Higher Order Nonlinear PDE’s
1. Travelling Wave Solutions
Consider the equation
/ t uÝx, tÞ + c / x uÝx, tÞ = 0,
If
then
x 5 R, t > 0.
Ý1.1Þ
uÝx, tÞ = fÝx ? atÞ
? a f v ÝzÞ + c f v ÝzÞ = 0,
Ý1.2Þ
z = x ? at
and this provides a solution if a = c. Similarly, the equation
/ tt uÝx, tÞ ? c 2 / xx uÝx, tÞ = 0,
x 5 R, t > 0,
Ý1.3Þ
has a solution of the form (1.2) for a = ± c. These are so called travelling wave solutions
since the wave form fÝzÞ propagates with speed equal to a. Since the wave form
propagates along the entire real line, it is natural to suppose that fÝzÞ is bounded for all z.
TWS’s for Linear Equations
Consider
/ t uÝx, tÞ + c / x uÝx, tÞ ? K / xx uÝx, tÞ = 0,
x 5 R, t > 0.
Ý1.4Þ
This equation, which is (1.1) with the addition of a diffusion term, has a solution of the form
(1.2) if f satisfies
?a f v ÝzÞ + c f v ÝzÞ ? K f ”ÝzÞ = 0.
For a = c this reduces to f ”ÝzÞ = 0, which leads to fÝzÞ = Az + B. Then
uÝx, tÞ = AÝx ? ctÞ + B, which is a pure convection solution with no diffusion. For a ® c, the
solution is
J = a?c.
fÝzÞ = C 0 e J z ,
K
This solution does not remain bounded for both z ¸ K and z ¸ ? K unless C 0 = 0. We
conclude that this diffusion equation has no interesting TW solutions.
Similarly,
/ t uÝx, tÞ + c / x uÝx, tÞ ? K / xxx uÝx, tÞ = 0,
x 5 R, t > 0.
Ý1.5Þ
which is (1.1) with the addition of a dispersive term, has a solution of the form (1.2) if f
satisfies
?a f v ÝzÞ + c f v ÝzÞ ? K f vvv ÝzÞ = 0.
1
For a = c this reduces to f vvv ÝzÞ = 0, which leads to fÝzÞ = Az 2 + Bz + C, which is again a
pure convection solution with no dispersion. For a ® c, the solution is
for
J2 = a ? c .
fÝzÞ = C 0 sinh Jz + C 1 cosh Jz,
K
This solution does not remain bounded for both z ¸ K and z ¸ ? K unless C 0 = C 1 = 0, and
we conclude that the dispersion equation has no interesting TW solutions.
In general, only hyperbolic linear PDE’s will have interesting TW solutions. However, for
nonlinear equations the effects of the nonlinear term and the higher order terms interact to
produce interesting results. We will begin by constructing simple solutions for several
examples of quasilinear equations of order higher than one.
Burger’s Equation (with Viscosity)
Consider the equation
/ t uÝx, tÞ + u / x uÝx, tÞ ? K / xx uÝx, tÞ = 0
produces shocks
Ý1.6Þ
smooths out the solution
We recall from previous results that the lower order part of the equation can lead to shock
type solutions while the higher order term has the effect of smoothing out the initial data to
produce a very smooth solution. We will now examine the interaction of these two
competing effects.
Assume a solution of the form (1.2). Then f must satisfy
i.e.,
Then
?a f v ÝzÞ + fÝzÞ f v ÝzÞ ? K f ”ÝzÞ = 0,
K f ”ÝzÞ = d
dz
K f v ÝzÞ =
or
1
2
1
2
fÝzÞ 2 ? a fÝzÞ .
fÝzÞ 2 ? a fÝzÞ + C 0 ,
f vÝzÞ = 1 ßf 2 ? 2af + C 1 à = 1 Ýf ? f 1 ÞÝf ? f 2 Þ.
2K
2K
Here
f 1 + f 2 = 2a
and f 1 f 2 = C 1
so
f 1,2 = a ±
Note that
a2 ? C1 ,
f v ÝzÞ = 0
f v ÝzÞ < 0
f v ÝzÞ > 0
0 < C1 < a2.
if f = f 1 or f = f 2
if f 1 < f < f 2 ,
if f < f 1 or f > f 2 .
