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Derivation of anomalously interacting lumps for the (2+1)-dimensional generalized Korteweg–de Vries equation via degeneracy of lump chains

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Abstract

This paper investigates two different novel approaches to degenerate from normally interacting lump chains to anomalously interacting lumps for the generalized Korteweg–de Vries equation. The first direct degeneracy method is to derive the anomalously interacting lumps by adjusting the appropriate phase parameter to make the periods of lump chains with close velocities trend to infinity. The second double degeneracy method is to first derive the anomalously interacting lump chains through the same parameters, and then the anomalously interacting lumps can be similarly obtained by adjusting its period to tend to infinity. Furthermore, anomalously interacting lump chains and lump’s asymptotic behavior are also investigated. These interacting lump chains can provide a more profound understanding of the properties of lump.

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Funding

This work is supported by National Natural Science Foundation of China under Grant Nos. 12301588, 12175111, 12235007, and K. C. Wong Magna Fund in Ningbo University.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, People’s Republic of China

    Wentao Li, Chongxia Lu & Biao Li

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  1. Wentao Li
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  3. Biao Li
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Contributions

WL: Conceptualization, methodology, software, investigation, formal analysis, writing—original draft. CL: Funding acquisition, investigation, supervision. BL: Conceptualization, funding acquisition, resources, supervision.

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Correspondence to Chongxia Lu.

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Appendix

Appendix

The functions \(h_{j}\) and \(h_{jk} \left( j,k=1,2,3\right) \) in Eq. (29) are expressed in the following form:

