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Complex nonlinearities of rogue waves in generalized inhomogeneous higher-order nonlinear Schrödinger equation

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Abstract

In this paper, the Nth-order rogue waves are investigated for an inhomogeneous higher-order nonlinear Schrödinger equation. Based on the Heisenberg ferromagnetic spin system, the higher-order nonlinear Schrödinger equation is generated. The generalized Darboux transformation is constructed by the Darboux matrix. The solutions of the Nth-order rogue waves are given in terms of a recursive formula. There are complex nonlinear phenomena in the rogue waves, add the first-order to the fourth-order rogue waves are discussed in some figures obtained by analytical solutions. It is shown that the general Nth-order rogue waves contain \(2n-1\) free parameters. The free parameters play a crucial role to affect the dynamic distributions of the rogue waves. The results obtained in this paper will be useful to understand the generation mechanism of the rogue wave.

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Acknowledgments

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSFC) through Grant Nos. 11290152, 11072008 and 10732020, the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHRIHLB).

Author information

Authors and Affiliations

  1. College of Mechanical Engineering, Beijing University of Technology, Beijing, 100022, People’s Republic of China

    N. Song, W. Zhang & M. H. Yao

  2. Department of Mathematics, North University of China, Taiyuan, 030051, Shanxi, People’s Republic of China

    N. Song

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  1. N. Song
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  2. W. Zhang
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  3. M. H. Yao
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Corresponding author

Correspondence to W. Zhang.

