Rogue waves in the three-level defocusing coupled Maxwell–Bloch equations

Abstract

The coupled Maxwell–Bloch (CMB) system is a fundamental model describing the propagation of ultrashort laser pulses in a resonant medium with coherent three-level atomic transitions. In this paper, we consider an integrable generalization of the CMB equations with the defocusing case. The CMB hierarchy is derived with the aid of a 3 × 3 matrix eigenvalue problem and the Lenard recursion equation, from which the defocusing CMB model is proposed as a special reduction of the general CMB equations. The n -fold Darboux transformation as well as the multiparametric n th-order rogue wave solution of the defocusing CMB equations are put forward in terms of Schur polynomials. As an application, the explicit rogue wave solutions from first to second order are presented. Apart from the traditional dark rogue wave, bright rogue wave and four-petalled rogue wave, some novel rogue wave structures such as the dark four-peaked rogue wave and the double-ridged rogue wave are found. Moreover, the second-order rogue wave triplets which contain a fixed number of these rogue waves are shown.

Lie symmetry analysis, optimal system and exact solutions for variable-coefficients Boiti-Leon-Manna-Pempinelli equation

“…Nonlinear partial differential equations (NLPDEs) are the significant models to study many nonlinear phenomena in modern mathematical physics. With the development of modern mathematics and physics, the exact solutions of NLPDEs, especially some high-dimensional NLPDEs [1,2], have become a hot topic for scholars. Integrable systems and soliton solutions have also been extensively studied in the field of NLPDEs [3].…”

Rogue waves in the massive Thirring model

“…Because of the underlying integrability, it was found that Peregrine soliton can be extended to higher order exact rogue wave solutions in many nonlinear wave systems such as the NLS equation, [10][11][12][13][14] the derivative NLS equation, [15][16][17][18] the Sasa-Satsuma equation, [19][20][21] the Manakov equations, [22][23][24][25] the long-wave-short-wave equations, [26][27][28][29] the three-wave resonant interaction equations, [30][31][32][33][34] the Davey-Stewartson equations 35,36 and many others. [37][38][39][40][41][42][43][44][45] These analytic rational solutions with the higher order polynomials also represent the localized structure in both space and time coordinates, and could possess multiple intensity peaks or higher peak amplitudes. In particular, differing from the scalar system, the coupled and vector integrable systems with the additional degrees of freedom could allow the novel counterpart of rogue wave such as dark and four-petaled types.…”

“…From the mathematical description, Peregrine soliton characterized by a kind of rational solutions for the focusing nonlinear Schrödinger (NLS) equation was discovered to act as the prototype of realistic rogue waves, 9 as it exhibits the local wave structure in temporal–spatial plane and the height of maximum peak at the center reaches to three times the finite background. Because of the underlying integrability, it was found that Peregrine soliton can be extended to higher order exact rogue wave solutions in many nonlinear wave systems such as the NLS equation, 10–14 the derivative NLS equation, 15–18 the Sasa–Satsuma equation, 19–21 the Manakov equations, 22–25 the long‐wave–short‐wave equations, 26–29 the three‐wave resonant interaction equations, 30–34 the Davey–Stewartson equations 35,36 and many others 37–45 . These analytic rational solutions with the higher order polynomials also represent the localized structure in both space and time coordinates, and could possess multiple intensity peaks or higher peak amplitudes.…”

Lie symmetry analysis, conservation laws and exact solutions for variable-coefficients (2+1)-dimensional dissipative long-wave system

“…But for any given nonlinear partial differential system, it is almost impossible to find all of its solutions [1][2][3]. However, by applying the symmetry theory to NPDEs, some exact solutions that keep certain transformations invariant can be obtained by the lowering of the dimension of the equations [4,5], which is also called the optimal system problem. For one dimensional optimal systems, its construction was first proposed by Ovisiannikov, who used the ensemble matrix of the adjoint representation [6].…”