$$\begin{aligned} h_{1}= & {} -2\left( \varepsilon ^{2}\left( \frac{\textrm{d}}{\textrm{d} \varepsilon }\xi _{1}\right) ^{2} +\varepsilon ^{2}\frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}} \xi _{1}+\varepsilon \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1} +\frac{1}{2}\right) , \end{aligned}$$
(A1)
$$\begin{aligned} h_{2}= & {} -2\left( \varepsilon ^{2}\left( \frac{\textrm{d}}{\textrm{d} \varepsilon }\xi _{1}\right) ^{2}-\varepsilon ^{2}\frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1} -\varepsilon \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1} +\frac{1}{2}\right) , \end{aligned}$$
(A2)
$$\begin{aligned} h_{3}= & {} 1, \end{aligned}$$
(A3)
$$\begin{aligned} h_{11}= & {} \frac{-486\varepsilon ^{10}-1377\varepsilon ^{8}-1026\varepsilon ^{6}-306\varepsilon ^{4}-12\varepsilon ^{2}-1}{(3\varepsilon ^{2}-1)^{5}}\nonumber \\{} & {} +\frac{2\varepsilon \left( 81\varepsilon ^{8}+135\varepsilon ^{6}+108\varepsilon ^{4}+15\varepsilon ^{2}+1\right) }{\left( 3\varepsilon ^{2}-1\right) ^{4}}\nonumber \\{} & {} \times \left[ \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}+\frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}^{*}\right] \nonumber \\{} & {} +\frac{2\varepsilon ^2\left( 27\varepsilon ^6-36\varepsilon ^4-1\right) }{\left( 3\varepsilon ^{2}-1\right) ^3} \left[ \left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _1\right) ^2+\frac{\textrm{d}^2}{\textrm{d}\varepsilon ^2}\xi _1\right. \nonumber \\{} & {} \left. +\left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _1^*\right) ^2+\frac{\textrm{d}^2}{\textrm{d}\varepsilon ^2}\xi _1^*\right] \nonumber \\{} & {} -\frac{4\varepsilon ^2\left( 54\epsilon ^6-9\varepsilon ^4+18\varepsilon ^2+1\right) }{\left( 3\varepsilon ^{2}-1\right) ^3} \left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}\frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}^{*}\right) \nonumber \\{} & {} +\frac{4\varepsilon ^{3}\left( 3\varepsilon ^{2}+1\right) }{(3\varepsilon ^{2}-1)^{2}}\left[ \frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1}\frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}^{*}+\frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}\frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1}^{*}\right. \nonumber \\{} & {} \left. +\left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}^{*}+\frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}\right) \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}\frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}^{*}\right] \nonumber \\{} & {} -\frac{4\varepsilon ^{4}}{3\varepsilon ^{2}-1}\left[ \frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1} \frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1}^{*}+\frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1} \left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}^{*}\right) ^{2}\right. \nonumber \\{} & {} \left. +\left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}\right) ^{2} \frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1}^{*}+\left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}\right) ^{2} \left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}^{*}\right) ^{2}\right] , \end{aligned}$$
(A4)
$$\begin{aligned} h_{12}= & {} \frac{729\varepsilon ^{10}+972\varepsilon ^{8}+1377\varepsilon ^{6}+18\varepsilon ^{4}+15\varepsilon ^{2}-1}{\left( 3\varepsilon ^{2}-1\right) ^{5}}\nonumber \\{} & {} -\frac{2\varepsilon \left( 243\varepsilon ^{6}+54\varepsilon ^{4}+12\varepsilon ^{2}-1\right) }{\left( 3\varepsilon ^{2}-1\right) ^{4}}\frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}\nonumber \\{} & {} +\frac{2\varepsilon ^{2}\left( 9\varepsilon ^{4}+9\varepsilon ^{2}-1\right) }{\left( 3\varepsilon ^{2}-1\right) ^{3}} \left[ \frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1}+\left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}\right) ^{2}\right] \nonumber \\{} & {} -\frac{2\varepsilon \left( 891\varepsilon ^{6}+99\epsilon ^{4}+15\varepsilon ^{2}-1\right) }{\left( 3\varepsilon ^{2}-1\right) ^{5}} \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}^{*}\nonumber \\{} & {} +\frac{2\varepsilon ^2\left( 54\varepsilon ^6-99\varepsilon ^4-6\varepsilon ^2-1\right) }{\left( 3\varepsilon ^{2}-1\right) ^4} \left[ \frac{\textrm{d}^2}{\textrm{d}\varepsilon ^2}\xi _1^*\right. \nonumber \\{} & {} \left. -\left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _1^*\right) ^2\right] \nonumber \\{} & {} +\frac{4\varepsilon ^{2}\left( 54\varepsilon ^{6}+63\varepsilon ^{4}+12\varepsilon ^{2}-1\right) }{\left( 3\varepsilon ^{2}-1\right) ^{4}} \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}^{*}\frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}\nonumber \\{} & {} -\frac{4\varepsilon ^{3}\left( 9\varepsilon ^{2}-1\right) }{\left( 3\varepsilon ^{2}-1\right) ^{3}}\left[ \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}^{*}\frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1}+\frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}^{*}\left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}\right) ^{2}\right] \nonumber \\{} & {} +\frac{4\varepsilon ^{3}\left( 9\varepsilon ^{2}+1\right) }{\left( 3\varepsilon ^{2}-1\right) ^{3}} \left[ \frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1}^{*}\frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1} -\frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}\left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}^{*}\right) ^{2}\right] \nonumber \\{} & {} +\frac{4\varepsilon ^{4}}{(3\varepsilon ^{2}-1)^{2}}\left[ \frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1}\left( \frac{\textrm{d}}{\textrm{d}\epsilon }\xi _{1}^{*}\right) ^{2}\right. \nonumber \\{} & {} \left. +\left( \frac{\textrm{d}}{\textrm{d}\epsilon }\xi _{1}^{*}\right) ^{2}\left( \frac{\textrm{d}}{\textrm{d}\epsilon }\xi _{1}\right) ^{2} -\frac{\textrm{d}^{2}}{\textrm{d}\epsilon ^{2}}\xi _{1}\frac{\textrm{d}^{2}}{\textrm{d}\epsilon ^{2}}\xi _{1}^{*}\right. \nonumber \\{} & {} \left. -\left( \frac{\textrm{d}}{\textrm{d}\epsilon }\xi _{1}\right) ^{2}\frac{\textrm{d}^{2}}{\textrm{d}\epsilon ^{2}}\xi _{1}^{*}\right] , \end{aligned}$$
(A5)
$$\begin{aligned} h_{13}= & {} \frac{-81\varepsilon ^{6}+198\varepsilon ^{4}+12\varepsilon ^{2}+1}{\left( 3\varepsilon ^{2}-1\right) ^{5}}\nonumber \\{} & {} -\frac{2\varepsilon \left( 15\varepsilon ^{2}+1\right) }{\left( 3\varepsilon ^{2}-1\right) ^{4}} \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}+\frac{2\varepsilon ^{2}}{\left( 3\varepsilon ^{2}-1\right) ^{3}}\nonumber \\{} & {} \times \left[ \left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}\right) ^{2}+\frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1}\right] , \end{aligned}$$
(A6)
$$\begin{aligned} h_{22}= & {} \frac{2187\varepsilon ^{12}+20898\varepsilon ^{10}-5670\varepsilon ^{8}+216\varepsilon ^{6}+252\varepsilon ^{4} -24\varepsilon ^{2}+1}{\left( 3\varepsilon ^{2}-1\right) ^{8}}\nonumber \\{} & {} -\frac{2\varepsilon \left( 1620\varepsilon ^{8}+513\varepsilon ^{6}-189\varepsilon ^{4}+21\varepsilon ^{2}-1\right) }{\left( 3\varepsilon ^{2}-1\right) ^{7}}\nonumber \\{} & {} \left[ \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}+\frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}^{*}\right] \nonumber \\{} & {} +\frac{2\varepsilon ^{2}\left( 54\varepsilon ^{6}+63\varepsilon ^{4}-18\varepsilon ^{2}+1\right) }{\left( 3\varepsilon ^{2}-1\right) ^{6}} \left[ \left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}\right) ^{2}\right. \nonumber \\{} & {} \left. -\frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1}+\left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}^{*}\right) ^{2} -\frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1}^{*}\right] \nonumber \\{} & {} +\frac{4\varepsilon ^{2}\left( 54\varepsilon ^{6}+207\varepsilon ^{4}-18\varepsilon ^{2}+1\right) }{\left( 3\varepsilon ^{2}-1\right) ^{6}} \frac{\textrm{d}}{\textrm{d}\epsilon }\xi _{1}^{*}\frac{\textrm{d}}{\textrm{d}\epsilon }\xi _{1}\nonumber \\{} & {} +\frac{4\varepsilon ^{3}\left( 15\varepsilon ^{2}-1\right) }{\left( 3\varepsilon ^{2}-1\right) ^{5}} \left[ \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}^{*}\frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1} +\frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1}^{*}\frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}\right. \nonumber \\{} & {} \left. -\left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _1\right) ^2\frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _1^{*} -\frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _1\left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _1^*\right) ^2\right] \nonumber \\{} & {} +\frac{4\varepsilon ^{4}}{\left( 3\varepsilon ^{2}-1\right) ^{4}}\left[ \frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1} \frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1}^{*}-\frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1} \left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}^{*}\right) ^{2}\right. \nonumber \\{} & {} \left. -\left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}\right) ^{2} \frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1}^{*}+\left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}^{*}\right) ^{2} \left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}\right) ^{2}\right] , \end{aligned}$$
(A7)
$$\begin{aligned} h_{23}= & {} \frac{-81\varepsilon ^{6}-144\varepsilon ^{4}+24\varepsilon ^{2}-1}{\left( 3\varepsilon ^{2}-1\right) ^{8}}\nonumber \\{} & {} +\frac{2\varepsilon \left( 21\epsilon ^{2}-1\right) }{\left( 3\varepsilon ^{2}-1\right) ^{7}}\frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1} +\frac{2\varepsilon ^{2}}{\left( 3\varepsilon ^{2}-1\right) ^{6}}\nonumber \\{} & {} \left[ \frac{\textrm{d}^{2}}{\textrm{d}\varepsilon ^{2}}\xi _{1}-\left( \frac{\textrm{d}}{\textrm{d}\varepsilon }\xi _{1}\right) ^{2}\right] , \end{aligned}$$
(A8)
$$\begin{aligned} h_{33}= & {} -\frac{1}{\left( 3\varepsilon ^{2}-1\right) ^9}. \end{aligned}$$
(A9)

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Li, W., Lu, C. & Li, B. Derivation of anomalously interacting lumps for the (2+1)-dimensional generalized Korteweg–de Vries equation via degeneracy of lump chains. Nonlinear Dyn 112, 7359–7375 (2024). https://doi.org/10.1007/s11071-024-09395-0

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