New Exact Solutions of Broer–Kaup Equations

Abstract

By a known transformation, (2+1)-dimensional Brioer–Kaup equations are turned to a single equation. The classical Lie symmetry analysis and similarity reductions are performed for this single equation. From some of reduction equations, new exact solutions are obtained, which contain previous results, and more exact solutions can be created directly by abundant known solutions of the Burgers equations and the heat equations.

Non-Lie Symmetry Groups and New Exact Solutions to the (2 + 1)-Dimensional Broer-Kaup System

“…In doing that we obtain the system's corresponding symmetry group theorem and the Backlund transformation formula of solutions finding, through which we are able to obtain some new exact solutions to the BK system. We therefore extend the results in paper [1][2][3][4].…”

“…Paper [1] uses the extended homogeneous balance method and separation of variables to discuss the localized coherent structure of system (1). Paper [2] by using the Lie-group optimized system classifies the solutions of system (1) and furthermore finds some new explicit solutions. Through the application of the extended homogeneous balance method, papers [3,4] obtain some exact solutions to system (1) and explore the system's induced phenomenon.…”

Lie symmetry analysis and conservation laws for the (2+1)-dimensional Mikhalev equation

“…Searching for solutions to partial differential equations (PDEs), which arise from physics, chemistry, economics and other fields, is one of the most fundamental and significant areas. A wealth of solving methods have been developed, such as the Lie symmetry analysis [5,8,11,15], the homogeneous balance method [13,18], Hirota's bilinear method [10], the Painlev's analysis method [6]. The Lie symmetry analysis is one of the most effective tools for solving partial differential equations and it was firstly traced back to the famous Norwegian mathematician Sophus Lie [12], who was influenced and inspired by the Galois theory founded in the early 18th century.…”