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Point-symmetry pseudogroup, Lie reductions and exact solutions of Boiti-Leon-Pempinelli system

Diana S. Maltseva, Roman O. Popovych

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.

Point-symmetry pseudogroup, Lie reductions and exact solutions of Boiti-Leon-Pempinelli system

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.

Paper Structure

This paper contains 13 sections, 7 theorems, 149 equations, 1 table.

Table of Contents

  1. Introduction
  2. Point-symmetry pseudogroup
  3. Classification of subalgebras
  4. Solutions by the method of differential constraints
  5. Lie reductions of codimension one
  6. Essential Lie reductions of codimension two
  7. On equivalent forms of the Boiti--Leon--Pempinelli system
  8. Laplace and Darboux transformations
  9. Generating new solutions with Laplace transformations
  10. Generating new solutions with Darboux transformations
  11. Conclusion
  12. Integration of ordinary differential equations related to elliptic functions
  13. Conservation laws

Key Result

Lemma 1

$\mathrm R_{\mathfrak g}=\langle P(g),Z(\beta)\rangle$, where $\mathrm R_{\mathfrak g}$ denotes the radical of $\mathfrak g$.

Theorems & Definitions (17)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Corollary 3
  • Corollary 4
  • Remark 5
  • Remark 6
  • Proposition 7
  • proof
  • ...and 7 more