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Generation of anomalously scattered lumps via lump chains degeneration within the Mel’nikov equation

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Abstract

This paper explores two distinct approaches for degenerating lump chains into anomalously scattered lumps within the Mel’nikov equation. The first approach involves directly degenerating lump chains with specific phase parameters by simultaneously modulating their periods. The second approach entails equalizing the parameters of distinct lump chains to obtain a degenerate lump chain, followed by adjusting its period to infinity to achieve anomalously scattered lumps. We also calculate the asymptotic form of the degenerate lump chains at large time and establishes the relationship between peak locations of the anomalously scattered lumps and certain polynomials.

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Data sharing does not apply to this article as no data sets were generated or analyzed during the current study.

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Acknowledgements

This project is supported by the National Natural Science Foundation of China (Grant Nos. 52171251, U21062251), LiaoNing Revitalization Talents Program (XLYC1907014) and the Fundamental Research Funds for the Central Universities (DUT21ZD205).

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Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Xiangyu Yang

  2. School of Mathematical Sciences, Beihang University, Beijing, 100191, China

    Zhen Wang

  3. Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices, South China Normal University, Guangzhou, 510631, China

    Zhao Zhang

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  1. Xiangyu Yang
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Correspondence to Zhen Wang.

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Appendix

Appendix

The auxiliary functions of three anomalously scattered lumps in Sect. 3.1 are

$$\begin{aligned}{} & {} f_{6a} = {X}^{12}+ \left( 6\,{y}^{2}+36 \right) {X}^{10}+600\,{X}^{9}t\\{} & {} \quad + \left( 15\,{y}^{4}+180\,{y}^{2}+270 \right) {X}^{8}+5040\,{X}^{7}t\\{} & {} \quad + \left( 20 \,{y}^{6}+360\,{y}^{4}+54000\,{t}^{2} \right. \\{} & {} \quad \left. +1440\,{y}^{2}+990 \right) {X}^{6 }-3600\,t \\{} & {} \quad \left( {y}^{4}{-}{\frac{57\,{y}^{2}}{5}}{-}{\frac{39}{10}} \right) {X}^{5}{+} \left( 15\,{y}^{8}{+}360\,{y}^{6}{+}2700\,{y}^{4}\right. \\{} & {} \quad \left. + \left( 810000\,{t}^{2}+4050 \right) {y}^{2} -918000\,{t}^{2}-6075 \right) {X}^{4} \\{} & {} \quad -10800000\,t \left( {\frac{{y}^{6}}{2250}}+{\frac{7 \,{y}^{4}}{3000}}+{t}^{2}+{\frac{3\,{y}^{2}}{1000}}\right. \\{} & {} \quad \left. +{\frac{19}{2000}} \right) {X}^{3}+ \left( 6\,{y}^{10}+180\,{y}^{8}+2160\,{y}^{6} \right. \\{} & {} \quad \left. + \left( -270000\,{t}^{2}+9450 \right) {y}^{4}\right. \\{} & {} \quad \left. + \left( 1620000\,{t}^{2} +28350 \right) {y}^{2}+7970400\,{t}^{2}+60750 \right) {X}^{2}\\{} & {} \quad +32400000 \, \left( -{\frac{{y}^{8}}{18000}} \right. \\{} & {} \quad \left. -{\frac{17\,{y}^{6}}{9000}}-{ \frac{47\,{y}^{4}}{6000}}+ \left( {t}^{2}-{\frac{1}{2000}} \right) {y }^{2}+{\frac{16\,{t}^{2}}{5}}+{\frac{11}{500}} \right) tX\\{} & {} \quad +{y}^{12}+36 \,{y}^{10}+630\,{y}^{8}+ \left( 126000\,{t}^{2}+6390 \right) {y}^{6}\\{} & {} \quad + \left( 2538000\,{t}^{2}+34425 \right) {y}^{4}+ \left( 11858400\,{t}^{ 2}+93150 \right) {y}^{2}\\{} & {} \quad +324000000\,{t}^{4}+9266400\,{t}^{2}+50625, \\{} & {} g_{6a}= {X}^{12}+{y}^{12}+24\,i{y}^{11}+12\,{X}^{10}+600\,{X}^{9}t\\{} & {} \quad + \left( 6\, {X}^{2}-228 \right) {y}^{10}+ \left( 120\,i{X}^{2}-960\,i \right) {y}^ {9}+270\,{X}^{8} \\{} & {} +5040\,{X}^{7}t + \left( 15\,{X}^{4}-900\,{X}^{2}-1800 \,Xt+630 \right) {y}^{8}\\{} & {} \quad + \left( 240\,i{X}^{4}-2880\,i{X}^{2}-28800\,i Xt-8640\,i \right) {y}^{7}+ \left( 54000\,{t}^{2} \right. \\{} & {} \quad \left. -4770 \right) {X}^{6} -322920\,{X}^{5}t+ \left( 20\,{X}^{6}-1320\,{X}^{4}-4800\,{X}^{3}t\right. \\{} & {} \quad \left. + 2160\,{X}^{2}+140400\,Xt+126000\,{t}^{2}+22230 \right) {y}^{6} \\{} & {} \quad + \left( 240\,i{X}^{6}-2880\,i{X}^{4}-57600\,i{X}^{3}t-12960\,i{X}^{2}\right. \\{} & {} \quad \left. + 129600\,iXt+1512000\,i{t}^{2}-24840\,i \right) {y}^{5}+ \left( - 4158000\,{t}^{2}\right. \\{} & {} \quad \left. -43875 \right) {X}^{4}+ \left( 15\,{X}^{8}-840\,{X}^{6 }-3600\,{X}^{5}t\right. \\{} & {} \quad \left. +2700\,{X}^{4}+262800\,{X}^{3}t+ \left( -270000\,{t}^{ 2}+52650 \right) {X}^{2} \right. \\{} & {} \quad \left. +826200\,Xt-5022000\,{t}^{2}+147825 \right) {y }^{4}+324000000\,{t}^{4}\\{} & {} \quad + \left( -10800000\,{t}^{3}-1182600\,t \right) {X}^{3}+ \left( 120\,i{X}^{8}-960\,i{X}^{6} \right. \\{} & {} \quad \left. -28800\,i{X}^{5}t+374400\,i{X}^{3}t+ \left( -2160000\,i{t}^{2}+97200\,i \right) {X}^{2}\right. \\{} & {} \quad \left. + 2116800\,iXt-1296000\,i{t}^{2}+91800\,i \right) {y}^{3}+ \left( - 11469600\,{t}^{2} \right. \\{} & {} \quad \left. -52650 \right) {X}^{2}+ \left( 6\,{X}^{10}-180\,{X}^{ 8}+1440\,{X}^{6}+127440\,{X}^{5}t\right. \\{} & {} \quad \left. + \left( 810000\,{t}^{2}+25650 \right) {X}^{4}+572400\,{X}^{3}t \right. \\{} & {} \quad \left. + \left( 8100000\,{t}^{2} -68850 \right) {X}^{2}-10173600\,{t}^{2}\right. \\{} & {} \quad \left. + \left( 32400000\,{t}^{3}-664200\,t \right) X+174150 \right) {y}^{2}+5896800\,{t}^{2} \\{} & {} \quad - \left( 25920000\, {t}^{3}+518400\,t \right) X+ \left( 24\,i{X}^{10}+4320\,i{X}^{6}\right. \\{} & {} \quad \left. + 336960\,i{X}^{5}t+ \left( 3240000\,i{t}^{2}+70200\,i \right) {X}^{4} \right. \\{} & {} \quad \left. +1944000\,i{X}^{3}t + \left( 19440000\,i{t}^{2}+16200\,i \right) {X}^{2}\right. \\{} & {} \quad \left. +129600000\,i \left( {t}^{2}+{\frac{23}{2000}} \right) Xt-9590400\,i{t }^{2}+178200\,i \right) y \\{} & {} \quad -30375, \end{aligned}$$

where \(X=x-7t\).

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Yang, X., Wang, Z. & Zhang, Z. Generation of anomalously scattered lumps via lump chains degeneration within the Mel’nikov equation. Nonlinear Dyn 111, 15293–15307 (2023). https://doi.org/10.1007/s11071-023-08615-3

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