Internat. J. Math. & Math. Sci.
Vol. I0 No. 4 (1987) 787-796
787
UNSTABLE PERIODIC WAVE SOLUTIONS OF
NERVE AXlON DIFFUSION EQUATIONS
RINA LING
Department of Mathematics and Computer Science
California State University,
Los Angeles, California 90032
(Received September 9, 1986)
BSTRAG’.
studied.
Unstable
periodic
solutions
of
systems
of
parabolic
equations
are
Special attention is given to the existence and stability of solutions.
KEY WORDS AD PHRASES.
Periodic wave solutions, Diffusion equations, existence and
stability analysis
1980 AS SUBJECT CLASSIFICATION ODE.
|.
IHTRDDUTION.
of partial differential equations are of great importance in
Diffusion systems
biosciences.
In this paper, unstable periodic solutions of systems of the form
u
are studied.
impulses
on
u
t
w
xx
+ F(u,w),
(I.I)
G(u,w),
t
Equations of this type arise in neurophysiology in the study of nerve
nerve axon,
see
[1,2].
involved in biology, see for example
2.
35k40, 35K57.
Other classes of diffusion equations are also
[3-9].
ZISTZNCE OF SOLUTIONS
It is known that for
u, if > 0 is sufficiently small, equation
G(u,w)
(I.I) has two types of wave solutions, namely, pulse travelling wave solutions and
A travelling wave solution is a solution of
periodic travelling wave solutions.
equation (I.I) of the form
[u(x,t), w(x,t)]
hence
[(z;c), (z;c)]
d2T
dz
2
[(z;c), (x;c)], z
x + ct,
satisfies the ordinary differential equation
c
__+d
F (,)
0,
dz
d
G(,)
c-+
0.
A pulse travelling wave solution is a non-constant solution of (2.1) satisfying
(2.1)
788
R. LING
[0,01,
[(z;c), (z;c)]
lim
and a periodic travelling wave solution is a periodic solution of (2.1).
In [I0], Evans showed that equation (1.1) has two pulse travelling solutions with
propagation speeds
different
and
c
c
On the existence of
2.
wave solutions, Hastings [II] showed that equation (2.1) with
constant
0
<
a
<I/2
solution if
is
sufficiently
[12]
eu
has a non-
small and the speed
studied the case in which
c
is
F(u,w)
u
for
u
u-I
for
a
<
<
a,
u,
Under this assumption, equation (2.1) has a non-constant periodic
smooth function of
the two cases when
and the period
c < c < c
2
They demonstrated the behavior of the function
limited in the range
p(c)
is a
c.
p(c)
under
is
c
0
only given by
u
F(u,w)
where
>
Rinzel and Keller
limited to a certain range.
is a function of
e
if
periodic solution
periodic travelling
G(u,w)
is not very small and when
a
is very small.
a
Dai
[13] proved
the existence and uniqueness of solutions for a general case and studied stability of
the solution.
3.
STABILITY ANALYSIS.
Stability of periodic travelling wave solutions is related to the eigenvalues of
a matrix in the following theorem.
Let
A(z;%,c)
be the matrix
0
0
A(z;I,c)
l-Fl
G
[#(z; c)
l[(z;c),
(z;c)]
(z;c)]
0
F.
and
(z:c)]
c
c
where
(z;c)]
-F2[#(z;c)
G2[(z;c),
c
denote the partial derivatives as usual, and let
G.
X (z;,c)
be a
matrix satisfying the differential equation
d
A X
d--X
with the initial condition
X (0;l,c)
I.
Suppose the functions
TaEOIM 3.1.
F
and
G
in equation (I.I)
staisfy (a)
the matrix X (p(c);/,c) has an eigenvalue of
0 and (c)
0, (b) G(0,0)
I, for some complex number k with Re k > 0, then a periodic travelling
wave solution [#(z;c), (z;c)] is unstable.
F(0,0)
modulus
PROOF.
With the change of variables,
Z
X
t
t,
+ Ct
[u(x,t), w(x,t)]
[(z,t), (z,t)],
UNSTABLE PERIODIC WAVE SOLUTIONS OF NERVE AXON DIFFUSION
EQUATIONS
789
equation (I.I) becomes
]
t
t
The
-c
+
z
F(],),
+
z
(3.1)
G(,).
linearized perturbation equation of the above system with respect to the solution
[(z;c), (z;c)]
is
t
t
where
c
zz
#(z;c)
c
ZZ
+ F
Z
+ G
C
Z
+F
+ G
[,]
(z;c),
and
[,]
since
2
2
[#,]
[#,]
(3.2)
G(0,O)
F(0,0)
Equation (3.2) has a
O.
solution of the form
where
(Yl’ Y2
U(z,t)
e
Yl
(z;A),
W(z,t)
e
Y2
(z;A),
satisfies the following system of linear ordinary differential equations
d2y
Ay
dz
AY2
dY
c
2
FI
d-- +
[ ’]
Yl
+
F2
[ ’]
+
G2
[#’]
Y2’
dY
c-a2 + G1
[#’]
Yl
Y2’
(3.3)
(z;c). Note that if equation (3.3) has a solution which
then equation
is bounded for all
z
for a number A with Re(A) > 0,
in (-,)
(3.2) has a solution
[U(z,t), W(z,t)] which grows exponentially, and hence, the
travelling wave solution [#(z;c), (z;c)] is unstable.
