xi
Preface
is is an introductory book about obtaining exact solutions to nonlinear partial differential
equations (NLPDEs). is book is based, in part, on my lecture notes from a graduate course I
have taught at the University of Central Arkansas (UCA) for over a decade.
In Chapter 1, I list several NLPDEs (systems of NLPDEs) that appear in science and
engineering and provide a springboard into the subject matter. As our world is essentially non-
linear, it is not surprising that the equations that govern most physical phenomena are nonlinear.
As Nobel Laureate Werner Heisenberg once said, “e progress of physics will to a large extent
depend on the progress of nonlinear mathematics, of methods to solve nonlinear equations.”
In Chapter 2, I introduce compatibility. I start with the method of characteristics, a topic
from a standard course in PDEs. I then move to Charpit’s method, a method that seeks com-
patibility between two first-order PDEs. From there, I consider the compatibility between first-
and second-order PDEs, particularly, a nonlinear diffusion equation, Burgers’ equation, and a
linear diffusion equation with a nonlinear source term. I then extend the method to the nonlinear
diffusion equations in .2 C 1/ dimensions and a system of nonlinear PDEs where we consider
the Cubic Schrödinger equation.
In Chapter 3, I introduce the idea of differential substitutions. e classic example is the
HopfCole transformation and Burgers’ equation; it is the example I use to introduce the topic.
I extend the results to generalized Burgers’ and KdV equations. I consider Matrix HopfCole
transformation and use this when I introduce Darboux transformations for linear diffusion and
wave equations.
In Chapter 4, I introduce point and contact transformations. ese are a generalization of
the usual change of variables one would find in an introductory course in PDEs. I introduce three
special transformations: the Hodograph transformation, the Legendre transformation, and the
Ampere transformation. I also introduce conditions, the contact conditions, which guarantee
that a transformation will be a contact transformation. With several examples considered, the
chapter ends with the famous Plateau problem from minimal surfaces.
In Chapter 5, we consider first integrals. Simply put, these are PDEs of a lower order (if
they exist) and yield exact solutions of a given equation. I first consider second-order quasilinear
PDEs in two independent variables and then general Monge-Ampere (MA) equations in two
independent variables. Oftentimes, PDEs admit very general classes of first integrals. When
this happens, it is sometimes possible to transform the given PDE to one that is very simple. I
then consider three classes of MA equations: a class of hyperbolic, parabolic, and elliptic MA
equations.
xii PREFACE
e final chapter, Chapter 6, is functional separability. is is a generalization of the usual
separation of variables that one usually finds in an introductory course when solving PDEs like
the heat equation, the wave equation, or Laplace’s equation. In Chapter 6, I turn the problem
into one of compatibility. I consider nonlinear diffusion equations, nonlinear dispersion equa-
tions in .1 C 1/ dimension, and a three-dimensional linear diffusion equation with a source
term.
It is fair to say that this book does not cover all techniques used to solve nonlinear PDEs
but rather represents techniques that I personally have interest in and have successfully used.
For example, techniques like the inverse scattering transform and Bäcklund transformations are
extremely important and should be added to one’s arsenal. It is also fair to say this is not the only
book on techniques for solving PDEs. For example, one should also consider A.R. Forsyth’s A
Treatise on Differential Equations, W. F. Ames’ Nonlinear Partial Differential Equations in En-
gineering, Vol. I and II, and L. Debnath’s Nonlinear Partial Differential Equations for Scientists
and Engineers.
Daniel J. Arrigo
April 2019
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