This leads to the following scenario for f versus z,
2
Figure 1
It is evident that the equation
f vÝzÞ = 1 Ýf ? f 1 ÞÝf ? f 2 Þ
2K
has an unstable critical point at f = f 2 and a stable critical point at f = f 1 . If we look at a
phase plane portrait for this equation
Figure 2
where f v is plotted against f , we see that the portion of the graph in Figure 1 that is
contained between the horizontal lines f = f 1 and f = f 2 corresponds to the heteroclinic orbit
shown in Figure 2, joining f 2 to f 1 .
In this example, an explicit analytical solution for fÝzÞ is possible. The equation implies
X
df
= 1 X dz
2K
Ýf ? f 1 ÞÝf ? f 2 Þ
leading to
?kz
fÝzÞ = f 1 + f 2 e?kz
1+e
Clearly
for
k = f 2 ? f 1 > 0.
2K
Ý1.7Þ
fÝzÞ ¸ f 1 as z ¸ + K
fÝzÞ ¸ f 2 as z ¸ ? K
fÝ0Þ = 12 Ýf 1 + f 2 Þ.
3
If we denote uÝx, t; KÞ = fÝx ? atÞ for f given by (1.7), with a = 12 Ýf 1 + f 2 Þ = fÝ0Þ, then as
K ¸ 0, uÝx, t; KÞ tends to the shock solution of the Riemann problem
/ t uÝx, tÞ + u / x uÝx, tÞ = 0,
uÝx, 0Þ =
f 2 if x < 0
f 1 if x > 0
The following figure illustrates how uÝx, t; KÞ approaches the shock solution as K decreases.
Evidently for large K the diffusion term overpowers the nonlinear term and the solution is
smooth but as K decreases, the nonlinear term begins to take over, inducing a steepening
front which becomes a shock discontinuity when K reaches zero.
Beta = 5, .5, .05
Note that the speed of the travelling wave, a = 12 Ýf 1 + f 2 Þ = fÝ0Þ, is the propagation speed
associated with the shock solution of the Riemann problem. It is an interesting difference
between the linear and nonlinear problems that the wave speed for the linear equations
(1.1) and (1.3) is determined by a coefficient in the equation while the wave speed for the
nonlinear equation (1.6) is determined by the initial state.
We have seen previously for the more general version of equation (1.6),
/ t uÝx, tÞ + / x FÝuÝx, tÞÞ ? K / xx uÝx, tÞ = 0
Ý1.8Þ
where F ”ÝuÞ > 0, that the PDE has a solution of the form (1.2) if f satisfies
f v ÝzÞ = 1 ßFÝfÝzÞÞ ? afÝzÞ + C 0 à.
K
For a TWS we must require that f v ÝzÞ ¸ 0, as | z| ¸ K and hence
a=
FÝfÝ+KÞÞ ? FÝfÝ?KÞÞ
FÝf 2 Þ ? FÝf 1 Þ
=
.
f2 ? f1
fÝ+KÞ ? fÝ?KÞ
We recognize the speed a of the travelling wave as the shock speed associated with the
shock solution of the Riemann problem for the equation,
/ t uÝx, tÞ + / x FÝuÝx, tÞÞ = 0.
Here, as for (1.6), the wave speed is determined by the initial state.
The assumption, F ”ÝuÞ > 0, implies that there are two values z = f 1 , f 2 where
f v ÝzÞ = 0. This is evident in the following figure where we have plotted
4
G 1 ÝfÞ = FÝfÝzÞÞ + C 0 and G 2 ÝfÞ = afÝzÞ versus f on the same axes,
It is evident that f v ÝzÞ = 1 ßG 1 ÝfÝzÞÞ ? G 2 ÝfÝzÞÞà < 0 for f 1 < f < f 2 and f v ÝzÞ > 0 otherwise.
K
This implies the existence of a stable critical point at f = f 1 and an unstable critical point at
f = f 2 and the TWS for the PDE corresponds to the heteroclinic orbit joining the unstable
point to the stable critical point.
