Fractional Consistent Riccati Expansion Method and Soliton-Cnoidal Solutions for the Time-Fractional Extended Shallow Water Wave Equation in (2 + 1)-Dimension

• Published in: Fractal and fractional Vol. 8; no. 10; p. 599
• Main Authors: Zhang, Lihua, Shen, Bo, Jia, Meizhi, Wang, Zhenli, Wang, Gangwei
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01-10-2024
• Subjects: Boiti–Leon–Manna–Pempinelli (BLMP) equation; Exact solutions; fractional consistent Riccati expansion (FCRE) method; fractional derivatives; Hypotheses; Korteweg-Devries equation; Methods; Shallow water; shallow water wave equation; Solitary waves; soliton-cnoidal solutions; Water waves; Wave equations; Wave interaction
• ISSN: 2504-3110; 2504-3110
• DOI: 10.3390/fractalfract8100599

In this work, a fractional consistent Riccati expansion (FCRE) method is proposed to seek soliton and soliton-cnoidal solutions for fractional nonlinear evolutional equations. The method is illustrated by the time-fractional extended shallow water wave equation in the (2 + 1)-dimension, which includes a lot of KdV-type equations as particular cases, such as the KdV equation, potential KdV equation, Boiti–Leon–Manna–Pempinelli (BLMP) equation, and so on. A rich variety of exact solutions, including soliton solutions, soliton-cnoidal solutions, and three-wave interaction solutions, have been obtained. Comparing with the fractional sub-equation method, G′/G-expansion method, and exp-function method, the proposed method gives new results. The method presented here can also be applied to other fractional nonlinear evolutional equations.