Bilinear Forms, N-soliton Solution for Extended Fifth-Order Korteweg-de Vries (eKdV), Breather

This paper investigates extended Korteweg-de Vries (eKdV) equations in shallow water with nonlinear and dispersive terms of the second order. With the Hirota method's and symbolic computation's help, the bilinear forms, N-soliton solutions, and graphs for the eKdV equations’ Breather are constructed. Asymptotic analysis demonstrates that the collisions for the profile are elastic. We can control collision types (head-on or taking lead collisions) by adjusting the sign of the velocity v. During collisions, the speeds of solitons are proportional to a4 and \(\alpha \) . In addition, there is a proportional relationship between the velocity v and amplitude a. When it comes to breather solutions, α affects the central trajectories of the solutions, whereas γ(t) affects the amplitudes of the solitons. The amplitude of solitons will experience a decrease whenever there is an increase in the positive coefficient of the line-damping term \(\gamma (t)\) . Breather and soliton are discussed and plotted graphically.