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Creation of anomalously interacting lumps by degeneration of lump chains in the BKP equation

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Abstract

On the bases of N-soliton solutions of Hirota’s bilinear method, anomalously interacting lump patterns of (2+1)-dimensional Bogoyavlenskii–Kadomtsev–Petviashvili equation can be generated through normally interacting lump chains by two different paths. In the first path, through tending the periods of M normally lump chains with close velocities to infinity, \(M(M+1)/2\) anomalously interacting lumps are obtained directly. In the second path, there are two steps to generate lump waves. Firstly, M normally interacting lump chains degenerate into M anomalously interacting lump chains with the same parameters as the first path. Secondly, the anomalously lump waves can be derived from the M anomalously interacting lump chains when the period tends to infinity. Moreover, the distance between anomalously interacting lump chains varies with time proportional to \(\ln {|t|}\).

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References

  1. Yu, X., Gao, Y.T., Sun, Z.Y., Meng, X.H., Liu, Y., Feng, Q., Wang, M.Z.: N-soliton solutions for the (2+1)-dimensional Hirota-Maccari equation in fluids, plasmas and optical fibers. J. Math. Anal. Appl. 378, 519–527 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. Zuo, D.W., Gao, Y.T., Xue, L., Feng, Y.J.: Lax pair, rogue-wave and soliton solutions for a variable-coefficient generalized nonlinear Schrödinger equation in an optical fiber, fluid or plasma. Opt. Quant. Electron. 48, 1–14 (2016)

    Article  Google Scholar 

  3. Heimburg, T.: The thermodynamic soliton theory of the nervous impulse and possible medical implications. Prog. Biophys. Mol. Biol. 173, 24–35 (2022)

    Article  Google Scholar 

  4. Appali, R., Petersen, S., Van Rienen, U.: A comparision of Hodgkin-Huxley and soliton neural theories. Adv. Radio Sci. 8, 75–79 (2010)

    Article  Google Scholar 

  5. Nezlin, M.V.: The soliton model for the Great Red Spot and other large vortices in planetary atmospheres. Soviet Astron. Lett. 10, 221–226 (1984)

    Google Scholar 

  6. Belyaeva, T.L., Serkin, V.N.: Wave-particle duality of solitons and solitonic analog of the Ramsauer-Townsend effect. Eur. Phys. J. D. 66, 1–9 (2012)

    Article  Google Scholar 

  7. Ablowitz, M.J., Satsuma, J.: Solitons and rational solutions of nonlinear evolution equations. J. Math. Phys. 19, 2180–2186 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  8. Satsuma, J., Ablowitz, M.J.: Two-dimensional lumps in nonlinear dispersive systems. J. Math. Phys. 20, 1496–1503 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  9. Liu, Y.K., Li, B.: Dynamics of rogue waves on multisoliton background in the Benjamin Ono equation. Pramana. J. Phys. 88, 1–6 (2017)

    Article  Google Scholar 

  10. Rao, J.G., Liu, Y.B., Qian, C., He, J.S.: Rogue waves and hybrid solutions of the Boussinesq equation. Z. Naturforsch. A. 72, 307–314 (2017)

    Article  Google Scholar 

  11. Zhang, Z., Qi, Z.Q., Li, B.: Fusion and fission phenomena for (2+1)-dimensional fifth-order KdV system. Appl. Math. Lett. 116, 107004 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  12. Yang, X.Y., Fan, R., Li, B.: Soliton molecules and some novel interaction solutions to the (2+1)-dimensional B-type Kadomtsev-Petviashvili equation. Phys. Scripta 95, 045213 (2020)

    Article  Google Scholar 

  13. Li, J.H., Chen, Q.Q., Li, B.: Resonance Y-type soliton solutions and some new types of hybrid solutions in the (2+1)-dimensional Sawada-Kotera equation. Commun. Theor. Phys. 73, 045006 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhang, Z., Yang, S.X., Li, B.: Soliton molecules, asymmetric solitons and hybrid solutions for (2+1)-dimensional fifth-order KdV equation. Chinese Phys. Lett. 36, 120501 (2019)

