A study on the (2+1)-dimensional first extended Calogero-Bogoyavlenskii- Schiff equation

Area of Science:

• Mathematical Physics
• Nonlinear Partial Differential Equations
• Integrable Systems

Background:

• The (2+1)-dimensional first extended Calogero-Bogoyavlenskii-Schiff equation is a recent addition to the study of nonlinear partial differential equations.
• Understanding the properties and solutions of such equations is crucial for various fields of physics and applied mathematics.

Purpose of the Study:

• To investigate the Lie symmetries of the (2+1)-dimensional first extended Calogero-Bogoyavlenskii-Schiff equation.
• To perform symmetry reductions and derive closed-form solutions.
• To compute conserved vectors of the equation.

Main Methods:

• Derivation of Lie symmetries.
• Symmetry reduction using translation symmetries.
• Application of Kudryashov and (G'/G)-expansion techniques.
• Multiplier approach and Noether's theorem for conserved vectors.

Main Results:

• A fourth-order ordinary differential equation was obtained through symmetry reduction.
• Closed-form solutions were successfully constructed using the Kudryashov and (G'/G)-expansion methods.
• Conserved vectors were computed, demonstrating the equation's conserved quantities.
• Graphical representations of the obtained solutions were provided.

Conclusions:

• The study successfully applied Lie symmetry analysis and established techniques to find exact solutions for the (2+1)-dimensional first extended Calogero-Bogoyavlenskii-Schiff equation.
• The findings contribute to the understanding of this nonlinear equation and its potential applications.
• The computation of conserved vectors highlights the integrability properties of the equation.