Explicit exact solutions and conservation laws in a medium with competing weakly nonlocal nonlinearity and parabolic law nonlinearity

In this paper, we studied a nonlinear wave equation that models the propagation of optical solutions in a weakly nonlocal and parabolic competing nonlinear medium. The exact traveling wave type solutions formulated in hyperbolic functions, rational and trigonometric functions multiplied by some exponential functions to the governing equation are determined explicitly by the extended form of the Kudryashov method. We have examined the behavior of the modulation instability (MI) growth rate. To substantiate the stability of the obtained dark and bell-shaped solitons, we use the split-step Fourier method. In addition, the conservation laws describing significant physical concepts of this equation are examined. Compared to the obtained results with Younis et al. (J Nonlinear Opt Phys Mater 24(04):1550049, 2015), Zhou et al. (Optik 124(22):5683-5686, 2013; Proc Rom Acad Ser A 16(2):152-159, 2015) and Akinyemi et al. (Optik 230:166281, 2021), we have shown the propagation of the solitonic waves which sometime tilt from right to left with stable shape.

Automated summary

In this paper, a nonlinear wave equation modeling optical solutions in a weakly nonlocal and parabolic competing nonlinear medium is studied. Exact traveling wave solutions in hyperbolic, rational, and trigonometric functions multiplied by exponential functions are determined using the extended form of the Kudryashov method. The modulation instability growth rate is analyzed, and the stability of dark and bell-shaped solitons is substantiated using the split-step Fourier method. The propagation of solitonic waves with stable shape, sometimes tilting from right to left, is demonstrated compared to previous studies.