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Dynamics of new higher-order rational soliton solutions of the modified Korteweg–de Vries equation

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Abstract

In this paper, we propose a generalised perturbation \((n, N-n)\)-fold Darboux transformation (DT) of the modified Korteweg–de Vries (mKdV) equation using the Taylor expansion and a parameter limit procedure. We apply the generalised perturbation \((1, N-1)\)-fold DT to find the new explicit higher-order rational soliton (RS) solutions in terms of determinants of the mKdV equation. These higher-order RS solutions are different from those known soliton results in terms of hyperbolic functions which are obtained from the classical iterated DT. The dynamics behaviours of the first-, second-, third-, and fourth-order RS solutions are shown graphically. The wave propagation characteristics and stability are also discussed using numerical simulations. We find that the initial constant seed solution plays an important role on the wave propagation stability of RS. Through Miura transformation, we give some complex higher-order rational solutions of the Korteweg–de Vries (KdV) equation which are different from the known results. The relevant structures also are discussed using some figures. The method used can also be extended to seek explicit rational solutions of other nonlinear integrable equations.

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Acknowledgements

This work has been partially supported by Qin Xin Talents Cultivation Program of Beijing Information Science and Technology University (QXTCP-B201704 and QXTCP-A201702), the NSFC under Grant Nos 11375030 and 61178091, the Beijing Natural Science Foundation under Grant No. 1153004 and China Postdoctoral Science Foundation under Grant No. 2015M570161.

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Authors and Affiliations

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

    Xiao-Yong Wen

  2. Key Laboratory of Mathematics Mechanization, Institute of Systems Science, AMSS, Chinese Academy of Sciences, Beijing , 100190, China

    Yong Chen

  3. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing , 100049, China

    Yong Chen

Authors
  1. Xiao-Yong Wen
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  2. Yong Chen
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Correspondence to Xiao-Yong Wen.

