Multi-soliton solutions and asymptotic analysis for the coupled variable-coefficient Lakshmanan-Porsezian-Daniel equations via Riemann-Hilbert approach

Abstract

In this paper, we study multi-soliton solutions and asymptotic analysis for the coupled variable-coefficient Lakshmanan-Porsezian-Daniel equations, which describe the simultaneous propagation of nonlinear waves in the inhomogeneous optical fibers. We analyze the spectrum of the Lax pair to establish the Riemann-Hilbert problem. Using such Riemann-Hilbert problem, we calculate various multi-soliton solutions without reflection, including breather-like and mixed solitons. We illustrate the propagation and interaction dynamics of the solitons through appropriate parameter selection and asymptotic analysis. We find that the interaction between solitons is elastic, the amplitudes of solitons are only determined by the initial velocity and interaction, and the soliton with lower energy always yields a position shift when elastic interaction occurs. In addition, we observe that the existence time of the wave changes with energy and that multiple elastic interactions between solitons can be obtained when we choose appropriate variable coefficients. Then, we investigate the influences of group velocity dispersions and fourth-order dispersions on the interactions of solitons through parameter modulation mode and asymptotic analysis. Furthermore, we present several new types of nonlinear phenomena graphically, including elastic interactions between parabolic solitons and hump-type solitons, elastic interactions between cubic solitons and hump-type solitons, and periodic-changing propagations.

Taking into consideration a fifth-order nonlinear Schrödinger equation in an optical fiber

“…Nonlinear studies in oceanography, acoustics, finance, fluid mechanics and nonlinear optics have been productive [1][2][3][4][5][6][7][8]. The propagation of the picosecond pulses in an optical fiber can be described by the following nonlinear Schrödinger equation:…”

Riemann–Hilbert Method Equipped with Mixed Spectrum for N-Soliton Solutions of New Three-Component Coupled Time-Varying Coefficient Complex mKdV Equations

“…However, the models solved by the RH method are basically isospectral, such as [27][28][29][30][31]. There is little research on the RH method for variable-coefficient equations [32][33][34][35], and even fewer cases [36,37] of non-isospectral or mixed spectral equations. Throughout all the existing literature, we know that Chen, Zhang, and Ye [36] successfully solved a non-isospectral Gross-Pitaevskii equation in 2021 by extending the RH method, followed by Zhang and Zhou [37] solving the variable-coefficient mixed spectral complex mKdV equation in 2023.…”

“…Throughout all the existing literature, we know that Chen, Zhang, and Ye [36] successfully solved a non-isospectral Gross-Pitaevskii equation in 2021 by extending the RH method, followed by Zhang and Zhou [37] solving the variable-coefficient mixed spectral complex mKdV equation in 2023. For all the isospectral equations with variable coefficients that have been studied by RH method, specifically, Li, Tian, Zhang, and Yang [32] obtained multi-soliton solutions containing distinct poles of arbitrary order to the fifth-order nonlinear Schrödinger equation (NLS); Xu and Zhang [33] derived N-soliton solutions of the generalized NLS equation; Zhou and Chen [34] gained high-order soliton solutions and analyzed their long-term asymptotic properties of the inhomogeneous Hirota equation; Ma, Li, Wang, Xie, and Du [35] constructed multi-soliton solutions and gave the corresponding asymptotic analysis for the coupled Lakshmanan-Porsezian-Daniel equations.…”

Multi-soliton solutions of coupled Lakshmanan–Porsezian–Daniel equations with variable coefficients under nonzero boundary conditions

“…[6][7][8] When dealing with non-uniform optical fibers, employing variable-coefficient NLS-type models offers a more accurate characterization of the system compared to their constantcoefficient counterparts. [9][10][11][12] With these insights, we study a coupled variable-coefficient LPDE to delineate the concurrent propagation of nonlinear waves in non-uniform optical media, [13][14][15][16][17][18] iq j,t + σ (t)q j,xx + ζ (t)…”

“…(1) under zero boundary conditions: Hirota's method has been used to analyze the interaction between solitons; [13] generalized Darboux transformations (DT) have been employed to obtain rogue waves and solitons; [14] binary DT and Hirota methods have been utilized to identify soliton solutions, respectively; [15] breathers, multi-peak solitons, and the transition from breathers to solitons have been analyzed through DT; [16,17] multi-soliton solutions have been investigated via parameter modulation and asymptotic analysis. [18] As far as we know, there have been no reports on investigations of soliton solutions for the LPDE under nonzero boundary conditions (NZBC). Thus, we investigate LPDE [19][20][21] −iq j,t + γ 8 (t)…”