The KdV Equation
If we replace the diffusive term in (1.6) by a dispersive term, we obtain
/ t uÝx, tÞ + u / x uÝx, tÞ + K / xxx uÝx, tÞ = 0
Ý1.9Þ
If we assume a solution of the form (1.2) then
i.e.,
and
?a f v ÝzÞ + fÝzÞ f v ÝzÞ + K f vvv ÝzÞ = 0
d ?a fÝzÞ +
dz
?a fÝzÞ +
1
2
1
2
fÝzÞ 2 + K f ”ÝzÞ
= 0,
fÝzÞ 2 + K f ”ÝzÞ = C 0 .
Now we impose on f the conditions
fÝzÞ, f v ÝzÞ, and f ”ÝzÞ all tend to 0 as | z| ¸ K.
Under these conditions a solution of the form (1.2) is called a solitary wave. Then
?a fÝzÞ +
1
2
fÝzÞ 2 + K f ”ÝzÞ = 0,
Ý1.10Þ
and
?a fÝzÞ f v ÝzÞ +
1
2
fÝzÞ 2 f v ÝzÞ + K f ”ÝzÞ f v ÝzÞ = 0.
5
i.e.,
d ?
dz
?
Then
1
2
1
2
a fÝzÞ 2 +
a fÝzÞ 2 +
1
6
1
6
fÝzÞ 3 +
fÝzÞ 3 +
1
2
1
2
K f v ÝzÞ 2
= 0.
K f v ÝzÞ 2 = C 1 = 0
where the solitary wave assumptions have been applied once more to conclude C 1 = 0.
Then
3K f v ÝzÞ 2 = 3a fÝzÞ 2 ? fÝzÞ 3 ,
and
f v ÝzÞ =
X
Then
leads to
1 fÝzÞ 3a ? fÝzÞ .
3K
df
=
f 3a ? f
1 X dz
3K
fÝzÞ = 3a Sech 2
a Ýz ? z 0 Þ
4K
and
uÝx, tÞ = fÝx ? atÞ = 3a Sech 2
a Ýx ? at ? z 0 Þ .
4K
One of the most novel features of this solution is that the wave speed, a, is proportional to
the wave amplitude which equals 3a.
u(x,0) for a=4
If we let u = f and v = f in the equation (1.10),
v
?a fÝzÞ +
1
2
fÝzÞ 2 + K f ”ÝzÞ = 0,
then
uv = v
and
v v = 1 uÝ2a ? uÞ.
2K
This 2-dimensional dynamical system has critical points at Ý0, 0Þ and Ý2a, 0Þ. The Jacobian
of the system is
6
JÝu, vÞ =
0
a?u
K
1
.
0
a and JÝ2a, 0Þ has eigenvalues V = ± i a , so that
K
K
Ý0, 0Þ is a saddle point and Ý2a, 0Þ is a center. This leads to the following phase plane
portrait
Then JÝ0, 0Þ has eigenvalues V = ±
The closed orbits represent periodic solutions to the first order dynamical system. The
homoclinic orbit that begins and ends at the origin represents the solitary wave solution to
the equation (1.10). The curves outside the homoclinic orbit are associated with unbounded
solutions to the equation.
Now suppose we release the solitary wave assumptions and try to find other types of
travelling wave solutions for the KdV equation. Returning to the equation
? a fÝzÞ +
1
2
fÝzÞ 2 + K f ”ÝzÞ = C 0
we do not suppose C 0 = 0, and we write
C 0 f v ÝzÞ = ?a fÝzÞ f v ÝzÞ +
1
2
fÝzÞ 2 f v ÝzÞ + K f ”ÝzÞ f v ÝzÞ
K
= d ? a fÝzÞ 2 + 1 fÝzÞ 3 + f v ÝzÞ 2 .
2
6
2
dz
Then
or
3K f v ÝzÞ 2 = ?fÝzÞ 3 + 3a fÝzÞ 2 + C 0 fÝzÞ + C 1 ,
7
f v ÝzÞ = 1 Ýf ? f 1 ÞÝf ? f 2 ÞÝf 3 ? fÞ = ± FÝfÞ .
3K
Now there are several cases to consider:
(a) F(f) has distinct, real roots f 1 < f 2 < f 3
(b) F(f) has one real root f 3 5 R, f 1,2 = J ± i b
(c) F has a double real root
f1 = f2 < f3
(d) F has a triple real root f 1 = f 2 = f 3 .
