The new exact solutions for the deterministic and stochastic (2+1)-dimensional equations in natural sciences

Abstract

This paper poses the Riccati–Bernoulli sub-ODE method in order to find the exact (random) travelling wave solutions for the (2+1)-dimensional cubic nonlinear Klein–Gordon (cKG) equation and the (2+1)-dimensional nonlinear Zakharov–Kuznetsov modified equal width (ZK-MEW) equation. The obtained travelling wave solutions are expressed by the hyperbolic, trigonometric and rational functions. Indeed, these solutions reflect some interesting physical interpretation for nonlinear phenomena. We discuss our method in deterministic case and in a random case. Additionally, we can show and discuss this method under some random distributions. Finally, some three-dimensional graphics of some solutions have been illustrated.

Keywords

• riccati–bernoulli sub-ode method
• nonlinear (stochastic) partial differential equations
• ckg equation
• zk-mew equation
• bäcklund transformation
• travelling wave solutions