On similarity solutions to (2+1)-dispersive long-wave equations

• Analytical solutions of (2+1)-coupled dispersive long wave equations (DLWEs) are obtainedby using Similarity transformations method via Lie group. • Some of the results derived in the previous findings are deduced from these results. • DLWEs can be observed in an open sea or in wide channels. • The analytical solutions show elastic multisolitons, single soliton, doubly solitons, stationary, kink and parabolic profiles. This work is devoted to get a new family of analytical solutions of the (2+1)-coupled dispersive long wave equations propagating in an infinitely long channel with constant depth, and can be observed in an open sea or in wide channels. The solutions are obtained by using the invariance property of the similarity transformations method via one-parameter Lie group theory. The repeated use of the similarity transformations method can transform the system of PDEs into system of ODEs. Under adequate restrictions, the reduced system of ODEs is solved. Numerical simulation is performed to describe the solutions in a physically meaningful way. The profiles of the solutions are simulated by taking an appropriate choice of functions and constants involved therein. In each animation, a frame for dominated behavior is captured. They exhibit elastic multisolitons, single soliton, doubly solitons, stationary, kink and parabolic nature. The results are significant since these have confirmed some of the established results of S. Kumar et al. (2020) and K. Sharma et al. (2020). Some of their solutions can be deduced from the results derived in this work. Other results in the existing literature are different from those in this work.