    Article  Google Scholar 

  15. Zhang, Z., Yang, X.Y., Li, B., Wazwaz, A.M., Guo, Q.: Generation mechanism of high-order rogue waves via the improved long-wave limit method: NLS case. Phys. Lett. A 450, 128395 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yang, X.Y., Zhang, Z., Wazwaz, A.M., Wang, Z.: A direct method for generating rogue wave solutions to the (3+1)-dimensional Korteweg-de Vries Benjamin-Bona-Mahony equation. Phys. Lett. A 449, 128355 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sun, Y.J., Wu, J.J., Li, B.: Rogue wave solutions of (3+1)-dimensional Kadomtsev-Petviashvili equation by a direct limit method. Commun. Theor. Phys. 75, 065002 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wu, J.J., Li, B.: Resonant collisions among localized waves in the (2+1)-dimensional Hirota-Satsuma-Ito equation. Mod. Phys. Lett. B 36, 2250148 (2022)

    Article  MathSciNet  Google Scholar 

  19. Li, X., Wang, Y., Chen, M., Li, B.: Lump solutions and resonance stripe solitons to the (2+1)-dimensional Sawada-Kotera equation. Adv. Math. Phys. 2017, 1743789 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Qi, Z.Q., Chen, Q.Q., Wang, M.M., Li, B.: New mixed solutions generated by velocity resonance in the (2+1)-dimensional Sawada-Kotera equation. Nonlinear Dyn. 108, 1617–1626 (2022)

    Article  Google Scholar 

  21. Zhang, Z., Yang, X.Y., Li, B.: Novel soliton molecules and breather-positon on zero background for the complex modified KdV equation. Nonlinear Dyn. 100, 1551–1557 (2020)

    Article  Google Scholar 

  22. Zhang, Z., Li, B., Chen, J.C., Guo, Q., Stepanyants, Y.: Peculiarities of resonant interactions of lump chains within the KP1 equation. Phys. Scripta 97, 115205 (2022)

    Article  Google Scholar 

  23. Dong, J.Y., Ling, L.M., Zhang, X.E.: Kadomtsev-Petviashvili equation: one-constraint method and lump pattern. Physica D 432, 133152 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yang, B., Yang, J.K.: Pattern transformation in higher-order lumps of the Kadomtsev-Petviashvili I equation. J. Nonlinear Sci. 32, 52 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  25. Han, G.F., Li, X., Zhao, Q., Li, C.: Interaction structures of multi localized waves within the Kadomtsev-Petviashvili I equation. Phys. D 446, 133671 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhang, Z., Yang, X.Y., Li, B., Guo, Q., Stepanyants, Y.: Multi-lump formations from lump chains and plane solitons in the KP1 equation. Nonlin. Dyn. 111, 1625–1642 (2023)

    Article  Google Scholar 

  27. Wu, J.J., Sun, Y.J., Li, B.: Degenerate lump chain solutions of (4+1)-dimensional Fokas equation. Results Phys. 45, 106243 (2023)

    Article  Google Scholar 

  28. Zhang, Z., Guo, Q., Stepanyants, Y.: Creation of weakly interacting lumps by degeneration of lump chains in the KP1 equation. Chaos Solit. Fract. 170, 113398 (2023)

    Article  MathSciNet  Google Scholar 

  29. Wang, C.J., Fang, H.: Various kinds of high-order solitons to the Bogoyavlenskii-Kadomtsev-Petviashvili equation. Phys. Scripta 95, 035205 (2020)

    Article  Google Scholar 

  30. Estévez, P.G., Lejarreta, J.D., Sardón, C.: Symmetry computation and reduction of a wave model in (2+1)-dimensions. Nonlinear Dyn. 87, 13–23 (2017)

    Article  MATH  Google Scholar 

  31. Estévez, P.G., Hernáez, G.A.: Non-isospectral problem in (2+1) dimensions. J. Phys. A Math. Gen. 33, 2131 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, C.J., Fang, H.: Non-auto Bäclund transformation, nonlocal symmetry and CRE solvability for the Bogoyavlenskii-Kadomtsev-Petviashvili equation. Comput. Math. Appl. 74, 3296–3302 (2017)

