Department of Mathematics, Bethune-Cookman University, USA
Received date: June 01, 2016; Accepted date: July 20, 2016; Published date: July 27, 2016
Citation: Tarver T (2016) Differential Equations: Determining Partial Integrability of Wave Theory. J Phys Math 7:186. doi:10.4172/2090-0902.1000186
Copyright: © 2016 Tarver T. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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Substantial research shows a number of methods to integrate ordinary and partial differential equations. However, methodology towards creating a theory for the Navier-Stokes equations is not obvious. One way to possibly go about creating a theory is gathering all empirical data relating to integrability of differential equations. There exist integrability methods for linear and non-linear differential equations. In the case of Navier-Stokes, we begin to look at non-linear integrability methods for creating a possible mathematical theory towards solutions.
One method that has been proven useful in differential systems is called Darboux’s equation for integrable systems. A common example of a Darboux integrable equation is one with an exponential component such as,
This method could be applied towards the Navier-Stokes equations from a classical geometric theoretical perception of differential equations developed by a number of mathematicians including Darboux . The reason for using this method is to find general algorithmic approaches towards the derivation of possible solutions .
Another concept to study exists within the integrability of Dispersive Wave Equations. There is a method within the integrable wave equations called the Korteweg-de Vries method or KdV. There is a general equation for possible integration to yield,
where [1,2]. This method of integrability is mentioned because it may have an implication of a correlation to the Navier- Stokes equations. The Korteweg-de Vries method of integrability may have a specific correlation to the next method I will discuss.
There is method for non-linear partial differential equations titled the Adomian of summation. Summation methodology is another way to integrate an equation depending on the necessity for it . There could exist a two-dimensional solution for vector field in the Navier- Stokes equation. This solution may be calculated by taking the form of
with a nonlinear term by
Where Ai are Adomian polynomials generated for all general forms of nonlinearity . Does the summation of the vector imply to exist?
We now go back to the concept of Korteweg-de Vries equation and integrability analysis of it’s inverse. There is a kdV nonlinear dynamical system to yield,
Where is an evolution parameter and The vector field being a proper subset is an implication towards the given divergent-free vector field on in the existence of Navier- Stokes equations. The vector field is also said to be an element of an infinite-dimensional periodic functional manifold, hence the M. Using this method, the element can be integrated such that
Lastly, I want to discuss a specific concept called Kink Waves. These waves are formally defined as traveling waves which rise or descend from one asymptotic form to another . It is said that one of these solutions approaches or converges to a constant. In the concept of dissipative waves, there is an equation by Burger to show,
Where v is the given viscosity. Since we are dealing with a viscosity and traveling waves, the Burger’s equation may have a direct correlation as to how the Navier-Stokes equations were developed. The Burger’s equation may also imply future solutions using integrability methodologies.