On the Application of the Double Reduction Theory to (n+1)-Dimensional Scalar PDEs and Systems of PDEs

Abstract

The generalization of the double reduction theory to higher-dimensional partial differential equations (PDEs) presents significant challenges, as highlighted by Sjoberg in prior studies. Despite these challenges, the application of the double reduction method to systems of PDEs and higher dimensional PDEs has been explored in the literature, albeit to a limited extent. Seminal contributions in this area include the work of Bokhari et al., who demonstrated that if a nontrivial conserved form exists with at least one associated symmetry in each reduction, a nonlinear system of qth order PDEs with n independent and m dependent variables can be systematically reduced to a nonlinear system of (q−1)th order ordinary differential equations (ODEs). This approach underscores the potential of the double reduction method in simplifying complex systems and advancing the analysis of higher-dimensional PDEs. In this thesis, we contribute to the existing body of literature on the double reduction method by applying it to a variety of (n+1)-dimensional PDEs and (1+1)-dimensional systems of PDEs. We also introduce a variation of the double reduction method, demonstrating that the explicit use of associated conservation laws is not always necessary. Instead, canonical variables derived from symmetry alone can suffice for the reduction process. We begin by illustrating the versatility of the double reduction method by applying it to specific (1+1)-dimensional PDEs, such as the Gibbons-Tsarev (GT) equation and the Hunter Saxton (HS) equation. We construct conservation laws using the multiplier method, compute Lie point symmetries, and identify symmetries associated with conservation laws. These steps lead to invariant solutions through the double reduction process. Furthermore, the introduced variation of the double reduction method is also applied to several (1+1)-dimensional scalar PDEs, highlighting its effectiveness in simplifying the reduction process. Additionally, we apply the double reduction method to (2+1)-dimensional scalar PDEs, including the Zakharov-Kuznetsov (ZK) equation. By identifying and utilizing inherited symmetries, we achieve a second symmetry reduction, which significantly simplifies the equations and leads to invariant solutions. Finally, we apply the generalized double reduction method to two (1+1)-dimensional systems of PDEs— the Broer-Kaup (BK) system and the Kaup-Boussinesq (K-B) system. In these cases, we identify symmetries, construct conservation laws using the multiplier method, and perform reductions, yielding exact solutions and demonstrating the practical applicability of the method.