1.1. EXERCISES 3
arises in the study of surfaces of constant negative curvature [28], and in the study of crystal
dislocations [29].
12. Equilibrium equations
@
xx
@x
C
@
xy
@y
C F
x
D 0
@
xy
@x
C
@
yy
@y
C F
y
D 0
(1.12)
arise in elasticity. Here,
xx
;
xy
and
yy
are normal and shear stresses, and F
x
and F
y
are body
forces [30]. ese have been used by Cox, Hill, and amwattana [31] (see also [32]) to model
highly frictional granular materials.
13. e Navier–Stokes equations
r u D 0
u
t
C u ru D
rP
C r
2
u
(1.13)
describe the velocity field and pressure of incompressible fluids. Here is the kinematic
viscosity,
u
is the velocity of the fluid parcel,
P
is the pressure, and
is the fluid density [33].
1.1 EXERCISES
1.1. Show solutions exist for the nonlinear diffusion equation
u
t
D
.
u
m
u
x
/
x
; m 2 R (1.14)
of the form u D kt
p
x
q
for suitable constants k; p; and q. Use these to obtain solutions
to
u
t
D
.
uu
x
/
x
and u
t
D
u
x
u
x
: (1.15)
1.2. Show that Fisher’s equation
u
t
D u
xx
C u.1 u/ (1.16)
admit solutions of the form u D f .x ct / where f satisfies the ordinary differential
equation (ODE)
f
00
C cf
0
C f f
2
D 0: (1.17)
Further, show exact solutions can be obtained in the form
f
D
1
a
C
be
kz
2
(1.18)
4 1. NONLINEAR PDES ARE EVERYWHERE
for suitable constants a; b; c; and k [34].
1.3. Show that the Fitzhugh–Nagumo equation
u
t
D u
xx
C u.1 u/.u C / (1.19)
admits solutions of the form u D f .x ct / where f satisfies the ODE
f
00
C cf
0
C f .1 f /.f C / D 0: (1.20)
Further, show exact solutions can be obtained in the form
f D
1
a C be
kz
(1.21)
for suitable constants a; b; c; ; and k.
1.4. Show that solutions exist of the form
u D
ax
x
2
C bt
(a and b constant) that satisfies Burgers’ equation (1.3).
1.5. Consider the PDE
u
t
D u
xx
C 2 sech
2
x u: (1.22)
Even though linear, exact solutions to equations of the form
u
t
D u
xx
C f .x/u (1.23)
can be difficult to find. If v D e
k
2
t
sinh k t or v D e
k
2
t
cosh k t (k is an arbitrary con-
stant), then
u D v
x
tanh x v
satisfies (1.22).
1.6. Show
u D ln
ˇ
ˇ
ˇ
ˇ
2f
0
.x/g
0
.y/
.f .x/ C g.y//
2
ˇ
ˇ
ˇ
ˇ
;
where f .x/ and g.y/ are arbitrary functions satisfies Liouville’s equation
u
xy
D e
u
:
1.1. EXERCISES 5
1.7. Show
u D 4 tan
1
e
axCa
1
y
;
where a is an arbitrary nonzero constant satisfies the Sine–Gordon equation
u
xy
D sin u:
1.8. Show that if
u D f .x C ct/
satisfies the KdV equation (1.6) then f satisfies
cf
0
C 6ff
0
C f
000
D 0 (1.24)
where prime denotes differentiation with respect to the argument of f . Show there is
one value of c such that f .r/ D 2 sech
2
r is a solution of (1.24).
1.9. e PDE
v
t
6v
2
v
x
C v
xxx
D 0
is known as the modified Korteweg de Vries (mKdV) equation. Show that if v is a
solution of the mKdV, then
u D v
x
v
2
is a solution of the KdV (1.6).
1.10. e Boussinesq equation
u
t t
C u
xx
u
2
xx
1
3
u
xxxx
D 0 (1.25)
under the substitution u D 2
.
ln F .t; x/
/
xx
(and integrating twice) becomes ([35])
FF
tt
F
2
t
C FF
xx
F
2
x
1
3
FF
xxxx
4F
x
F
xxx
C 3F
2
xx
D 0: (1.26)
Further, show that (1.26) admits solutions of the form F D ax
2
C bt
2
C c for suitable
a; b; and c (see [36]).
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