A novel expansion method and its application to two nonlinear evolution equations arising in ocean engineering

Abstract Nonlinear evolution equations are unavoidable for precisely modelling and understanding nonlinear wave phenomena. The study of nonlinear waves enriches our comprehension of natural phenomena and supports technological advancements across various disciplines. In this work, we have proposed a new expansion method to find the travelling wave solutions of nonlinear evolution equations. This method is named as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:mi mathvariant="normal">F</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">μ</mml:mi> <mml:mi mathvariant="normal">F</mml:mi> <mml:mo>+</mml:mo> <mml:mi mathvariant="normal">G</mml:mi> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:mfenced> </mml:mrow> <mml:mo>−</mml:mo> </mml:math> expansion method. We applied the proposed technique to construct the exact travelling wave solutions to two well-known nonlinear equations arising in ocean engineering. These equations are extended (2 + 1)-dimensional Boussinesq equation and (3 + 1)-dimensional generalized shallow water wave equation. Propagation of obtained travelling wave solutions are illustrated by surface plots and two-dimensional graphs plotted for suitable parametric values. We observed soliton, kink, breather, lump and periodic wave structures. The results show efficiency and reliability of the proposed method.