Numerical solutions and conservation laws for nonlinear evolution equations

Abstract This paper presents numerical solutions of nonlinear evolution equations using a hybrid collocation method. Nonlinear evolution equations, including the regularized long wave (RLW) equation and the modified regularized long wave (MRLW) equation, play a crucial role in modeling various physical phenomena. A hybrid collocation technique is used for estimating and examining the characteristics of the solitary waves, including their shape, structure, and propagation. The Crank–Nicolson method is used for time discretization and the hybrid collocation method for space discretization. The Fourier series analysis has been used to analyze the stability of the proposed method, and it is established that the hybrid collocation method is unconditionally stable. The accuracy of the proposed scheme is checked by computing the error norm L ∞ and the three invariants. The novelty of the method lies in deriving new approximations for the second derivative and applying it on time-dependent nonlinear partial differential equations. A comparison with existing techniques in the literature is conducted to check the improvements in results. The numerical outcomes show that the proposed scheme effectively depicts the conservation laws of solitary waves. The values of three invariants at different time levels have been shown to coincide with their analytical values. The propagation of one, two, and three solitary waves, development of the Maxwellian initial condition into one, two, and more solitary waves, and wave undulations have been illustrated graphically. The method captures the collisions between solitary waves very accurately. Our findings demonstrate that the new cubic B-spline approach offers an accurate and effective solution for the nonlinear evolution equations.