Now, real solutions for f v ÝzÞ = ± FÝfÞ exist only when FÝfÞ ³ 0. In each of the 4 cases then
this leads to real solutions for the following ranges of f values.
(a) The darkened intervals indicate that solutions exist for f < f 1 and f 2 < f < f 3 :
(b) Solution exists for f < f 3 :
(c) Solution exists for f < f 1 or f < f 3
8
(d) Solution exists for f < f 1
Consider, for example, case (b). At some point z 0 where fÝz 0 Þ < f 3 , either we have
f v Ýz 0 Þ < 0, in which case fÝzÞ continues to decrease for all z > z 0 so that fÝzÞ must tend to
minus infinity as z ¸ +K, or else we have f v Ýz 0 Þ > 0, in which case fÝzÞ increases until it
reaches f 3 and then it stops because f v Ýz 3 Þ = 0. In this case it must be that fÝzÞ must tend
to minus infinity as z ¸ ?K. Then it follows that in case (b), the solutions are either constant
or else they are unbounded. The same result occurs in case (d), in case (c) when f ² f 1 ,
and in the part of the solution interval in case (a) when f ² f 1 . On the other hand, the part of
case (c) where f 1 = f 2 < f < f 3 corresponds to the solitary wave solution, since f satisfies
f v ÝzÞ =
1 fÝzÞ f 3 ? fÝzÞ , where we chose f 1 = f 2 = 0.
3K
There remains the part of case (a) where f 2 < f < f 3 , and we have 3K f v ÝzÞ 2 = FÝfÝzÞÞ, or
differentiating,
6K f v ÝzÞ f”ÝzÞ = F v ÝfÝzÞÞ f v ÝzÞ.
Then
f”ÝzÞ = 1 F v ÝfÝzÞÞ
6K
and we see from the sketch that at a point
z 1 where fÝz 1 Þ = f 2 , we have F v Ýf 2 Þ > 0 so that f ”Ýz 1 Þ > 0 and f 2 is a min for fÝzÞ
9
z 2 where fÝz 2 Þ = f 3 , we have F v Ýf 3 Þ < 0 so that f ”Ýz 1 Þ < 0 and f 3 is a max for fÝzÞ.
Then the solution fÝzÞ must oscillate between the values f 2 and f 3 .
This periodic solution has been seen to exist in a qualitative sense. In order to discover
quantitative information about it, we must integrate the differential equation for fÝzÞ which
requires use of elliptic functions. The solutions in this case are referred to as cnoidal waves.
The Sine Gordon Equation
Consider a solution of the form (1.2) for the equation
/ tt uÝx, tÞ ? / xx uÝx, tÞ = sin uÝx, tÞ
Ý1.11Þ
a 2 f ”ÝzÞ ? f ”ÝzÞ = sinÝfÝzÞÞ,
Then
or,
f ”ÝzÞ = A sinÝfÝzÞÞ
where
A=
1 .
a ?1
2
We can rewrite this as
or
and
f ”ÝzÞ f v ÝzÞ = A sinÝfÝzÞÞ f v ÝzÞ
d
dz
1
2
f v ÝzÞ 2
= ?A d cosÝfÝzÞÞ
dz
f v ÝzÞ 2 = C 1 ? 2A cosÝfÝzÞÞ,
f v ÝzÞ =
C 1 ? 2A cosÝfÝzÞÞ .
Integration of this last equation is possible but leads to elliptic functions. Instead, a
qualitative examination of the solution can be carried out by letting
10
f v ÝzÞ = gÝzÞ,
g v ÝzÞ =
sinÝfÝzÞÞ
.
a2 ? 1
Then this dynamical system (whose equations are formally the same as the pendulum
equations) has singular points at Ýn^, 0Þ and it is easy to show that when a 2 > 1 the even
integer multiples of ^ are saddle points while the odd integer multiples of ^ are centers.
When a 2 < 1 the odd integer multiples of ^ are saddle points and the even integer multiples
of ^ are centers. This leads to the usual phase plane picture associated with the pendulum.
The figure below shows the phase plane portrait for a = .1.
