A Generalization of the Modified Liouville Equation

Corresponding Author: Ruben-Dario Ortiz-Ortiz Faculty of Exact and Natural Sciences, University of Cartagena, Cartagena de Indias, Colombia Email: rortizo@unicartagena.edu.co Abstract: We study the modified Liouville equation using various transformations to build dynamical systems and we use Dulac’s criterion for give sufficient conditions of the non-existence of periodic orbits in the dynamical systems generated of the modified Liouville equation.


Introduction
The Bendixson-Dulac criterion consists of a sufficient number of conditions for the nonexistence of periodic orbits in planar dynamical systems (Farkas, 1994). The modified Liouville equation (Abdelrahman et al., 2015;Salam et al., 2012) plays an important role in various areas of mathematical physics, from plasma physics and field theoretical modeling to fluid dynamics, using various transformations the differential equation can be written as a dynamic system that under some conditions does not have periodic orbits 2013a;Osuna and Villaseñor, 2011). The system in (Marin-Ramirez et al., 2015) coincides to our system. A generalization of a dynamical system was made in (Yan-Min et al., 2016;Qiu-Peng et al., 2015;Xiangwei et al., 2016). A Dulac function for a quadratic system was found in (Marin et al., 2013b). A Dulac function and a geometric method for a quadratic system was studied in (Marin-Ramirez et al., 2014). In this article our objective is construct dynamical systems that does not have periodic orbits using Dulac functions and we use the following criterion to show the non-existence of periodic orbits. The Dulac criterion was used in (Rana, 2015).
does not change sign in D and vanishes at most on a set of measure zero. Then the system: Does not have periodic orbits in D. We need to find a function h(x 1 , x 2 ), which satisfies the conditions of the theorem of Bendixson-Dulac, that is called a Dulac function. does not change sign and vanishes only on a measure zero set}.

Techniques to Construction of Dulac Functions
Theorem 2.2 If there exist c(x 1 , x 2 )∈Ω such that h is a solution of the system: with h∈Ω, then for Equation 1 h is a Dulac function on D. (Osuna and Villaseñor, 2011).

The Modified Liouville Equation
where, a, b and β are non zero and arbitrary coefficients.
Using the wave transformation u(x, t) = u(ξ) ξ = kx+wt with: and: can be reduced to: where, We obtain: . Integrating with respect to v: As v′(ξ) = dv/dξ we obtain: . and: , then: It follows that: If δc 1 = C and c 2 = B then the general solution of this differential equation is: where, C and B are constants, k 2 a 2 -w 2 ≠ 0. From Equation 4 and C = 2b then: From Equation 3 and u(x, t) = u(ξ) we obtain: Integrating and taking the constant of integration equal to 0:

Dynamical System
From Equation 5 and making a change of variables: . We obtain the following system: Supposing that then Equation 10 becomes: where, c(x 1 , x 2 )< 0 for b, δ>0 then some of the plane regions are:

Main Results
Theorem 4.1 The system of Equation 9 can be generalized as: Solving the previous differential equation by integrating factor, we have: is defined, then we have the generalized modified Liouville equation: If c 1 (x 1 ) = 0 and c 2 (x 2 ) = x 2 we have the modified Liouville equation we have an equivalent system: Proof. Replacing in Equation 2, we obtain: Solving the previous differential equation, we have:

Conclusion
Several solutions were obtained taking different values of the constant of integration. The corresponding system of the modified Liouville equation was generalized. Using travelling waves, the modified Liouville equation was transformed into a dynamical system and, with the use of Dulac's criterion, we gave sufficient conditions for the nonexistence of periodic orbits in four domains. By differentiable transformations other dynamical systems can be obtained first set of equations. Here, we can get a new generalization of this system. These results are important for the study of nonlinear partial differential equations. Very interesting future work is the generalization of the original partial differential equation to the modified Liouville equation in time and space. Also, we can consider a family of Dulac functions h = exp (ax 2 ) for different values of the parameter a. In this study, we worked with a = 1.