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Dynamics of loop soliton solutions of $${\pmb {\mathcal {PT}}}$$PT -symmetric nonlocal short pulse equation
Nonlinear Dynamics ( IF 5.6 ) Pub Date : 2020-03-06 , DOI: 10.1007/s11071-020-05548-z
Y. Hanif , H. Sarfraz , U. Saleem

Abstract

In this article, we present a system of coupled short pulse equations as integrability condition of associated generalized linear eigenvalue problem. We obtain system of \(\mathcal {PT}\)-symmetric short pulse equation (SPE) by using suitable reduction condition. In order to investigate the dynamics of loop solitons for nonlocal SPE, we use hodograph transformation to transform standard SPE into an equivalent equation of motion and associated linear system. We construct multiple-soliton solutions for \(\mathcal {PT}\)-symmetric nonlocal SPE by employing Darboux transformation on the matrix-valued solution to the transformed linear system. We obtain a generalized formula for multiple solutions in terms of ratio of determinants. Families of exact loop and breather soliton solutions are computed by using the generalized formula. We analyze the impact of nonlocality on loops, breathers and interacting solitons for incoherent spectral parameters. We also characterize the conduct of the loop soliton interactions. Nonlocality-induced instabilities for loop and breather solutions are also illustrated. We also recover multiple loop soliton solutions of the classical SPE under local symmetry reduction.



中文翻译:

$$ {\ pmb {\ mathcal {PT}}} $$ PT对称非局部短脉冲方程的环孤子解的动力学

摘要

在本文中,我们提出了一个耦合短脉冲方程组的系统,作为关联广义线性特征值问题的可积性条件。通过使用合适的归约条件,我们获得了\(\ mathcal {PT} \)-对称短脉冲方程(SPE)的系统。为了研究非局部SPE回路孤子的动力学,我们使用了Hodograph变换将标准SPE转换为等效的运动方程和相关的线性系统。我们为\(\ mathcal {PT} \)构造多孤子解在变换后的线性系统的矩阵值解上采用Darboux变换实现非对称非局部SPE。我们根据行列式的比率获得了多个解决方案的通用公式。确切的回路和通气孤子解的族通过使用广义公式来计算。我们分析了非局域性对环,呼吸器和相互作用孤子的影响,以获取非相干光谱参数。我们还描述了回路孤子相互作用的行为。还说明了回路和通气解决方案的非局部性引起的不稳定性。我们还恢复了局部对称性降低下经典SPE的多回路孤子解。

更新日期:2020-03-06
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