Painlevé Test

IMPORTANT MESSAGE:

It can happen that our software for the PainleveTestV2 fails to  find all possible dominant behaviors. In particular, this can happen when one or more of the leading order exponents (alphas)  remain undetermined.  In such cases
our software has no straightforward way of finding (or trying) values for the undetermined alphas.  

WHEN IN DOUBT, the software should be run with DominantBehaviorMin ->integer1  and/or  DominantBehaviorMax -> integer2, where  integer1 and integer2 can be specified  by the user. For example, DominantBehaviorMin -> -3 and/or DominantBehaviorMax -> 4 
were choosen, the the code will consider the following choices for the free alphas: -3,-2,-1,0,1,2,3, and 4.

Default values:  DominantBehaviorMin is set to -3 and  DominantBehaviorMax is set to -1 by default. So, if not values are selected, the code will run the test for the default values for the alphas starting  from the default DominantBehaviorMin to the default DominantBehaviorMax value. Thus, for each undetermined alpha, the code will consider the values -3,-2 and -1. 

Selecting appropriate values for DominantBehaviorMax is important when one or more of the alphas are positive and the 
other alphas remain undetermined. Examples are given at the bottom of this notebook.

Why did we not design more sophisticated code that would automatically compute bounds for the undetermined values?  This is largely due to the lack of reliable Mathematica functions that handle inequalities.

To load the package, you must set the directory to where the file is located, say c:\myDirectory, and load the package as follows:

(* SetDirectory["c:\\myDirectory"]; *)
(* For example, SetDirectory["d:\\demoPainlevetest"] *)

SetDirectory["c:\\data\\demoPainleveTest"]

Get["PainleveTestV4-2018.m"]

The companion paper by Douglas Baldwin and Willy Hereman, entitled ``Symbolic Software for the Painleve Test of Nonlinear Ordinary and Partial
Differential Equations" has been published in the Journal of Nonlinear Mathematical Physics, vol. 13, number 1, February 2006,
pages 90-110.  

A short description of how to use the package can be obtained by entering:

??PainleveTest

As a first example, we consider the equation due to Korteweg and de Vries (abbreviated as the KdV equation) :

PainleveTest[{D[u[x,t], t]+6*u[x,t]*D[u[x,t], x]+D[u[x,t], {x,3}]==0}, 
u[x,t], {x,t} , Verbose -> True ]

As a second example, run the Painleve test for KdV equation but this time using the Kruskal simplication:

PainleveTest[{D[u[x,t], t]+6*u[x,t]*D[u[x,t], x]+D[u[x,t], {x,3}]==0}, 
u[x,t], {x,t} , KruskalSimplification -> x, Verbose -> True ]

(* Equation of Abdul-Majid Wazwaz, sent on June 27, 2018 *)

WaveEquationWazwaz= D[D[u[x,t],{x,3}],t]+4*D[u[x,t],x]*D[D[u[x,t],x],t]+2*D[u[x,t],{x,2}]*D[u[x,t],t]+D[u[x,t],{x,2}]==0

PainleveTest[{WaveEquationWazwaz}, 
u[x,t], {x,t} , KruskalSimplification -> x,Verbose -> True ]