Symmetry reduction and exact solutions of two higher-dimensional nonlinear evolution equations

Area of Science:

• Mathematical Physics
• Nonlinear Dynamics
• Differential Equations

Background:

• Nonlinear evolution equations (NLEEs) are crucial in modeling complex phenomena across various scientific disciplines.
• Understanding the symmetries and exact solutions of NLEEs is fundamental for analyzing their behavior and applications.
• Lie group analysis provides a powerful framework for systematically reducing the complexity of differential equations.

Purpose of the Study:

• To investigate the symmetries and perform symmetry reduction for two specific higher-dimensional nonlinear evolution equations.
• To derive exact analytical solutions for these important NLEEs using established mathematical techniques.
• To construct a diverse set of explicit function solutions, including rational, exponential, trigonometric, hyperbolic, and elliptic forms.

Main Methods:

• Lie group method was employed to identify and analyze the symmetries inherent in the NLEEs.
• The [Formula: see text]-expansion method was utilized to systematically generate exact solutions.
• A complex method was also applied to complement the solution-finding process.

Main Results:

• Symmetries and symmetry reductions were successfully obtained for the two higher-dimensional NLEEs.
• Five distinct types of explicit function solutions were successfully constructed.
• The derived solutions encompass rational, exponential, trigonometric, hyperbolic, and elliptic function forms.

Conclusions:

• The study successfully demonstrated the application of Lie group analysis and the [Formula: see text]-expansion method for solving complex NLEEs.
• The discovery of multiple explicit function solutions enriches the understanding of the behavior of these equations.
• The findings contribute to the advancement of analytical techniques in nonlinear sciences and the study of differential equations.