[Paper Review] Point-symmetry pseudogroup, Lie reductions and exact solutions of Boiti-Leon-Pempinelli system

Diana S. Maltseva, Roman O. Popovych|arXiv (Cornell University)|Mar 15, 2021

Nonlinear Waves and Solitons48 references4 citations

TL;DR

This paper presents a comprehensive symmetry analysis of the (1+2)-dimensional Boiti-Leon-Pempinelli system, computing its complete point-symmetry (pseudo)group via a novel megaideal-based algebraic method and constructing extensive families of exact solutions through Lie reductions and differential constraints. The key contribution is the identification of new, non-trivial exact solutions beyond previously known ones, significantly extending the solution space using Laplace and Darboux transformations.

ABSTRACT

We carry out extended symmetry analysis of the (1+2)-dimensional Boiti-Leon-Pempinelli system, which corrects, enhances and generalizes many results existing in the literature. The point-symmetry pseudogroup of this system is computed using an original megaideal-based version of the algebraic method. A number of meticulously selected differential constraints allow us to construct families of exact solutions of this system, which are significantly larger than all known ones. After classifying one- and two-dimensional subalgebras of the entire (infinite-dimensional) maximal Lie invariance algebra of this system, we study only its essential Lie reductions, which give solutions beyond the above solution families. Among reductions of the Boiti-Leon-Pempinelli system using differential constraints or Lie symmetries, we identify a number of famous partial and ordinary differential equations. We also show how all the constructed solution families can significantly be extended by Laplace and Darboux transformations.

Motivation & Objective

• To perform an enhanced and extended symmetry analysis of the (1+2)-dimensional Boiti-Leon-Pempinelli system.
• To compute the complete point-symmetry (pseudo)group using a novel megaideal-based version of the algebraic method.
• To construct larger families of exact solutions via carefully selected differential constraints and Lie reductions.
• To classify one- and two-dimensional subalgebras of the infinite-dimensional maximal Lie invariance algebra and identify essential reductions yielding new solutions.
• To demonstrate how Laplace and Darboux transformations can significantly extend the constructed solution families.

Proposed method

• A modified megaideal-based version of the algebraic method is applied to compute the complete point-symmetry (pseudo)group of the system.
• The method involves pushing forward a finite-dimensional subalgebra of the infinite-dimensional maximal Lie invariance algebra instead of the entire algebra.
• Differential constraints are meticulously selected to generate large families of exact solutions, including non-Lie solutions.
• One- and two-dimensional subalgebras of the maximal Lie invariance algebra are exhaustively classified to perform optimal Lie reductions.
• Laplace and Darboux transformations are applied to extend the solution families beyond those obtained via direct reduction.
• Conservation laws are analyzed by computing reduced cosymmetries up to order four, identifying conserved currents associated with parameter functions.

Experimental results

Research questions

• RQ1 What is the complete point-symmetry (pseudo)group of the Boiti-Leon-Pempinelli system, and how can it be computed efficiently despite the system’s infinite-dimensional maximal Lie invariance algebra?
• RQ2 How can families of exact solutions be systematically constructed using differential constraints and Lie reductions, and what is their scope relative to previously known solutions?
• RQ3 Which Lie reductions yield solutions outside the families obtained via differential constraints, and how can they be classified optimally?
• RQ4 To what extent can Laplace and Darboux transformations extend the set of exact solutions of the system?
• RQ5 What is the structure of the conservation laws and cosymmetries of the system, and how do they relate to the system’s integrability and solution generation?

Key findings

• The complete point-symmetry (pseudo)group of the Boiti-Leon-Pempinelli system is computed using a novel megaideal-based algebraic method, providing a rigorous foundation for further symmetry analysis.
• A large family of exact solutions is constructed via differential constraints, significantly expanding the known solution space beyond trivial or incorrect solutions found in prior literature.
• The classification of one- and two-dimensional subalgebras of the maximal Lie invariance algebra leads to essential Lie reductions that yield new solutions not captured by differential constraint methods.
• The general solutions of the degenerate ordinary differential equation arising from Lie reductions are explicitly derived in terms of elementary functions, including rational, exponential, and trigonometric forms.
• Laplace and Darboux transformations are shown to extend the constructed solution families, enabling the generation of new exact solutions from existing ones.
• The space of reduced cosymmetries and associated conservation laws is fully characterized up to order four, with conserved currents identified for parameter functions of time and space.