _{{2}}+24\,{\alpha }^{5}\sqrt{3}a\alpha _{{3}}{\eta }^{2}\right. \\&\quad \left. -\,9720\,{\alpha }^{4}\sqrt{3}{a}^{4}\alpha _{{3}}{\tau }^{2}-19440\,{\alpha }^{4}\sqrt{3}{a}^{4}\tau \alpha _{{2}}\right. \\&\quad \left. -\,2160\,{\alpha }^{4}\sqrt{3}\eta {a}^{3}\alpha _{{2}}-12\,i{\alpha }^{6}\sqrt{3}\alpha _{{3}}\tau \right. \\&\quad \left. -\,1080\,{\alpha }^{4}{a}^{2}\alpha _{{3}}\tau -90\,{\alpha }^{4}\alpha _{{3}}a\eta +1080\,{a}^{3}{\alpha }^{3}\alpha _{{3}}\tau \right. \\&\quad \left. +\,162\,{a}^{2}{\alpha }^{3}\alpha _{{3}}\eta +1620\,{a}^{4}{\alpha }^{2}\alpha _{{3}}\tau +234\,{a}^{3}{\alpha }^{2}\alpha _{{3}}\eta \right. \\&\quad \left. -\,324\,{a}^{5}\alpha \,\alpha _{{3}}\tau -1080\,{\alpha }^{4}{a}^{2}\alpha _{{2}} \right) , \end{aligned}$$

and \(\alpha _{{1 }}, \alpha _{{2}}, \alpha _{{3 }}\) are arbitrary constants.

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Yu, F. Matter rogue waves and management by external potentials for coupled Gross–Pitaevskii equation. Nonlinear Dyn 80, 685–699 (2015). https://doi.org/10.1007/s11071-015-1898-3

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