Solutions, group analysis and conservation laws of the (2+1)-dimensional time fractional ZK–mZK–BBM equation for gravity waves

Based on the investigation of (2+1)-dimensional ZK–mZK–BBM equation, it describes the gravity water waves in a long-wave regime. With the help of the semi-inverse method and the variational method, the time fractional ZK–mZK–BBM equation is derived in the sense of Riemann–Liouville fractional derivatives, which opens a new window for understanding the features of gravity water waves. Further, the symmetry of the (2+1)-dimensional time fractional ZK–mZK–BBM equation is studied by fractional order symmetry. Meanwhile, based on the new conservation theorem, the conserved laws of (2+1)-dimensional time fractional ZK–mZK–BBM equation are constructed. Finally, we show how to derive the solutions of the time fractional ZK–mZK–BBM equation by a bilinear method and the radial basis functions (RBFs) meshless approach.