    MathSciNet  MATH  Google Scholar 

  33. Wang, C.J., Fang, H.: Various kinds of high-order solitons to the Bogoyavlenskii-Kadomtsev-Petviashvili equation. Phys. Scripta 95, 035205 (2020)

    Article  Google Scholar 

  34. Zhao, Z.L., Yue, J., He, L.C.: New type of multiple lump and rogue wave solutions of the (2+1)-dimensional bogoyavlenskii-kadomtsev-Petviashvili equation. Appl. Math. Lett. 133, 108294 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  35. Yang, J.Y., Ma, W.X.: Lump solutions to the BKP equation by symbolic computation. Int. J. Mod. Phys. B 30, 1640028 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  36. Yang, J.Y., Ma, W.X., Qin, Z.: Abundant mixed lump-soliton solutions to the BKP equation. Anal. Math. Phys. 8, 224–232 (2018)

    MATH  Google Scholar 

Download references

Funding

This work is supported by National Natural Science Foundation of China under Grant Nos. 12175111, 12275144, 12235007 and 11975131 and K.C.Wong Magna Fund in Ningbo University.

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

    Yujie Sun & Biao Li

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  1. Yujie Sun
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Appendices

Appendix A

Functions \(h_j\) and \(h_{jk}\) \((j,k=1,2,3)\) in Eq. (26) are determined as follows:

