1
C H A P T E R 1
Nonlinear PDEs are
Everywhere
Outside of Quantum mechanics, the world around us is modeled by nonlinear partial differen-
tial equations (NLPDEs). Here is just a short list of places that one may find NLPDEs.
1. e nonlinear diffusion equation
u
t
D
.
D.u/u
x
/
x
(1.1)
is a NLPDE that models heat transfer in a medium where the thermal conductivity may depend
on the temperature. e equation also arises in numerous other fields such as soil physics,
population genetics, fluid dynamics, neurology, combustion theory, and reaction chemistry, to
name just a few (see [1] and the references within).
2. e nonlinear wave equation
u
tt
D
c.u/
2
u
x
x
(1.2)
essentially models wave propagation and appears in applications involving one-dimensional
gases, shallow water waves, longitudinal threadlines, finite nonlinear strings, elastic-plastic
materials, and transmission lines, to name a few (see [2] and the references within).
3. Burgers’ Equation
u
t
C uu
x
D u
xx
(1.3)
is a fundamental partial differential equation that incorporates both nonlinearity and diffusion.
It was first introduced as a simplified model for turbulence [3] and appears in various areas of
applied mathematics, such as soil-water flow [4], nonlinear acoustics [5], and traffic flow [6].
4. Fisher’s equation
u
t
D u
xx
C u.1 u/ (1.4)
is a model proposed for the wave of advance of advantageous genes [7] and also has appli-
cations in early farming [8], chemical wave propagation [9], nuclear reactors [10], chemical
kinetics [11], and theory of combustion [12].
2 1. NONLINEAR PDES ARE EVERYWHERE
5. e Fitzhugh–Nagumo equation
u
t
D u
xx
C u.1 u/.u C / (1.5)
models the transmission of nerve impulses [13], [14] and arises in population genetics mod-
els [15].
6. e Korteweg deVries equation (KdV)
u
t
C 6uu
x
C u
xxx
D 0 (1.6)
describes the evolution of long water waves down a canal of rectangular cross section. It has also
been shown to model longitudinal waves propagating in a one-dimensional lattice, ion-acoustic
waves in a cold plasma, waves in elastic rods, and used to describe the axial component of
velocity in a rotating fluid flow down a tube [16] .
7. e Boussinesq equation
u
t t
C u
xx
C
.
2uu
x
/
x
C
1
3
u
xxxx
D 0 (1.7)
was introduced by Boussinesq in 1871 [17], [18] to model shallow water waves on long channels.
It also arises in other applications such as in one-dimensional nonlinear lattice-waves [19], [20],
vibrations in a nonlinear string [21], and ion sound waves in a plasma [22], [23].
8. e Eikonial equation
jruj D F .x/; x 2 R
n
(1.8)
appears in ray optics [24].
9. e Gross–Pitaevskii equation
i
t
D r
2
C
V .x/ C j j
2
(1.9)
is a model for the single-particle wavefunction in a Bose–Einstein condensate [25], [26].
10. Plateau’s equation
1 C u
2
y
u
xx
2u
x
u
y
u
xy
C
1 C u
2
x
u
yy
D 0 (1.10)
arises in the study of minimal surfaces [27].
11. e Sine–Gordon equation
u
xy
D sin u (1.11)
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