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Solitary Wave Solutions of Delayed Coupled Higgs Field Equation

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Abstract

This paper is devoted to the study of the solitary wave solutions for the delayed coupled Higgs field equation

$$\left\{{\matrix{{{u_{tt}} - {u_{xx}} - \alpha u + \beta f \ast u{{\left| u \right|}^2} - 2uv - \tau u{{({{\left| u \right|}^2})}_x} = 0,} \hfill \cr {{v_{tt}} + {v_{xx}} - \beta {{({{\left| u \right|}^2})}_{xx}} = 0.} \hfill \cr}} \right.$$

We first establish the existence of solitary wave solutions for the corresponding equation without delay and perturbation by using the Hamiltonian system method. Then we consider the persistence of solitary wave solutions of the delayed coupled Higgs field equation by using the method of dynamical system, especially the geometric singular perturbation theory, invariant manifold theory and Fredholm theory. According to the relationship between solitary wave and homoclinic orbit, the coupled Higgs field equation is transformed into the ordinary differential equations with fast variables by using the variable substitution. It is proved that the equations with perturbation also possess homoclinic orbit, and thus we obtain the existence of solitary wave solutions of the delayed coupled Higgs field equation.

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Acknowledgements

The authors sincerely thank the anonymous referees for very careful reading and valuable comments on our paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, and Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, 130024, P. R. China

    Shu Guan Ji

  2. School of Mathematics, Jilin University, Changchun, 130012, P. R. China

    Shu Guan Ji & Xiao Wan Li

  3. College of Mathematics and System Science, Xinjiang University, Urumqi, 830046, P. R. China

    Xiao Wan Li

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  1. Shu Guan Ji
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  2. Xiao Wan Li
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Correspondence to Xiao Wan Li.

Additional information

Supported by NSFC (Grant Nos. 12071065 and 11871140) and the National Key Research and Development Program of China (Grant No. 2020YFA0713602)

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Ji, S.G., Li, X.W. Solitary Wave Solutions of Delayed Coupled Higgs Field Equation. Acta. Math. Sin.-English Ser. 38, 97–106 (2022). https://doi.org/10.1007/s10114-022-0268-6

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  • DOI: https://doi.org/10.1007/s10114-022-0268-6

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