$$\begin{aligned}&\ h_{1}=2\,{\epsilon }^{2} \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \right) ^{2}\nonumber \\&\quad -2\,{\epsilon }^{2}{\frac{ {\textrm{d}}^{2}}{{\textrm{d}}{\epsilon }^{2}}}\xi _{{1}} \left( \epsilon \right) -2\,\epsilon \,{\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) +1\end{aligned}$$
(A-1)
$$\begin{aligned}&\quad \ h_{2}=-2\,{\epsilon }^{2} \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \right) ^{2}\nonumber \\&\quad +2\,{\epsilon }^{2}{\frac{ {\textrm{d}}^{2}}{{\textrm{d}}{\epsilon }^{2}}}\xi _{{1}} \left( \epsilon \right) \nonumber \\&\quad +2\,\epsilon \,{\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) +1\end{aligned}$$
(A-2)
$$\begin{aligned}&\quad \ h_{3}=1\end{aligned}$$
(A-3)
$$\begin{aligned}&\quad h_{11}=6\,{\epsilon }^{6}+12\,{\epsilon }^{4}+9\,{\epsilon }^{2}+12\,{\epsilon }^{5}+4\,{\epsilon }^{3}\nonumber \\&\quad +\left( 6\,{\epsilon }^{5}+18\,{\epsilon }^{3} +2\,\epsilon \right) \Bigg [{ \frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) +{ \frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}}^* \left( \epsilon \right) \Bigg ]\nonumber \\&\quad +\left( -2\,{\epsilon }^{6}+6\,{\epsilon }^{4}+2\,{\epsilon }^{2} \right) \nonumber \\&\quad \Bigg [ \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \right) ^{2}+{\frac{{\textrm{d}}^{2}}{{\textrm{d}}{\epsilon }^ {2}}}\xi _{{1}} \left( \epsilon \right) +{\frac{{\textrm{d}}^{2}}{{\textrm{d}}{ \epsilon }^{2}}}\xi _{{1}}^* \left( \epsilon \right) \nonumber \\&\quad + \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}}^* \left( \epsilon \right) \right) ^{ 2}\Bigg ]\nonumber \\&\quad +\left( 8\,{\epsilon }^{6}+36\,{\epsilon }^{4}+4\,{\epsilon }^{2} \right) \Bigg [ {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}}^* \left( \epsilon \right) \Bigg ]\nonumber \\&\quad + \left( 4\,{\epsilon }^{6}+4\,{\epsilon }^{4} \right) \Bigg [ \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \right) ^{2} \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }} \xi _{{1}}^* \left( \epsilon \right) \right) ^{2}\Bigg ]\nonumber \\&\quad +\left( 12\,{\epsilon }^{5}+4\,{\epsilon }^{3} \right) \nonumber \\&\quad \Bigg [ {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}}^* \left( \epsilon \right) \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \right) ^{2}+ \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1 }}^* \left( \epsilon \right) \right) ^{2}{\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \nonumber \\&\quad + {\frac{{\textrm{d}}^{2}}{ {\textrm{d}}{\epsilon }^{2}}}\xi _{{1}} \left( \epsilon \right) { \frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}}^* \left( \epsilon \right) + {\frac{{\textrm{d}}^{2}}{{\textrm{d}}{\epsilon }^{2}}}\xi _{{1}}^* \left( \epsilon \right) {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \Bigg ]\nonumber \\&\quad +\left( 4\,{\epsilon }^{6}+4\,{\epsilon }^{4} \right) \nonumber \\&\quad \Bigg [ {\frac{ {\textrm{d}}^{2}}{{\textrm{d}}{\epsilon }^{2}}}\xi _{{1}} \left( \epsilon \right) \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}}^* \left( \epsilon \right) \right) ^{2}+ {\frac{{\textrm{d}}^{2}}{{\textrm{d}}{ \epsilon }^{2}}}\xi _{{1}}^* \left( \epsilon \right) \left( { \frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \right) ^{2}\nonumber \\ {}&\quad + {\frac{{\textrm{d}}^{2}}{{\textrm{d}}{\epsilon }^{2}}}\xi _ {{1}} \left( \epsilon \right) {\frac{{\textrm{d}}^{2}}{{\textrm{d}}{ \epsilon }^{2}}}\xi _{{1}}^* \left( \epsilon \right) \Bigg ]\end{aligned}$$
(A-4)