Evidently, the heteroclinic orbit joining the saddle at Ý?^, 0Þ to the saddle at Ý^, 0Þ
corresponds to a travelling wave solution for the Sine-Gordon equation where the shape of
the wave varies slightly as the wave speed a varies, TWS’s exist for every wave speed
a 2 < 1. The closed orbits inside the separatrix correspond to periodic solutions. When
a 2 > 1, there are saddle points at 0 and 2^ and we get TWS’s for every wave speed
a 2 > 1 as well, but connecting u = 0 to u = 2^ instead of connecting u = ?^ to u = ^.
The Nonlinear Schrodinger Equation
Consider the equation
i / t uÝx, tÞ + / xx uÝx, tÞ + u |u| 2 = 0,
Ý1.12Þ
where u is complex valued. This equation arises in applications where it is expressed
relative to a frame of reference that is moving with the group velocity of a travelling wave.
Therefore the TW solution of this equation appears as a standing wave. That is, we look for
a solution of the form,
11
uÝx, tÞ = fÝxÞ e iK t .
Then
or
?K fÝxÞ + f ”ÝxÞ + fÝxÞ 3 = 0,
f ”ÝxÞ = fÝxÞ K ? fÝxÞ 2 .
Since explicit integration leads to elliptic functions, instead we let
f v ÝxÞ = gÝxÞ,
and
g v ÝxÞ = fÝxÞ K ? fÝxÞ 2 .
This system has critical points at Ý0, 0Þ and ± K , 0 . It is straightforward to show that the
origin is a saddle point while the other two critical points are centers. The closed curves in
the phase plane portrait are periodic solutions while the separatrix is a homoclinic orbit
corresponding to a solitary wave that begins with f = f v = 0 at x = ?K, it increases to a point
of maximum intensity (where f v = 0, f > 0) and then decreases to f = f v = 0 at x = +K.
A Diffusion-Dispersion Example
Consider the example
/ t uÝx, tÞ + u / x uÝx, tÞ ? J / xx uÝx, tÞ + K / xxx uÝx, tÞ = 0
where J > 0. If we suppose uÝx, tÞ = vÝx ? ctÞ then
?cv v ÝzÞ + vÝzÞv v ÝzÞ ? J v”ÝzÞ + K v vvv ÝzÞ = 0.
12
Integrating once leads to
C 0 ? c vÝzÞ +
1
2
vÝzÞ 2 ? J v v ÝzÞ + K v”ÝzÞ = 0.
If we suppose
v v ÝzÞ and v”ÝzÞ ¸ 0 as | z| ¸ K then vÝzÞ tends to a constant value as z
tends to infinity. However, vÝzÞ need not tend to the same constant value in both directions.
Suppose, then that
vÝzÞ ¸ 0 as z ¸ K,
and vÝzÞ ¸ V 0
as z ¸ ?K.
Then
C 0 ? c vÝKÞ +
C 0 ? c vÝ?KÞ +
1
2
vÝKÞ 2 ? J v v ÝKÞ + K v”ÝKÞ = C 0 = 0
1
2
vÝ?KÞ 2 ? J v v Ý?KÞ + K v”Ý?KÞ = C 0 ? c V 0 +
1
2
V 20 = 0.
Evidently, these conditions at infinity can hold only if C 0 = 0 and
c = 12 V 0 ; that is, the
speed of the travelling wave solution is determined by the ”initial value” V 0 . In this case the
wave form of the TWS, vÝzÞ satisfies,
K v”ÝzÞ = c vÝzÞ ?
1
2
vÝzÞ 2 + J v v ÝzÞ.
Treating this as a dynamical system by letting UÝzÞ = vÝzÞ, VÝzÞ = v v ÝzÞ leads to
d
dt
U
V
=
V
1U c? 1U + JV
K
K
2
This system has a saddle point at Ý2c, 0Þ and a stable node or stable focus at Ý0, 0Þ
depending on whether J 2 ? 4cK is positive or negative respectively. In either case there is a
heteroclinic orbit joining the saddle point to the origin. In the case that the origin is a stable
node, the travelling wave form decreases monotonically to zero. In the case that the origin
is a stable focus, the function UÝzÞ assumes both positive and negative values as z tends to
plus infinity. We can give a physical motivation for this behavior as follows.
If we write
K v”ÝzÞ = c vÝzÞ ?
