Higher-order algebraic soliton solutions of the Gerdjikov-Ivanov equation: Asymptotic analysis and emergence of rogue waves

Abstract In this paper, the inverse scattering approach is applied to the Kundu-Eckhaus equation with two cases of zero boundary condition (ZBC) and nonzero boundary conditions (NZBCs) at infinity. Firstly, we obtain the exact formulae of soliton solutions of three cases of N simple poles, one higher-order pole and multiple higher-order poles via the associated Riemann-Hilbert problem (RHP). Moreover, given the initial data that allow for the presence of discrete spectrum, the higher-order rogue waves of the equation are presented. For the case of NZBCs, we can construct the infinite order rogue waves through developing a suitable RHP. Finally, by choosing different parameters, we aim to show some prominent characteristics of this solution and express them graphically in detail. Our results should be helpful to further explore and enrich the related nonlinear wave phenomena.

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Asymptotic analysis of the crack tip stress field (consideration of higher order terms)

Сибирский журнал вычислительной математики ◽

10.15372/sjnm20190307 ◽

2019 ◽

Keyword(s):

Stress Field ◽

Asymptotic Analysis ◽

Crack Tip ◽

Higher Order ◽

Higher Order Terms ◽

Crack Tip Stress

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N-soliton solutions and the Hirota conditions in (1 + 1)-dimensions

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2020-0214 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Wen-Xiu Ma

Keyword(s):

Common Factors ◽

Soliton Solutions ◽

Higher Order ◽

Kdv Equations ◽

The Common ◽

Bilinear Formulation ◽

Generalized Kdv Equations ◽

Hirota Bilinear

Abstract We analyze N-soliton solutions and explore the Hirota N-soliton conditions for scalar (1 + 1)-dimensional equations, within the Hirota bilinear formulation. An algorithm to verify the Hirota conditions is proposed by factoring out common factors out of the Hirota function in N wave vectors and comparing degrees of the involved polynomials containing the common factors. Applications to a class of generalized KdV equations and a class of generalized higher-order KdV equations are made, together with all proofs of the existence of N-soliton solutions to all equations in two classes.

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Optical soliton solutions to a higher-order nonlinear Schrödinger equation with Kerr law nonlinearity

Results in Physics ◽

10.1016/j.rinp.2021.104515 ◽

2021 ◽

pp. 104515

Author(s):

B. Günay

Keyword(s):

Schrödinger Equation ◽

Nonlinear Schrödinger Equation ◽

Schrodinger Equation ◽

Soliton Solutions ◽

Higher Order ◽

Nonlinear Schrodinger Equation ◽

Optical Soliton ◽

Nonlinear Schrödinger ◽

Kerr Law ◽

Kerr Law Nonlinearity

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The Higher Order Asymptotic Fields for Plane Interfacial Crack with Physical Weak-Discontinuity

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.217-218.1314 ◽

2011 ◽

Vol 217-218 ◽

pp. 1314-1318

Author(s):

Yao Dai ◽

Lei Zhang ◽

Peng Zhang ◽

Jun Feng Liu

Keyword(s):

Experimental Investigation ◽

Numerical Analysis ◽

Asymptotic Analysis ◽

Theoretical Basis ◽

Engineering Application ◽

Interfacial Crack ◽

Higher Order ◽

Weak Discontinuity ◽

Displacement Method ◽

Asymptotic Fields

The higher order discontinuous asymptotic fields which are similar to the Williams’ solutions of homogenous material are obtained by the displacement method and asymptotic analysis for a plane crack at the physical weak-discontinuous interface in non-homogeneous materials. The results provide a theoretical basis for the numerical analysis, experimental investigation and the engineering application of physical weak-discontinuous fracture.

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Tunnelling Effects of Solitons in Optical Fibers with Higher-Order Effects

Zeitschrift für Naturforschung A ◽

10.5560/zna.2012-0033 ◽

2012 ◽

Vol 67 (6-7) ◽

pp. 338-346

Author(s):

Chao-Qing Dai ◽

Hai-Ping Zhu ◽

Chun-Long Zheng

Keyword(s):

Optical Fibers ◽

Schrodinger Equation ◽

Soliton Solutions ◽

Higher Order ◽

Order Effects ◽

Bright Solitons ◽

Dark Solitons ◽

Tunnelling Effect ◽

Tunnelling Effects ◽

Higher Order Effects

We construct four types of analytical soliton solutions for the higher-order nonlinear Schrödinger equation with distributed coefficients. These solutions include bright solitons, dark solitons, combined solitons, and M-shaped solitons. Moreover, the explicit functions which describe the evolution of the width, peak, and phase are discussed exactly.We finally discuss the nonlinear soliton tunnelling effect for four types of femtosecond solitons

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Interplay Between Dispersion and Nonlinearity in Femtosecond Soliton Management

Zeitschrift für Naturforschung A ◽

10.1515/zna-2010-6-710 ◽

2010 ◽

Vol 65 (6-7) ◽

pp. 549-554

Author(s):

Ramaswamy Radha ◽

Vaduganathan Ramesh Kumar

Keyword(s):

Pulse Propagation ◽

Group Velocity Dispersion ◽

Soliton Solutions ◽

Higher Order ◽

Order Effects ◽

Nls Equation ◽

Linear Eigenvalue Problem ◽

Third Order Dispersion ◽

Femtosecond Regime ◽

Soliton Management

In this paper, we investigate the inhomogeneous higher-order nonlinear Schr¨odinger (NLS) equation governing the femtosecond optical pulse propagation in inhomogeneous fibers using gauge transformation and generate bright soliton solutions from the associated linear eigenvalue problem. We observe that the amplitude of the bright solitons depends on the group velocity dispersion (GVD) and the self-phase modulation (SPM) while its velocity is dictated by the third-order dispersion (TOD) and GVD. We have shown how the interplay between GVD, SPM, and TOD can be profitably exploited to change soliton width, amplitude (intensity), shape, phase, velocity, and energy for an effective femtosecond soliton management. The highlight of our paper is the identification of ‘optical similaritons’ arising by virtue of higher-order effects in the femtosecond regime.

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