$$\begin{aligned}&\quad h_{12}=-3\,{\epsilon }^{8}+7\,{\epsilon }^{6}+12\,{\epsilon }^{4}+1\nonumber \\&\quad + \left( 14\,{ \epsilon }^{7}+24\,{\epsilon }^{5}+2\,\epsilon \right) {\frac{\textrm{d}}{ {\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \nonumber \\&\quad + \left( 16\, {\epsilon }^{7}+30\,{\epsilon }^{5}-2\,\epsilon \right) {\frac{\textrm{d}}{ {\textrm{d}}\epsilon }}\xi _{{1}}^* \left( \epsilon \right) \nonumber \\&\quad + \left( 8\,{ \epsilon }^{8}+28\,{\epsilon }^{6}-4\,{\epsilon }^{2} \right) {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}}^* \left( \epsilon \right) \nonumber \\&\quad +\left( 4\,{\epsilon }^{8}-14\,{\epsilon }^{6}-12\,{\epsilon }^{4}-2\,{\epsilon }^ {2}\right) \nonumber \\&\quad \Bigg [{{{\frac{{\textrm{d}}^{2}}{{\textrm{d}}{\epsilon }^{2}}}\xi _{{1}}^* \left( \epsilon \right) - \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}}^* \left( \epsilon \right) \right) ^{2}}}\Bigg ]+\left( 2\,{\epsilon }^{8}+8\,{ \epsilon }^{6}+2\,{\epsilon }^{2}\right) \nonumber \\&\quad \Bigg [{{{\frac{{\textrm{d}}^{2}}{{\textrm{d}}{ \epsilon }^{2}}}\xi _{{1}} \left( \epsilon \right) + \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \right) ^{ 2}}}\Bigg ]\nonumber \\&\quad + \left( 20\,{\epsilon }^{7}+24\,{\epsilon }^{5}+4\,{\epsilon }^{3} \right) \Bigg [ \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}}^* \left( \epsilon \right) \right) ^{2}{\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1} } \left( \epsilon \right) \nonumber \\&\quad - \left( {\frac{{\textrm{d}}^{2}}{{\textrm{d}}{ \epsilon }^{2}}}\xi _{{1}}^* \left( \epsilon \right) \right) {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \Bigg ]\nonumber \\ +&\quad \left( 4 \,{\epsilon }^{7}-4\,{\epsilon }^{3} \right) \Bigg [ \left( {\frac{\textrm{d}}{ {\textrm{d}}\epsilon }}\xi _{{1}}^* \left( \epsilon \right) \right) \left( { \frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \right) ^{2}\nonumber \\ {}&\quad + \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}}^* \left( \epsilon \right) \right) {\frac{{\textrm{d}}^{2}}{{\textrm{d}}{ \epsilon }^{2}}}\xi _{{1}} \left( \epsilon \right) \Bigg ]+ \left( 4\,{ \epsilon }^{8}+8\,{\epsilon }^{6}+4\,{\epsilon }^{4} \right) \nonumber \\&\quad \Bigg [ \left( { \frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}}^* \left( \epsilon \right) \right) ^{2} \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \right) ^{2}\nonumber \\&\quad + \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon } }\xi _{{1}}^* \left( \epsilon \right) \right) ^{2}{\frac{{\textrm{d}}^{2}}{ {\textrm{d}}{\epsilon }^{2}}}\xi _{{1}} \left( \epsilon \right) \nonumber \\&\quad - \left( { \frac{{\textrm{d}}^{2}}{{\textrm{d}}{\epsilon }^{2}}}\xi _{{1}}^* \left( \epsilon \right) \right) {\frac{{\textrm{d}}^{2}}{{\textrm{d}}{\epsilon }^{2}}}\xi _{{1} } \left( \epsilon \right) \nonumber \\&\quad - \left( {\frac{{\textrm{d}}^{2}}{{\textrm{d}}{ \epsilon }^{2}}}\xi _{{1}}^* \left( \epsilon \right) \right) \left( { \frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \right) ^{2}\Bigg ]\end{aligned}$$
(A-5)
$$\begin{aligned}&\quad \ h_{13}=(\epsilon ^2+1)\Bigg \{3\,{\epsilon }^{6}-16\,{\epsilon }^{4}-8\,{\epsilon }^{2}-1\nonumber \\&\quad - \left( 14\,{ \epsilon }^{5}+16\,{\epsilon }^{3}+2\,\epsilon \right) \nonumber \\&\quad {\frac{\textrm{d}}{ {\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) - \left( 2\,{ \epsilon }^{6}+4\,{\epsilon }^{4}+2\,{\epsilon }^{2} \right) \nonumber \\&\quad \Bigg [ \left( { \frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \right) ^{2}+{\frac{{\textrm{d}}^{2}}{{\textrm{d}}{\epsilon }^{2}}}\xi _{{1}} \left( \epsilon \right) \Bigg ]\Bigg \}\end{aligned}$$
(A-6)
