Wronskian, Gramian, Pfaffian and periodic-wave solutions for a (3+1)-dimensional generalized nonlinear evolution equation arising in the shallow water waves

• Published in: Nonlinear dynamics Vol. 108; no. 2; pp. 1599 - 1616
• Main Authors: Liu, Fei-Yan, Gao, Yi-Tian, Yu, Xin, Ding, Cui-Cui
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.04.2022; Springer Nature B.V
• Subjects: Automotive Engineering; Classical Mechanics; Control; Dynamical Systems; Engineering; Environmental engineering; Hydraulic engineering; Mechanical Engineering; Nonlinear evolution equations; Original Paper; Qualitative analysis; Shallow water; Solitary waves; Vibration; Water waves; Pfaffian solitons; Gramian solutions; (3+1)-dimensional generalized; Periodic-wave solutions; Shallow water waves; Wronskian solutions
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-022-07249-1

Application of the shallow water waves in environmental engineering and hydraulic engineering is seen. In this paper, a (3+1)-dimensional generalized nonlinear evolution equation (gNLEE) for the shallow water waves is investigated. The N th-order Wronskian, Gramian and Pfaffian solutions are proved, where N is a positive integer. Soliton solutions are constructed from the N th-order Wronskian, Gramian and Pfaffian solutions. Moreover, we analyze the second-order solitons with the influence of the coefficients in the equation and illustrate them with graphs. Through the Hirota-Riemann method, one-periodic-wave solutions are derived. Relationship between the one-periodic-wave solutions and one-soliton solutions is investigated, which shows that the one-periodic-wave solutions can approach to the one-soliton solutions under certain conditions. We reduce the (3+1)-dimensional gNLEE to a two-dimensional planar dynamic system. Based on the qualitative analysis, we give the phase portraits of the dynamic system.