Symmetry reductions and conservation laws of the short pulse equation

Fakhar, Kamran; Wang, Gangwei; Kara, A. H.

doi:10.1016/j.ijleo.2016.08.013

Cited by 6 publications

(3 citation statements)

References 34 publications

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“…Determined the symmetries of SPE and travelling wave solution by parametric representation and power series process, respectively, [11]. Evaluated the symmetry reductions and conservation laws by using the direct method for SPE, [12]. Authors also determined the Lie symmetries for SPE through the non-local system.…”

Section: Introductionmentioning

confidence: 99%

Lie symmetries and reductions via invariant solutions of general short pulse equation

2023

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Around 1880, Lie introduced an idea of invariance of the partial differential equation (PDE) under one-parameter Lie group of transformation to find the invariant, similarity, or auto-model solutions. Lie symmetry analysis (LSA) provides us an algorithm to search for point symmetries for solving related linear systems for infinitesimal generators. Actually, point symmetries lead us to one-parameter family of solutions from a known solution. LSA is a program that provides us the exact solutions for the non-linear differential equations (DEs) in analogy of the program designed by Galois for algebraic polynomial equations. In this paper, we have carried out the LSA for computing the similarity solutions (symmetries) of the non-linear short pulse equation (SPE) for the cases when h(u) = eu, k(u) = uxx, h(u)=eun, and k(u) = uxx. In addition, an optimal system of one-dimensional sub-algebra has been set up. The reductions and invariant solutions for the generalized SPE are calculated corresponding to this optimal system as well. Reductions reduce the non-linear PDE or system of PDEs into non-linear reduced ordered ODE or system of PDEs. This helps to solve these systems of PDEs to reduced form. Graphical behavior of the transformed points of the 1-parameter solution functions have drawn.

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Section: Introductionmentioning

confidence: 99%

Lie symmetries and reductions via invariant solutions of general short pulse equation

2023

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“…In the field of optical communications, nonlinear mathematical physical models that can describe the pulse propagation in optical fibers play an important role. The famous nonlinear Schrödinger (NLS) equation [1,2], the short pulse (SP) equation [3,4], and the Wadati-Konno-Ichikawa equation [5,6] have been established respectively to describe different pulse propagations in optical fibers. Based on the SP equation, Feng proposed a new model.…”

Section: Introductionmentioning

confidence: 99%

Conservation laws of the complex short pulse equation and coupled complex short pulse equations

Zhang¹,

Shen²,

Han³

et al. 2022

Commun. Theor. Phys.

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In this paper, the complex short pulse equation and the coupled complex short pulse equations that can describe the ultra-short pulse propagation in optical fibers are investigated. The two complex nonlinear models are turned into multi-component real models by proper transformations. Lie symmetries are obtained via the classical Lie group method, and the results for the coupled complex short pulse equations contain the existing results as particular cases. Based on the linearizing operator and adjoint linearizing operator for the two real systems, adjoint symmetries can be obtained. Explicit conservation laws is constructed using the symmetry/adjoint symmetry pair (SA) method. Relationships between the nonlinear self-adjointness method and the SA method are investigated.

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“…Moreover, we study this new equation with the aid of Lie symmetry analysis method. This method is one of the most important and efficient techniques to find the exact solutions [12][13][14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

On exact solutions for new coupled nonlinear models getting evolution of curves in Galilean space

2019

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Symmetry reductions and conservation laws of the short pulse equation

Cited by 6 publications

References 34 publications

Lie symmetries and reductions via invariant solutions of general short pulse equation

Lie symmetries and reductions via invariant solutions of general short pulse equation

Conservation laws of the complex short pulse equation and coupled complex short pulse equations

On exact solutions for new coupled nonlinear models getting evolution of curves in Galilean space

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