N-Soliton and Other Analytic Solutions for a ( $$3 + 1$$ )-Dimensional Korteweg–de Vries–Calogero–Bogoyavlenskii–Schiff Equation with the Time-Dependent Coefficients for the Shallow Water Waves

Abstract

Shallow water waves are seen in magnetohydrodynamics, atmospheric science, oceanography and so on. In this article, we study a (\(3+1\))-dimensional Korteweg–de Vries–Calogero–Bogoyavlenskii–Schiff equation with the time-dependent coefficients for the shallow water waves. N-soliton solutions are obtained via the simplified Hirota method. Via the N-soliton solutions, we present the elastic interactions between the two solitons and among the three solitons. Some other analytic solutions are constructed through the tanh method and \((\frac{G'}{G^{2}})\)-expansion method.