$$\begin{aligned}&\quad h_{22}=(\epsilon ^2+1)\Bigg \{ 9\,{\epsilon }^{10}+69\,{\epsilon }^{8}\nonumber \\&\quad +63\,{\epsilon }^{6}+{\epsilon }^{4 }+1\nonumber \\&\quad + \left( 24\,{\epsilon }^{9}+66\,{\epsilon }^{7}-2\,{\epsilon }^{5}+2 \,{\epsilon }^{3}-2\,\epsilon \right) \nonumber \\&\quad \Bigg [{\frac{\textrm{d}}{{\textrm{d}}\epsilon }} \xi _{{1}} \left( \epsilon \right) +{\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}}^* \left( \epsilon \right) \Bigg ]\nonumber \\&\quad + \left( 4\,{\epsilon }^{10}+26\,{ \epsilon }^{8}+18\,{\epsilon }^{6}-2\,{\epsilon }^{4}+2\,{\epsilon }^{2} \right) \nonumber \\&\quad \Bigg [ \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \right) ^{2}\nonumber \\&\quad + \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon } }\xi _{{1}}^* \left( \epsilon \right) \right) ^{2}\nonumber \\&\quad -{\frac{{\textrm{d}}^{2}}{ {\textrm{d}}{\epsilon }^{2}}}\xi _{{1}} \left( \epsilon \right) -{\frac{ {\textrm{d}}^{2}}{{\textrm{d}}{\epsilon }^{2}}}\xi _{{1}}^* \left( \epsilon \right) \Bigg ]\nonumber \\&\quad + \left( 8\,{\epsilon }^{10}+52\,{\epsilon }^{8}+36\,{\epsilon }^{6}-4\, {\epsilon }^{4}+4\,{\epsilon }^{2} \right) \nonumber \\&\quad \left( {\frac{\textrm{d}}{ {\textrm{d}}\epsilon }}\xi _{{1}}^* \left( \epsilon \right) \right) {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) + \left( 12 \,{\epsilon }^{9}+20\,{\epsilon }^{7}+4\,{\epsilon }^{5}-4\,{\epsilon }^{3 } \right) \nonumber \\&\quad \Bigg [ \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \right) \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}}^* \left( \epsilon \right) \right) ^{2}\nonumber \\ +&\quad \left( {\frac{\textrm{d}}{ {\textrm{d}}\epsilon }}\xi _{{1}}^* \left( \epsilon \right) \right) \left( { \frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \right) ^{2}\nonumber \\&\quad - \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \right) {\frac{{\textrm{d}}^{2}}{{\textrm{d}}{ \epsilon }^{2}}}\xi _{{1}}^* \left( \epsilon \right) \nonumber \\&\quad - \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}}^* \left( \epsilon \right) \right) { \frac{{\textrm{d}}^{2}}{{\textrm{d}}{\epsilon }^{2}}}\xi _{{1}} \left( \epsilon \right) \Bigg ]\nonumber \\&\quad + \left( 4\,{\epsilon }^{10}+12\,{\epsilon }^{8}+12\,{\epsilon }^{6}+4\,{ \epsilon }^{4} \right) \nonumber \\&\quad \Bigg [ \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1 }} \left( \epsilon \right) \right) ^{2} \left( {\frac{\textrm{d}}{{\textrm{d}} \epsilon }}\xi _{{1}}^* \left( \epsilon \right) \right) ^{2}\nonumber \\&\quad - \left( { \frac{{\textrm{d}}^{2}}{{\textrm{d}}{\epsilon }^{2}}}\xi _{{1}} \left( \epsilon \right) \right) \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}}^* \left( \epsilon \right) \right) ^{2}\nonumber \\&\quad - \left( {\frac{{\textrm{d}}^{2}}{ {\textrm{d}}{\epsilon }^{2}}}\xi _{{1}}^* \left( \epsilon \right) \right) \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _{{1}} \left( \epsilon \right) \right) ^{2}\nonumber \\&\quad + \left( {\frac{{\textrm{d}}^{2}}{{\textrm{d}}{\epsilon }^ {2}}}\xi _{{1}} \left( \epsilon \right) \right) {\frac{{\textrm{d}}^{2}}{ {\textrm{d}}{\epsilon }^{2}}}\xi _{{1}}^* \left( \epsilon \right) \Bigg ]\Bigg \}\end{aligned}$$
(A-7)
$$\begin{aligned}&\quad h_{23}= \left( {\epsilon }^{2}+1 \right) ^{4} \Bigg \{-3\,{\epsilon }^{6}-22\,{ \epsilon }^{4}+4\,{\epsilon }^{2}-1\nonumber \\&\quad - \left( 10\,{\epsilon }^{5}+8\,{ \epsilon }^{3}-2\,\epsilon \right) {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _ {{1}} \left( \epsilon \right) \nonumber \\&\quad +2\,{\epsilon }^{2} \left( {\epsilon }^{2} +1 \right) ^{2}\nonumber \\&\quad \Bigg [{\frac{{\textrm{d}}^{2}}{{\textrm{d}}{\epsilon }^{2}}}\xi _{{1}} \left( \epsilon \right) - \left( {\frac{\textrm{d}}{{\textrm{d}}\epsilon }}\xi _ {{1}} \left( \epsilon \right) \right) ^{2}\Bigg ] \Bigg \}\end{aligned}$$
(A-8)
$$\begin{aligned}&\quad \ h_{33}=\left( {\epsilon }^{2}+1 \right) ^{9} \end{aligned}$$
(A-9)

In addition, \(h_{21}=h_{12}^*\),  \(h_{31}=h_{13}^*\)  and  \(h_{32}=h_{23}^*\).

Appendix B

Table 1 Symbol annotation table
Full size table

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Sun, Y., Li, B. Creation of anomalously interacting lumps by degeneration of lump chains in the BKP equation. Nonlinear Dyn 111, 19297–19313 (2023). https://doi.org/10.1007/s11071-023-08857-1

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