Appendix A

Appendix A

$$\begin{aligned} \phi ^{(1)}= & {} (1/6) \sqrt{2/c} (-114 c^2 t+12 c b_1\\&+\,12 i cd_1+3 x+4 c^2 x^3-72 c^4 x^2 t\\&+\,432 c^6 x t^2-864 c^8 t^3),\\ \psi ^{(1)}= & {} -(1/6) \sqrt{2} (15 c x-186 c^3 t+12 c^2 x^2\\&-\,144 c^4 x t+432 c^6 t^2+12 c^2 b_1\\&+12 i c^2 d_1+4 c^3 x^3-72 c^5 x^2 t+432 c^7 x t^2\\&-\,864 c^9 t^3+3)/c^{3/2},\\ \phi ^{(2)}= & {} (1/240) \sqrt{2} (-15 x-2790 c^2 t+120 c^2 x^3\\&+120 c b_1-164160 c^8 t^3-6000 c^4 x^2 t\\&+\,59040 c^6 x t^2+480 c^2 b_2+16 c^4 x^5\\&-124416 c^{14} t^5+120 i c d_1+480 i c^2 d_2\\&-34560 c^{10} x^2 t^3+480 c^3 x^2 b_1+17280 c^7 t^2 b_1\\&-480 c^6 x^4 t+103680 c^{12} x t^4+5760 c^8 x^3 t^2\\&+480 i c^3 x^2 d_1-5760 i c^5 x t d_1-5760 c^5 x t b_1\\&+\,17280 i c^7t^2 d_1)/c^{3/2}, \\ \psi ^{(2)}= & {} -(1/240)\sqrt{2} (-15+80 c^4 x^4+360 c^2 x^2\\&+103680 c^{12} t^4-69120 c^{10} x t^3+17280 c^8 x^2 t^2\\&-1920 c^6 x^3 t+960 c^3 b_1 x-5760 c^5 t b_1\\&-12000 c^4 x t+59040 c^6t^2-5760 i c^5 d_1 t\\&+\,480 i c^3 d_2-5760 i c^6 x t d_1+105 c x+600 c^2 b_1\\&-\,7350 c^3 t+280 c^3 x^3-198720 c^9 t^3+480 c^3 b_2\\&+\,16 c^5 x^5-124416 c^{15} t^5-8880 c^5 x^2t\\&+\,76320 c^7 x t^2-34560 c^{11} x^2 t^3+480 c^4 x^2 b_1\\&+\,17280 c^8 t^2 b_1-480 c^7 x^4 t+103680 c^{13} x t^4\\&+\,5760 c^9 x^3 t^2-5760 c^6 x t b_1+960 i c^3 d_1 x\\&+\,600 i c^2 d_1+17280 i c^8 t^2 d_1\\&+480 i c^4 x^2 d_1)/c^{5/2},\\ \phi ^{(3)}= & {} (1/20160) \sqrt{2} (315 x+10080 i c^2 d_2\\&-\,17915904 c^{20} t^7-32130 c^2 t+1260 c^2 x^3\\&-1260 c b_1-30270240 c^8 t^3-425880 c^4 x^2 t\\&+7554960 c^6 x t^2+1451520 i c^8t^2 d_2\\&+10080 c^2 b_2+1680 c^4 x^5-82736640 c^{14} t^5\\&-15240960 c^{10} x^2 t^3+30240 c^3 x^2 b_1\\&+4959360 c^7 t^2 b_1-104160 c^6 x^4 t\\&+57335040 c^{12} x t^4+1895040 c^8 x^3 t^2\\&- 1008000 c^5 x t b_1+64 c^6 x^7+40320 c^3 b_3\\&+\,40320 c^4 x^2 b_2+1451520 c^8 t^2 b_2+40320 c^4 b_1^2 x\\&+6720 c^5 x^4 b_1-\,241920 c^6 b_1^2 t\\&+8709120 c^{13} t^4 b_1-483840 c^{12} x^4t^3\\&+48384 c^{10} x^5 t^2-40320 c^4 d_1^2 x\\&+\,241920 c^6 d_1^2 t-2688 c^8 x^6 t+20901888 c^{18} t^6 x\\&-\,10450944 c^{16} t^5 x^2+2903040 c^{14} x^3 t^4\\&-\,483840 c^6 x t b_2-161280 c^7x^3 t b_1\\&+\,1451520 c^9 x^2 t^2 b_1\\&-\,5806080 c^{11} x t^3 b_1+30240 i c^3 x^2 d_1\\&-\,483840 i c^6 b_1 t d_1+4959360 i c^7 t^2 d_1\\&-\,5806080 i c^{11} x t^3 d_1+40320 i c^3d_3\\&+\,80640 i c^4 b_1 x d_1+8709120 i c^{13} t^4 d_1\\&+\,40320 i c^4 x^2 d_2-483840 i c^6 x t d_2\\&-\,1008000 i c^5 x t d_1-161280 i c^7 x^3 t d_1\\&+\,1451520 i c^9 x^2 t^2 d_1-1260 i c d_1\\&+\,6720 i c^5 x^4 d_1)/c^{5/2},\\ \psi ^{(3)}= & {} -(1/20160) \sqrt{2} (315+6720 i c^6 x^4 d_1\\&-\,17915904 c^{21} t^7+8709120 i c^{14} t^4 d_1+64 c^7 x^7\\&+\,448 c^6 x^6+80640 i c^4 x d_2-483840 i c^7 x^2 t d_1\\&-\,1491840 i c^6 x t d_1+8400 c^4 x^4+3780 c^2 x^2\\&+\,57335040 c^{12} t^4-30481920 c^{10} x t^3\\&+\,5685120 c^8 x^2 t^2-416640 c^6 x^3 t+60480 c^3 b_1 x\\&-\,1008000 c^5 t b_1-851760 c^4 xt+7554960 c^6 t^2\\&-\,483840 c^7 x t b_2-161280 c^8 x^3 t b_1\\&+\,1451520 c^{10} x^2 t^2 b_1-5806080 c^{12} x t^3 b_1\\&-\,483840 i c^7 b_1 t d_1+40320 c^4 b_3+60480 i c^3 d_1 x\\&+\,8820 ic^2 d_1-483840 c^6 t b_2+40320 c^5 x^2 b_2\\&+\,1451520 c^9 t^2 b_2+40320 c^5 b_1^2 x+6720 c^6 x^4 b_1\\&-241920 c^7 b_1^2 t+8709120 c^{14} t^4 b_1\\&-\,483840 c^{13} x^4 t^3+48384 c^{11}x^5 t^2\\&-\,40320 c^5 d_1^2 x+241920 c^7 d_1^2 t-2688 c^9 x^6 t\\&+\,20901888 c^{19} t^6 x-10450944 c^{17} t^5 x^2\\&+\,2903040 c^{15} x^3 t^4+241920 c^{10} x^4 t^2\\&+\,80640 c^4 x b_2+40320 ic^4 d_3+80640 i c^5 b_1 x d_1\\&+\,50400 i c^3 d_2+1451520 i c^{10} x^2 t^2 d_1\\&-40320 c^4 d_1^2+2903040 i c^9 x t^2 d_1-945 c x\\&+8820 c^2 b_1-266490 c^3 t\\&+11340 c^3 x^3-44059680 c^9 t^3\\&+\,50400 c^3 b_2+3024 c^5 x^5-93187584 c^{15} t^5\\&-\,929880 c^5 x^2 t+12514320 c^7 x t^2\\&-\,18144000 c^{11} x^2 t^3+70560 c^4 x^2 b_1\\&+\,6410880 c^8 t^2 b_1-144480 c^7 x^4t\\&+\,66044160 c^{13} x t^4+2378880 c^9 x^3 t^2\\&-\,1491840 c^6 x t b_1+20901888 c^{18} t^6\\&+40320 c^4 b_1^2-483840 c^7 x^2 t b_1\\&+2903040 c^9 x t^2 b_1-16128 c^8 x^5 t\\&-\,20901888 c^{16}t^5 x-1935360 c^{12} x^3 t^3\\&+\,8709120 c^{14} x^2 t^4+26880 c^5 x^3 b_1\\&-\,5806080 c^{11} t^3 b_1-1008000 i c^5 d_1 t\\&-\,5806080 i c^{12} x t^3 d_1-483840 i c^6 t d_2\\&+\,80640 ic^4 b_1 d_1+1451520 i c^9 t^2 d_2\\&-\,5806080 i c^{11} t^3 d_1+40320 i c^5 x^2 d_2\\&+\,70560 i c^4 x^2 d_1-483840 i c^7 x t d_2\\&+\,6410880 i c^8 t^2 d_1-161280 i c^8 x^3 td_1\\&+\,26880 i c^5 x^3 d_1)/c^{7/2}. \end{aligned}$$

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Wen, XY., Chen, Y. Dynamics of new higher-order rational soliton solutions of the modified Korteweg–de Vries equation. Pramana - J Phys 91, 23 (2018). https://doi.org/10.1007/s12043-018-1592-3

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