Excitation control for three-dimensional Peregrine solution and combined breather of a partially nonlocal variable-coefficient nonlinear Schrödinger equation


A (2 + 1)-dimensional variable-coefficient partially nonlocal nonlinear Schrödinger equation is considered, and analytical Peregrine solution (PS) and combined Akhmediev breather (AB) are presented from a reduced transformation and the Darboux transformation method. Based on these analytical solutions, the excitation control for three-dimensional PS and combined AB is investigated by comparing values between the maximum value of the effective propagation distance \(\zeta _{\max }\) and the crest position \(\zeta _{0}\) (for the first-order PS and second-order PS) or the crest position \(\zeta _n(n=1,2,3)\) (for the combined AB). If \(\zeta _{\max }<\zeta _{0}\), \(\zeta _{\max }=\zeta _{0}\) and \(\zeta _{\max }>\zeta _{0}\), the anterior excitation, crest excitation, and tail excitation of the first-order PS and second-order PS can be studied, respectively. Similarly, if \(\zeta _{\max }<\zeta _{n}\), \(\zeta _{\max }=\zeta _{n}\) and \(\zeta _{\max }>\zeta _{n}\), the anterior excitation, crest excitation, and tail excitation of the first first-order PS, the second-order PS and the second first-order PS in the combined AB can be discussed, respectively. The \(k+1\)-layer excited structures for the first-order PS, second-order PS, and the combined AB can be constructed for the fixed value of the Hermite polynomial parameter k.

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This work was supported by the National Natural Science Foundation of China (Grant No. 11775185) and the higher school visiting scholar development project (Grant No. FX2013103).

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Correspondence to Yi-Xiang Chen.

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Chen, Y., Xu, F. & Hu, Y. Excitation control for three-dimensional Peregrine solution and combined breather of a partially nonlocal variable-coefficient nonlinear Schrödinger equation. Nonlinear Dyn 95, 1957–1964 (2019). https://doi.org/10.1007/s11071-018-4670-7

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  • (2 + 1)-Dimensional variable-coefficient nonlinear
  • Partially nonlocal nonlinearity
  • Excitation control
  • Peregrine solution
  • Combined breather