Appendix

Appendix

Analytical expressions of the coefficients in Eq. (21) are given as

$$\begin{aligned} \varphi _1^{[2]}= & {} \left( \frac{1}{\sqrt{a}}\left( \frac{i}{96}\left( 27648a^{9}t^{3}{\varepsilon }^{2}+2304a^{7}t^{3}{\varepsilon }\right. \right. \right. \nonumber \\&-\,96a^{3}xt^{2}-192a^{3}x^{2}t+64a^{5}t^{3}-240ta\nonumber \\&-\,12t^{3}a^{3}1152ia^{6}t^{3}{\varepsilon }+27648ia^{8}xt^{2}{\varepsilon }^{2}\nonumber \\&+\,4608ia^{6}xt^{2}{\varepsilon }-1152a^{5}xt^{2}{\varepsilon }-2304a^{5}x^{2}t{\varepsilon }\nonumber \\&\times \,110592a^{11}t^{3}{\varepsilon }^{3}+192ia^{4}xt^{2}+6912ia^{8}t^{3}{\varepsilon }^{2}\nonumber \\&-\,48ia^{2}x^{2}t-12ia^{2}xt^{2}-5952t{\varepsilon }a^{3}-192aa_1\nonumber \\&-\,48 i x-12it-144a^{5}t^{3}{\varepsilon }-192iab_1 \nonumber \\&\left. -\,64ia^{2}x^{3}-ia^{2}t^{3}+48ia^{4}t^{3}\right) \nonumber \\&-\,\frac{1}{32}\sqrt{a}\left( {-4x+48it{\varepsilon }a^{3}-t+4ita} \right) ^{2}\nonumber \\&+\,1/16a^{-3/2}+\frac{1}{768}\sqrt{a}\big ( -4x+48it{\varepsilon }a^{3}\nonumber \\&-\,t+4ita \big )\big (-1152a^{6}t^{3}{\varepsilon }-192a^{4}xt^{2}\nonumber \\&+\,192x+48t-48a^{4}t^{3}-4608a^{6}xt^{2}{\varepsilon }\nonumber \\&-\,1152ia^{5}xt^{2}{\varepsilon }-2304ia^{5}x^{2}t{\varepsilon }\nonumber \\&-\,27648a^{8}xt^{2}{\varepsilon }^{2}+64a^{2}x^{3}+a^{2}t^{3}-6912a^{8}t^{3}{\varepsilon }^{2}\nonumber \\&+\,48a^{2}x^{2}t+12a^{2}xt^{2}-960ita-12ia^{3}t^{3}\nonumber \\&\times \,64ia^{5}t^{3}-768iaa_1 +768ab_1 -23808it{\varepsilon }a^{3}\nonumber \\&-\,144ia^{5}t^{3}{\varepsilon }-96ia^{3}xt^{2}+27648ia^{9}t^{3}{\varepsilon }^{2}\nonumber \\&\times \,2304ia^{7}t^{3}{\varepsilon }-192ia^{3}x^{2}t\nonumber \\&\left. \left. +\,110592ia^{11}t^{3}{\varepsilon }^{3}\big )\right) \right) e^{\frac{i}{2}\theta },\nonumber \\ \end{aligned}$$
$$\begin{aligned} \varphi _1^{[2]}= & {} \left( {-\frac{1}{\sqrt{a}}\left( \frac{i}{96}\right. \left( 110592a^{11}t^{3}{\varepsilon }^{3}+27648a^{9}t^{3}\right. {\varepsilon }^{2}} \right. \nonumber \\&+\,2304a^{7}t^{3}{\varepsilon }-96a^{3}xt^{2}+192a^{3}x^{2}t\nonumber \\&+\,27648ia^{8}xt^{2}{\varepsilon }^{2}-48ia^{2}x^{2}t+64a^{5}t^{3}\nonumber \\&-240ta-12t^{3}a^{3}+1152ia^{6}t^{3}{\varepsilon } 12ia^{2}xt^{2}\nonumber \\&+\,6912ia^{8}t^{3}{\varepsilon }^{2}+192ia^{4}xt^{2}-48ix-12it\nonumber \\&-\,ia^{2}t^{3}-192iab_1+48ia^{4}t^{3}-64ia^{2}x^{3}\nonumber \\&-\,1152a^{5}xt^{2}{\varepsilon }-2304a^{5}x^{2}t{\varepsilon }-192aa_1 \nonumber \\&\left. \left. -\,5952t{\varepsilon }a^{3}-144a^{5}t^{3}{\varepsilon }4608ia^{6}xt^{2}{\varepsilon } \right) \right) \nonumber \\&-\frac{1}{32}\sqrt{a}\Big ( {-4x+48it{\varepsilon }a^{3}-t+4ita} \Big )^{2}\nonumber \\&+\,1/16a^{-3/2}+ \frac{1}{768}\sqrt{a}\Big ( -4x+48it{\varepsilon }a^{3}\nonumber \\&-\,t+4ita\Big )\Big (-96ia^{3}xt^{2}-192ia^{3}x^{2}t\nonumber \\&+\,192x+48t+64ia^{5}t^{3}+110592ia^{11}t^{3}{\varepsilon }^{3}\nonumber \\&+\,27648ia^{9}t^{3}{\varepsilon }^{2}-768iaa_1-12ia^{3}t^{3}\nonumber \\&-\,960ita+12a^{2}xt^{2}+48a^{2}x^{2}t-192a^{4}xt^{2}\nonumber \\&+\,768ab_1 -144ia^{5}t^{3}{\varepsilon }+2304ia^{7}t^{3}{\varepsilon }\nonumber \\&-23808it{\varepsilon }a^{3}-4608a^{6}xt^{2}{\varepsilon }-27648a^{8}xt^{2}{\varepsilon }^{2}\nonumber \\&-48a^{4}t^{3}+a^{2}t^{3}+64a^{2}x^{3}-6912a^{8}t^{3}{\varepsilon }^{2}\nonumber \\&-1152t^{3}a^{6}{\varepsilon } \quad -1152ia^{5}xt^{2}{\varepsilon }\nonumber \\&\left. -\,2304ia^{5}x^{2}t{\varepsilon } \Big ) \right) e^{-\frac{i}{2}\theta }. \end{aligned}$$
(38)

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Song, N., Zhang, W. & Yao, M.H. Complex nonlinearities of rogue waves in generalized inhomogeneous higher-order nonlinear Schrödinger equation. Nonlinear Dyn 82, 489–500 (2015). https://doi.org/10.1007/s11071-015-2170-6

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