Construction of Multi-soliton Solutions of the N-Coupled Hirota Equations in an Optical Fiber^*

Abstract

This paper focuses on investigation of the N -coupled Hirota equations arising in an optical fiber. Starting from analyzing the spectral problem, a kind of matrix Riemann-Hilbert problem is formulated strictly on the real axis. Then based on the resulting matrix Riemann-Hilbert problem under the constraint of no reflection, multi-soliton solutions to the N -coupled Hirota equations are presented explicitly.

Riemann-Hilbert problems of a non-local reverse-time AKNS system of six-order and dynamical behaviours of $N$-soliton

“…where all the involved functions are defined as follows: [1] = 0, a [2] = − β α 2 T 0,0 , a [3] = −i β α 3 (T 0,1 − T 1,0 ),…”

“…They have many applications in real life, for example in optical fiber, vortex filaments, ocean and water waves, and gravitational fields [1]- [2]. The Korteweg deVries, nonlinear Schrödinger equation and Sine-Gordon are all well known examples of integrable systems, where their intrinsic solutions are solitons.…”

Construction of multiple soliton solutions of the quintic nonlinear Schrodinger equation

“…In order to derive soliton solutions explicitly, we take G = I 2 in (17), which indicates that no reflection exists in the scattering problem. Therefore, the solutions [24] for this special Riemann-Hilbert problem can be given by…”

“…By now, many efficient approaches have been available for finding soliton solutions, some of which include inverse scattering transformation [1,2], Darboux transformation [3][4][5][6], Bäcklund transformation [7], Riemann-Hilbert method [8], and Hirota's bilinear method [9][10][11][12]. In recent years, there has been an increasing interest in exploring abundant multiple soliton solutions of NLEEs via the Riemann-Hilbert method, including the coupled derivative Schrödinger equation [13], the Kundu-Eckhaus equation [14], and others [15][16][17][18][19][20][21].…”

On multi-soliton solutions to the Heisenberg ferromagnetic spin chain equation in (2+1)-dimensions

“…To date, a number of methods have confirmed the effectiveness in investigating solutions to NLPDEs, some of which include the inverse scattering transformation [1] , the Hirota bilinear method [2] , the Darboux transformation [3] and the Bäcklund transformation [4] . In recent years, researchers have shown an increased interest in study of NLPDEs utilizing the Riemann-Hilbert technique, such as the coupled nonlinear Schrödinger equation [5] , the Kundu-Eckhaus equation [6] , the coupled mKdV system [7] and the 𝑁 -coupled Hirota equation [8] .…”