1
2
vÝzÞ 2 + J v v ÝzÞ = ? d
dz
1
6
v3 ?
1
2
cv 2 + Jv v ÝzÞ
then we can interpret this as the equation for the motion of a nonlinear spring with potential
function
PÝvÞ =
1
6
v3 ?
1
2
cv 2
and viscous damping equal to Jv v ÝzÞ. Since J > 0 the parameter z must be interpreted as
negative time. If the spring state is given by v = v v = 0 at t = ?K Ýi.e., z = +KÞ then the
particle will tend to the state v = 2c, v v = v” = 0 as t ¸ K. For J small relative to K the
spring will execute oscillations on the way to the steady state, whereas if J is large relative
to K, then the spring does not oscillate as it travels to the steady state. In the context of the
nonlinear spring, these two scenarios are called the underdamped and overdamped cases,
respectively. The following figures illustrate
13
The path to steady state
A Reaction-Diffusion Example
We will consider now an example in which a travelling wave occurs for precisely one value
of the wave speed. The equation
/ t uÝx, tÞ ? / xx uÝx, tÞ = fÝuÝx, tÞÞ
Ý1.13Þ
where
14
fÝ0Þ = fÝaÞ = fÝ1Þ = 0
fÝxÞ < 0, 0 < x < a,
fÝxÞ > 0, a < x < 1,
v
v
f vÝ0Þ < 0, f ÝaÞ > 0, f Ý1Þ < 0
i.e.,
1
X 0 fÝsÞds > 0,
is called a reaction diffusion equation. We will consider these equations in more detail
later. For now, we suppose uÝx, tÞ = vÝx ? ctÞ,
where vÝzÞ ¸ 0 as z ¸ ?K,
vÝzÞ ¸ 1 as z ¸ +K,
v v ÝzÞ ¸ 0 as |z| ¸ K.
v”ÝzÞ = ?cv v ÝzÞ ? fÝvÞ,
Then
or,
v v ÝzÞ = wÝzÞ
w v ÝzÞ = ?cwÝzÞ ? fÝvÝzÞÞ.
Ý1.14Þ
The existence of a travelling wave solution for (1.13) will now follow from a careful
examination of the dynamical system (1.14). Under the assumptions on f(u), there are
critical points at Ý0, 0Þ, Ý0, aÞ and Ý0, 1Þ and it is not hard to determine that the eigenvalues
of the Jacobian of the system are given by
at Ý0, 0Þ
V ±0 ÝcÞ = ? 12 c ±
at Ý0, aÞ
V ±a ÝcÞ = ? 12 c ±
at Ý0, 1Þ
V ±1 ÝcÞ
1
2
= ? c±
1
2
1
2
1
2
c 2 ? 4f v Ý0Þ ,
a saddle point,
c 2 ? 4f v ÝaÞ ,
stable node or stable focus,
v
c ? 4f Ý1Þ ,
2
a saddle point.
There is a trajectory w = w 0 Ýv; cÞ which leaves (0,0) tangent to the vector V +0 ÝcÞ and there is
also a trajectory w = w 1 Ýv; cÞ that approaches the critical point (1,0) along the vector V ?1 ÝcÞ.
Our aim is to find a value for c such that
w 0 Ýv; cÞ = w 1 Ýv; cÞ for all Ýv, wÞ such that 0 < v < 1, w > 0.
15
Define
EÝv, wÞ =
Then
d
dz
1
2
v
w 2 + X fÝzÞ dz.
0
EÝv, wÞ = wÝzÞw v ÝzÞ + fÝvÞv v ÝzÞ
= wÝ?cw ? fÝvÞÞ + fÝvÞ w = ?cw 2 .
Then, for c < 0, EÝvÝzÞ, wÝzÞÞ is a nondecreasing function as we move along trajectories of
the system in the direction of increasing z. The level sets of EÝv, wÞ, which are orbits
corresponding to c = 0, are shown in the following figure,
Note particularly that EÝv, w 0 Ýv, 0ÞÞ = 0 ; i.e., E = 0 along w = w 0 Ýv, 0Þ
and
1
EÝv, w 1 Ýv, 0ÞÞ = X fÝzÞ dz > 0; i.e., E > 0 along w = w 1 Ýv, 0Þ.
