Dynamical study of different types of soliton solutions with bifurcation, chaos and sensitivity analysis to the non-linear coupled Schrödinger model

Abstract

This study presents an analytical and dynamical examination of the non-linear coupled Schrödinger model, which describes non-linear wave propagation in dispersive and memory-dependent media. The model involves M-truncated and Beta derivatives. By applying the modified [Formula: see text]-expansion function method, we obtained linear and rational forms of trigonometric and hyperbolic trigonometric solutions. The behavior of solutions under different parameters is further analyzed using the bifurcation technique. Additionally, the study demonstrates chaotic behavior, non-linear coupled Schrödinger model. Sensitivity analysis is also conducted to illustrate the influence of small variations in system parameters on the overall dynamics. This model yields various types of solutions, including singular complexiton, singular periodic, singular bell or singular bright, singular shape, kink, anti-kink, bright and dark soliton waves. These solutions are graphically illustrated using 2D, 3D and contour plots for suitable parameter values to reflect the physical behavior of the system. The results of this study are expected to be highly beneficial in diverse scientific fields and complex investigations.

Keywords: Bright and dark solitons; Chaotic dynamics; Modified [Formula: see text]-expansion function approach; Parameter analysis; Singular soliton solutions.

Fig. 1

The singular bell pattern or singular bright soliton wave is the 3D animation for produced by Eq. (38), 2D line visualizations of at different values of t and Interconnected contour display when and .

Fig. 2

The singular periodic soliton wave is the 3D animation for produced by Eq. (39), 2D line visualizations of at different values of t, and interconnected contour display when and .

Fig. 3

(a) The singular shape soliton wave is the 3D animation for produced by Eq. (40), (b) 2D line visualizations of at different values of t and (c) Interconnected contour display when and .

Fig. 4

The singular complexion pattern soliton wave is the 3D animation for produced by Eq. (41), 2D line visualizations of at different values of t and Interconnected contour display when and .

Fig. 5

The singular shape soliton wave is the 3D animation for produced by Eq. (42), 2D line visualizations of at different values of t and Interconnected contour display when and .

Fig. 6

Phase portrait of dynamical system for case 1: when and .

Fig. 7

Phase portrait of dynamical system for case 2: when and .

Fig. 8

Chaotic behaviour of Eq. (46) for , , and . 3D phase projection, 2D phase projection and Time series.

Fig. 9

Chaotic behaviour of Eq. (46) for , , and . (a) 3D phase projection, (b) 2D phase projection and (c) Time series.

Fig. 10

Chaotic behaviour of Eq. (46) for , , and . (a) 3D phase projection, (b) 2D phase projection and (c) Time series.

Fig. 11

Sensitive behaviour of Eq. (43) with initial values for red curve and for blue curve.

Fig. 12

Sensitive behaviour of Eq. (43) with initial values for red curve and for green curve.

Fig. 13

Sensitive behaviour of Equation (43) with initial values for blue curve and for green curve.

Fig. 14

Sensitive behaviour of Eq. (43) with initial values for red curve, for blue curve and for green curve.

Fig. 15

The kink wave pattern is the 3D animation for produced by Eq. (51) for , , , , and and , 2D line visualizations for positive at different values of s.

Fig. 16

(a) The anti-kink wave pattern is the 3D animation for produced by Eq. (51) for , , , , and and , (b) 2D line visualizations for negative at different values of s.

Fig. 17

The bright soliton wave is the 3D animation for produced by Eq. (55) for , , , , , and , 2D line visualizations for positive at different values of s.

Fig. 18

The dark soliton wave is the 3D animation for produced by Eq. (55) for , , , , , and , 2D line visualizations for negative at different values of s.