Solitary waves and shock waves for double-layered fluid flow with dispersion triplet: Zaremaoghaddam and Gear–Grimshaw models (KdV equation)

Abstract Background The Zaremaoghaddam model studies internal waves, fluid dynamics, and nonlinear wave equations. In shallow water, internal solitons, or stratified fluids, the procedure may involve modifying or applying nonlinear wave models like Korteweg–de Vries equation, Boussinesq, or Gear–Grimshaw. The Gear–Grimshaw system simulates internal waves in a two-layer stratified fluid, such as an ocean with two density layers, using interconnected nonlinear evolution equations. The Korteweg–de Vries equation is extended to include wave mode interactions. Results The paper recovers solitary waves and shock waves for double-layered fluid flow that is modeled whose basic platform is the Korteweg–de Vries equation. This retrieval is made possible with the usage of the generalized exponential differential rational function approach. Two models for the double-layered flow are taken into consideration, namely the Zaremaoghaddam model and the Gear–Grimshaw model. The parameter restrictions for the existence of such solutions are also enumerated. Conclusions This paper has many implications and opens up many future opportunities. Since double-layered fluid flow never addresses rogue wave features, the paper’s results would be the foundation for studying them. Additionally, viscosity can be considered in the two models in this paper. A practical perspective would result since viscosity is inevitable in any fluid or airflow. Additionally, these models are new. Thus, such models must be investigated by identifying conservation laws and studying soliton perturbation theory.