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Solutions by quadratures of complex Bernoulli differential equations and their quantum deformation

Rutwig Campoamor-Stursberg, Eduardo Fernandez-Saiz, Francisco J. Herranz

Abstract

It is shown that the complex Bernoulli differential equations admitting the supplementary structure of a Lie-Hamilton system related to the book algebra $\mathfrak{b}_2$ can always be solved by quadratures, providing an explicit solution of the equations. In addition, considering the quantum deformation of Bernoulli equations, their canonical form is obtained and an exact solution by quadratures is deduced as well. It is further shown that the approximations of $k^{th}$-order in the deformation parameter from the quantum deformation are also integrable by quadratures, although an explicit solution cannot be obtained in general. Finally, the multidimensional quantum deformation of the book Lie-Hamilton systems is studied, showing that, in contrast to the multidimensional analogue of the undeformed system, the resulting system is coupled in a non-trivial form.

Solutions by quadratures of complex Bernoulli differential equations and their quantum deformation

Abstract

It is shown that the complex Bernoulli differential equations admitting the supplementary structure of a Lie-Hamilton system related to the book algebra can always be solved by quadratures, providing an explicit solution of the equations. In addition, considering the quantum deformation of Bernoulli equations, their canonical form is obtained and an exact solution by quadratures is deduced as well. It is further shown that the approximations of -order in the deformation parameter from the quantum deformation are also integrable by quadratures, although an explicit solution cannot be obtained in general. Finally, the multidimensional quantum deformation of the book Lie-Hamilton systems is studied, showing that, in contrast to the multidimensional analogue of the undeformed system, the resulting system is coupled in a non-trivial form.
Paper Structure (8 sections, 1 theorem, 101 equations)

This paper contains 8 sections, 1 theorem, 101 equations.

Table of Contents

  1. Introduction
  2. Lie--Hamilton systems on the book algebra revisited
  3. Exact solutions of complex Bernoulli equations
  4. Deformed Lie--Hamilton systems from the quantum book algebra
  5. On the approximations of order $k$ in the deformation parameter
  6. Deformed complex Bernoulli differential equations
  7. Higher dimensional quantum deformations of $\mathfrak{b}_2$-LH systems
  8. Conclusions and final remarks

Key Result

Proposition 2.1

Any first-order system such that the associated $t$-dependent vector field admits the decomposition for some functions $f(t),g(t)$ and vector fields ${\bf X}_1,{\bf X}_2$ satisfying with $\lambda,\mu$ constants, can be solved by quadratures.

Theorems & Definitions (2)

  • Proposition 2.1
  • proof