New solutions of the soliton type of shallow water waves and superconductivity models

Abstract

For uni-directional wave transmission in the smooth bottom of shallow sea water and the superconductivity of nonlinear media with dispersion systems, the (1 + 1)-dimensional Camassa-Holm and Landau-Ginzburg-Higgs equations are of particular interest in research to the academics. Analytical wave solutions to the stated models have been successfully constructed in this study, which might have considerable implications in describing the nonlinear dynamical behavior associated with the phenomena. The models we aim to uncover have been put into the form of differential equations with one characteristic variable expending the wave transformation coordinate and, thereupon, the rational (G′/G)-expansion technique is executed. Using the considered technique, diverse soliton solutions in suitable forms arrayed to trigonometric, rational, and hyperbolic functions have been determined. The achieved solutions are figured out in 3D profiles, assigning the free parameters involved in solutions to particular values and discussed their physical significance to bring out the inner context of the tangible incidents in the natural domain. The rational (G′/G)-expansion approach is efficient, concise, and capable of finding analytical solutions to other nonlinear models that can be considered in subsequent studies.

Keywords

• Rational -expansion technique
• Modified camassa-holm equation
• (1+1)-Dimensional Landau-Ginzburg-Higgs equation
• Characteristic variable
• Soliton