0
Now for some P > 0, draw a vertical line L at v = a + P, and draw w = w 0 Ýv, 0Þ from (0,0)
to L. Let the region bounded by this curve on the bottom and by L and the w-axis on the
right and left be denoted by R.
For c < 0, w = w 0 Ýv, cÞ cannot leave the region R through the left side, the bottom or the top
of R; i.e.,
16
i) along the w-axis v v ÝzÞ = wÝzÞ > 0 so trajectories enter R
from left to right through v = 0.
ii) trajectories cannot exit through the bottom since
v
d w 0 Ýv, cÞ = w ÝzÞ = ?c ? fÝvÞ > ? fÝvÞ = d w 0 Ýv, 0Þ
v
w
w
dv
dv
v ÝzÞ
iii) w 0 Ýv, cÞ must meet L at some finite w > 0, since
d 2 w 0 Ýv, cÞ < 0
for 0 < v < a + P
dv 2
Then for each c < 0, w 0 Ýv, cÞ meets L at some point Ýa + P, w 0 Ýa + P, cÞÞ. Similarly, w 1 Ýv, cÞ
meets L at some point Ýa + P, w 1 Ýa + P, cÞÞ. From looking at the level curves of EÝv, wÞ we
see that w 0 Ýa + P, 0Þ < w 1 Ýa + P, 0Þ. Moreover, for c sufficiently negative,
w 0 Ýa + P, cÞ > w 1 Ýa + P, cÞ. To see this, note that at each Ýv, wÞ 5 0 < v < a + P, w > 0 we
have
v
d w 0 Ýv, cÞ = w ÝzÞ = ?c ? fÝvÞ .
w
dv
v v ÝzÞ
In particular, for a fixed K > 0, and w > Kv
d w Ýv, cÞ > ?c ? fÝvÞ .
Kv
dv 0
Now
hence
fÝvÞ
v
² M for 0 < v < a + P
d w Ýv, cÞ > ?c ? M > K
K
dv 0
for c < 0, sufficiently negative.
Then for c sufficiently negative, (depending on K), w 0 Ýa + P, cÞ > KÝa + PÞ.
Also
1
EÝv, w 1 Ýv, 0ÞÞ = 0 + X fÝzÞ dz > 0,
0
and since E is constant on this orbit (i.e., c = 0 ) we have
EÝa + P, w 1 Ýa + P, 0ÞÞ =
Then
w 1 Ýa + P, 0Þ 2 = 2 X
or
w 1 Ýa + P, 0Þ =
1
w 1 Ýa + P, 0Þ 2 + X
a+P
0
1
fÝzÞ dz = X fÝzÞ dz.
0
1
a+P
2X
1
2
fÝzÞ dz > X fÝzÞ dz > 0,
1
a+P
0
fÝzÞ dz .
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Now w 1 Ý1, cÞ = 0 for c < 0 and E is increasing along orbits (in the direction of increasing z)
hence
1
EÝa + P, w 1 Ýa + P, cÞÞ < EÝ1, w 1 Ý1, cÞÞ = EÝ1, 0Þ = X fÝzÞ dz.
0
That is,
and
1
2
w 1 Ýa + P, cÞ 2 + X
w 1 Ýa + P, cÞ <
a+P
0
2X
1
fÝzÞ dz < X fÝzÞ dz,
1
a+P
0
fÝzÞ dz = w 1 Ýa + P, 0Þ.
Now we have shown that
w 0 Ýa + P, 0Þ < w 1 Ýa + P, 0Þ,
w 0 Ýa + P, cÞ > KÝa + PÞ, for c < 0 sufficiently negative
w 1 Ýa + P, cÞ < w 1 Ýa + P, 0Þ for all c < 0,
and the orbits w 0 Ýv, cÞ, w 1 Ýv, cÞ vary smoothly with c. Then it follows that for some ĉ < 0
and K > 0 sufficiently large,
w 0 Ýa + P, ĉÞ = w 1 Ýa + P, ĉÞ.
For precisely this value of c = ĉ, there is a heteroclinic orbit joining Ý0, 0Þ to Ý1, 0Þ and there
is a corresponding TWS to Ý1.13Þ having wave speed ĉ.
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