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Singular-loop rogue wave and mixed interaction solutions with location control parameters for Wadati–Konno–Ichikawa equation

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Abstract

This paper is devoted to studying the complete integrable Wadati–Konno–Ichikawa equation, which is an important integrable model with physical background. Based on the known hodograph transformation, we give an alternative two-component nonlinear system of this equation. By constructing its special generalized \((m, N-m)\)-fold Darboux transformation, we obtain various location-manageable localized wave solutions, like higher-order rogue wave and periodic wave solutions with smooth, singular and singular-loop structures. It is found that the rogue wave can show a singular-loop structure when the special parameters are selected. For the first-order exact solutions, we analyze and summarize the reasons for singular structures when the plane wave amplitude reaches a certain value. Furthermore, we also discuss and summarize mixed interaction structures of diverse localized waves. In particular, these abundant structures can be managed to an arbitrary location by adjusting some control parameters.

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

    Zhe Lin & Xiao-Yong Wen

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  1. Zhe Lin
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Appendix

Appendix

The expressions of \(E_0...E_3, F_0...F_4\) in the solutions(19):

$$\begin{aligned} E_0&=-16\, \left( {c}^{2}+1 \right) ^{3} \bigg [ -\frac{1}{2}+2\,{a}^{2}{c}^{5}e_{{0}}d_{{0 }}\\&\quad + {a}^{2}{c}^{4}\left( -3\,{d_{{0}}}^{2}+{e_{{0}}}^{2} \right) -2\,{a}^{2}{c}^{3}e_{{0}}d_{{0}}\\&\quad -{a}^{2}{c}^{2} \left( {d_{{0}}}^{2}+{e_{{0}}}^{2} \right) + acd_{{0}} \left( {c}^{2}+1 \right) ^{3/2} \bigg ] \\&\quad -32c\,\textrm{i} \left( {c}^{2}+1 \right) ^{3} \bigg [ a \sqrt{{c}^{2}+1} \\&\quad \times \left( {a}^{3}{c}^{3}d _{{0}}^{3} +{a}^{3}{c}^{3}e_{0}^{2}d_{{0}} +{a}^{2}{c}^{2}e_{{0}}{d_{{0}}}^{2}+{a}^{2}{c}^{2}e_{{0}}^{3}+\frac{1}{2}\,e_{{0}} \right) \\&\quad +{a}^{2}{c}^{3}e_{{0}}d_{{0}}+{a}^{2}{c}^{2} \left( -{d_{{0}}}^{2}+{e_{{0}}}^{2} \right) -{a}^{2}ce_{{0 }}d_{{0}} +\frac{1}{4}\bigg ] ,\\ E_1&=16\,ac \left( {c}^{2}+1 \right) ^{2} \left( ac^2\sqrt{{c}^{2}+1}+a\sqrt{{c}^{2}+1}\right. \\&\quad \left. +{a}^{2}{c}^{5}d_{{0}}+3\,{a}^{2}{c}^{4}e_{ {0}}-7\,{a}^{2}{c}^{3}d_{{0}}-3\,{a}^{2}{c}^{2}e_{{0}}-2\,{a}^{2}cd_{{0 }} \right) T\\&\quad +16\,a \left( {c}^{2}+1 \right) ^{2}c \left[ a\sqrt{ {c}^{2}+1} \left( 3\,{c}^{3}d_{{0}}+{c}^{2}e_{{0}}+cd_{{0}}\right. \right. \\&\quad \left. \left. -{c} ^{4}e_{{0}}\right) -\frac{1}{2}\,{c}^{4}-{c}^{2}-\frac{1}{2} \right] y\\&\quad +\textrm{i}\Bigg \{ 8\,{a}^{2}{c}^{2} \left( {c}^{2}+1 \right) ^{2}\bigg [ \left( 4\,{a}^ {2}{c}^{3}e_{{0}}d_{{0}}+14{a}^{2}{c}^{2}{d_{{0}}}^{2}\right. \\&\quad \left. +10{a}^{2}{c}^{2}{e_{{0}}}^{2} +4\,{a}^{2}ce_{{0}}d_{{0}}+1 \right) \sqrt{{c} ^{2}+1}\\&\quad +a \left( {c}^{3}d_{{0}}+4\,{c}^{2}e_{{0}}-5\,cd_{{0}}-2\,e_{{0 }} \right) \bigg ] T \\&\quad +16\,{a}^{2}{c}^{2} \left( {c}^{2}+1 \right) ^{2} \bigg [ \left( -{c}^{2}e_{{0}}+2\,cd_{{0}}+e_{{0}} \right) \sqrt{{c} ^{2}+1}\\&\quad -a{c}^{4} \left( 3\,{d_{{0}}}^{2}+{e_{{0}}}^{2} \right) \\&\quad -2a\,{c}^{3}d_{{0}}e_{{0}}-a{c}^{2} \left( 3\,{d_{{0}}}^{2}+{e_{{0}}}^{2} \right) \\&\quad -2ac\,e_{{0}}d_{{0}} \bigg ] y\Bigg \}, \\ E_2&=-4\,{a}^{2}{c}^{2} \left( {c}^{2}+1 \right) \left( 5\,{a}^{2}{c}^{4}- 17\,{a}^{2}{c}^{2}-4\,{a}^{2} \right) {T}^{2}\\&\quad +8\,{a}^{3}{c}^{2} \left( {c}^{2 }+1 \right) ^{3/2}\left( {c}^{4}-7\,{c}^{2}-2 \right) Ty \\&\quad +4\,{a }^{2}{c}^{2} \left( {c}^{2}+1 \right) \\&\left( 3\,{c}^{4}+4\,{c}^{2}+1 \right) {y}^{2}+\textrm{i} \bigg \{ -8\,{a}^{2} \left( {c}^{2}+1 \right) {c}^{3} \left[ {a}^{3}\sqrt{{c}^{2}+1} \right. \\&\quad \left. \left( {c}^{3}d_{{0}}+7\,{c}^{2}e_{{0}}+16\,cd_{{0}}+4\,e_{{0 }} \right) - \left( -3\,{c}^{2}+6 \right) {a}^ {2} \right] {T}^{2}\\&\quad -8\,{a}^{2} \left( {c}^{2}+1 \right) {c}^{3} \left[ a \left( -{c}^{2}+5 \right) \sqrt{{c}^{2}+1}\right. \\&\quad \left. -2{a}^{2}\, \left( {c}^{ 4}e_{{0}}+7\,{c}^{3}d_{{0}}+3\,{c}^{2}e_{{0}}+7\,cd_{{0}}+2\,e_{{0}} \right) \right] Ty\\&\quad -8\,{a}^{2} \left( {c}^{2}+1 \right) \\&\quad {c}^{3} \left[ a \left( 3\,{c}^{3}d_{{0}}+{c}^{2}e_{{0}}+3\,cd_{{0}}+e_{{0}} \right) \sqrt{{c}^{2}+1}-{c}^{2}-1 \right] {y}^{2} \bigg \},\\ E_3&=\textrm{i} \left[ 12{a}^{6}{c}^{4}\, \left( {c}^{2}+4 \right) \sqrt{{c}^{2}+1} {T}^{3}\right. \\&\quad \left. -4{a}^{5}{c}^{4}\, \left( {c}^{2}+1 \right) \left( {c}^{2}+16 \right) y{T}^{2}\right. \\&\quad \left. +28\,{a}^{4} {c}^{4}\left( {c}^{2}+1 \right) ^{3/2}{y}^{2}T-4{a}^{3}{c}^{4}\, \left( {c}^{2}+1 \right) ^{2}{y}^{3}\right] ,\\ F_0&=4 \left( {c}^{2}+1 \right) ^{3}\, \Bigg \{ 4\,ac\sqrt{{c}^{2}+1} \\&\quad \left[ 2\,{a}^{2}{c}^{3}e_{{0}} \left( {d_{{0}}}^{2}+{e_{{0} }}^{2} \right) \right. \\&\quad \left. - \left( 2{a}^{2}{c}^{2}{d_{{0}}}^{3} +2{a}^{2}{c}^{2}{e_{{0}}}^{2}d_{{0}} +d_{{0}} \right) +ce_{{0}}-d_{{0}} \right] + \left( {c}^{2}+1 \right) \\&\quad \left[ 1+4\,{a}^{4} \left( {d_{{0}}}^{2}+{e_{{0}}}^{2} \right) ^{2}{c}^{4}-8\,{a}^{2}{c}^{3}d_{{0}}e_{{0}}\right. \\&\quad \left. +4\,{a}^{2} \left( 2\,{d_{{0}}}^{2}+{e_{{0}}}^{2} \right) {c}^{2} \right] \Bigg \} , \end{aligned}$$
$$\begin{aligned} F_1&=-8\,ac\left( {c}^{2}+1 \right) ^{2} \bigg \{ a\sqrt{{c}^{2}+1} \Big [ 2\, \left( {d_{{0}}}^{2}+3\,{e_{{0}}}^{2} \right) {a}^{2}{c}^{4}\\&\quad +4\,{a}^{2}{c}^{3}d_{{0}}e_{{0}}- \left( 12\,{a}^{2}{c}^{2}{d_{{0}}}^{2} +4\,{a}^{2}{c}^{2} {e_{{0}}}^{2} +1 \right) -2 \Big ] \\&\quad +\Big [ 4\,{a}^{4}{c}^{4}e_{{0}} \left( {c}^{2}+1 \right) \left( {d_{{0}}}^{2}+{e_{{0}}}^{2} \right) \\&\quad +2{a}^{2}{c}^{3}d_{{0}}\, \left( 4{a}^{2}{d_{{0}}}^{2}+4{a}^{2}{e_{{0}}}^{2} -1 \right) -2\,{a}^{2}{c}^{2}e_{{0}}\\&\quad +8\,{a}^{2}cd_{{0}} \Big ] \bigg \} T+8\,ac\left( {c}^{2}+1 \right) ^2 \\&\quad \bigg \{ a\sqrt{{c}^{2}+1} \Big [ 4\,{a}^{2}{c}^{5}d_{{0}} \left( {d_{{0}}}^{2}+{e_{{0}}}^{2} \right) \\&\quad -2\,{c}^{4}e_{{0}}+4d_{{0}}{c}^{3}\, \left( {a}^{2}e_0^{2}+{a}^{2}d_0^{2} +1 \right) \\&\quad -2\,{c}^{2}e_{{0}}+4\,cd_{{0}} \Big ] + \Big [ -4\,{a}^{2}{c}^{3}d_{{0}}e_{{0}}\\&\quad +{c}^{2}\left( 6a^2\,{d_{{0}}}^{2}+a^2{e_{{0}}}^{2}+1\right) +1 \Big ] \left( {c}^{2}+1 \right) \bigg \}y, \\ F_2&=4\,{a}^{4} {c}^{2} \left( {c}^{2}+1 \right) \bigg [ 2{a}^{2} {c}^{2}\left( {c}^{2}+1 \right) \, \left( 3\,{c}^{2}{e_{{0}}}^{2}\right. \\&\quad \left. +4{e_{{0}}}^{2} +8\,d_{{0}}ce_{{0}}+{d_{{0}}}^{2}{c}^{2}+12{d_{{0}}}^{2} \right) \\&\quad +6\,ac\sqrt{{c}^{2}+1} \left( {c}^{3}e_{{0}}+{c}^{2}d_{{0}}-4\,d_{{0}} \right) \\&-3\,{c}^{4}+5\,{c}^{2}+8 \bigg ] {T}^{2}-8\,{a}^{3}\left( {c}^{2}+1 \right) ^{2}{c}^{2} \bigg [ 4{a}^{2}{c}^{2}\,\sqrt{{c}^{2}+1}\\&\quad \left( d_{{0}}ce_{{0}}+3\,{d_{{0}}}^{2}+{e_{{0}}}^{2} \right) + 2a\,{c}^{2}e_{{0}}+2ac^3\,d_{{0}}-12acd_{{0}}\\&\quad -\sqrt{{c}^{2}+1} \left( {c}^{2}-4 \right) \bigg ] yT+8\,{a}^{2}{c}^{2} \left( {c}^{2}+1 \right) ^{2} \\&\quad \left[ {a}^{2}{c}^{2} \left( {c}^{2}+1 \right) \left( 3\,{d_{{0}}}^{2}+{e_{{0}}}^{2} \right) \right. \\&\quad \left. + ac\left( ce_{{0}}-3\,d_{{0}} \right) \sqrt{{c}^{2}+1}+{c}^{2}+1 \right] {y}^{2}, \\ F_3&=-4\,{a}^{3}{c}^{3} \bigg [ {a}^{3}\left( {c}^{4}+2\,{c}^{2}-8 \right) \sqrt{{c}^{2}+1}\\&\quad +2\, {a}^{4}c \left( {c}^{2}+1 \right) \left( {c}^{2}+4\right) \left( ce_{{0}}+2\,d_{{0}} \right) \bigg ] {T}^{3}\\&\quad -4\,{a}^{3}{c}^{3} \bigg [ -2\,{a}^{3} \left( {c}^{5}d_{{0}}+4\,{c}^{4}e_{{0}}\right. \\&\quad \left. +13\,{c}^{3}d_{{0}}+4\,{c}^{2}e_{{0}}+12\,cd_{{0}} \right) \sqrt{{c}^{2}+1}\\&\quad +{a}^{2} \left( {c}^{2}+1 \right) \left( -3\,{c}^{2}+12\right) \bigg ] {T}^{2}y-4\,{a}^{3}{c}^{3}\\ {}&\quad \times \bigg [ -a \left( {c}^{2}+1 \right) ^{3/2} \left( -{c}^{2}+6 \right) \\&\quad +2\,{a}^{2} \left( {c}^{2}+1 \right) \left( {c}^{4}e_{{0}}+6\,{c}^{3}d_{{0}}+{c}^{2}e_{{0}}+6\,c d_{{0}} \right) \bigg ] T{y}^{2}\\&\quad -4\,{a}^{3}{c}^{3}\bigg [ -2\, a \left( {c}^{2}+1 \right) ^{3/2} \left( {c}^{3}d_{{0}}+cd_{{0}} \right) + \left( {c}^{2}+1 \right) ^{2} \bigg ] {y}^{3}, \\ F_4&={a}^{8}{c}^{4}\left( {c}^{2}+4 \right) ^{2}{T}^{4} -8\,{a}^{7}{c}^{4} \left( {c}^{2}+4 \right) \sqrt{{c}^{2}+1}y{T}^{3}\\&\quad +2\,{a}^{6}{c}^{4} \left( {c}^{2}+12 \right) \left( {c}^{2}+1 \right) {y}^{2}{T}^{2}\\ {}&\quad -8 \,{a}^{5}{c}^{4} \left( {c}^{2}+1 \right) ^{3/2}{y}^{3}T\\&\quad +{a}^{4}{c}^{4} \left( {c}^{2}+1 \right) ^{2}{y}^{4}. \end{aligned}$$

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Lin, Z., Wen, XY. Singular-loop rogue wave and mixed interaction solutions with location control parameters for Wadati–Konno–Ichikawa equation. Nonlinear Dyn 111, 3633–3651 (2023). https://doi.org/10.1007/s11071-022-07984-5

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