Using Floquet’s theory, we can show that equation (3.3) has a bounded non-trivial
solution if and only if one of the eigenvalues of X( p( c) A ,c)
is a modulus I. Equa-
where
tion
#(z;c)
and
(3.3) can be rewritten as
A-El [0’] Yl
d- \dz
c
dY2
d-
GI
[’]
Yl
+
(G2[#’]
+ c
-a F2
-A)
Y2’
and so can be represented by the matrix differential equation
d
d---v
A(z;A,c) __v,
’
Y2’
R. LING
790
where
v
dz
Y2
I
J
Now, since the coefficient matrix A (z;k,c)
z, Floquet’s theory yields that equation (3.3) has a
and the matrix A is as defined before.
is a
p(c)-periodic function of
bounded
non-trivial
X(p(c) ;k ,c)
if
solution
oaly
and
defined before is of modulus I.
In the following lemma,
X(p(c);0,c)
eigenvalue of
if one of the eigenvalues of the matrix
The proof is now complete.
it is shown that under the special case
0,
one
is unity and the product of the other two eigenvalues is
greater than one.
L 3.1.
i
.(l,c),
l
G2(u,w) _>
(a)
Suppose
I, 2, 3,
0
all
for
denote the eigenvalues of
and
u
w
X (p(c);k,c),
and
(b)
k
0,
let
then one eigenvalue,
say
I,
i(0,c)
2(0’c) 3(0,c) >
and
PROOF.
I.
Differentiation of equation (2.1) leads to
dO
d
d)
(c-z
where
O
O(z;c)
G
[0,] d0 +
(z;c).
and
z G2
satisfies the matrix equation
d-that is,
Wz
A (z;0,c)
d,
-z
Therefore the vector
dO
d
[0,]
Wz,
d
(3.4)
EQUATIONS
UNSTABLE PERIODIC WAVE SOLUTIONS OF NERVE AXON DIFFUSION
791
We know that (see for example, Sanchez)
w
and since
w
Z
(z;c)
p(c)
is a
w
X(z;0,c)
(z;c)
Z
Z
(O;c)
w
w
Z
(O;c)
periodic function of
Z
(p(c);c)
z, it follows that
X (p(c);0,c) w
Z
(0;c).
(3.5)
Thus there is an eigenvalue, say
(O,c)
I
I.
Further, by Jacobi’s formula,
{X(z;k,c)}
det
f
{det X(O;%,c)} exp
-
tr
0
G
z
(I) exp
(c+
2 #
,
C
0
In particular,
det
{X(p(c);0,c)}
{A(;l,c)}
exp [c p(c)] exp
d$
j(c)
P
G2[#,]
0
C
d
>
since
But
c
>
O, p(c)
det {X
>
0
(p(c);O,c)}
I
Note
that
[123 (k,c[ >
L(c)
is
under
k
for
(0,c)
I,
the
_>
2 (u,w)
(0,c)
I
for all u,w.
0
(0,c)
2
hence
2
of
assumptions
L’(c)
<
0,
then
(0,c)
Suppose (a) p’(c)
<
B1
actually holds.
[k=O
-p’ (c)
(0,c)
>
3.1,
either
(l,c)
I.
[2 (%,c)[
-
is increasing at
Vl (x’c)
I (l,c) >
We claim that the following equality
Tf- a (), ,c)
and
>
or
In the next theorem, we will see that if
0, then
the assumptions in Lemma 3.1 also hold, then
3
Lemma
IX=0 > 0.
TIiEORKM 3.2.
PROOF:
(0 ,c)
3
sufficiently small.
decreasing, i.e.
a-- l(X ,e)
G
and
X=0
for
>
0,
O,
i.e.
and hence if (b)
sufficiently small.
R. LING
792
Recall the vector
w
(z;c),
namely,
which satisfies the periodicity
(p(c);c)
w
w (0;c).
Differentiation of the above equation with respect to
w
_v
_v
Let
Z
[Yl
(p(c);c) p’(c) + w C (p(c);c)
(z;%,c),
Y2
(z;%,c)]
w
(0;c) + k w C (0;c),
w
c
C
leads to
(0;c).
(3.6)
be a solution of equation (3.3) satisfying
the initial condition
v
(0;%,c)
Z
(3.7)
[dz
(z;c),
_v (z;k,c) is the vector defined before. We have observed before that
(z;c), which satisfies equation (3.4), is a solution of equation (3.3) under %=0.
In view of the condition (3.7) and by uniqueness of solutions, we have
where
-d]
v
(z;0,c)
w
(3.)
(z;c).
Z
-- -
Differentiation of equation (3.3) with respect to
Under
%
O,
Yl
+ %
Y2
+ k
and
Yl d2
dz
Y2
d (z;c)
dz
where
Yi
respect to
d* (z;c)
d-
Yi
---(z;O,c)
c, we get
now.
YP
d
-c- (-)
[YI’ Y2
replacing
(3.8) becomes
c
2
d
.Yl.
2
dz2
c
-)
d
d
+ G
by
d
+
[’]
[YI’ Y2 ]’
.Yl.
Cz ---)
Y2.
t--) +
leads to
GI[*,*I
FI[,]
+
G2
Yl
Yl + G2[*
Y2
---,
Y2
[’] k---"
(3.9)
equation
+ V I[0,]
+ F 2 [,]
aYl
*]
+
F2[O
Y2
---,
(3.9)
by equality
]
aY2
-,
(3 lO)
On the other hand, differentiating equation (2.1) with
_
UNSTABLE PERIODIC WAVE SOLUTIONS OF NERVE AXON DIFFUSION
d2
dz
d
dz
# (z;c)
where
b-7 (z;c), cc
d#
z
c
2
c
d
+
c
+
G1 [#
(z;c).
and
(z;c)]
d
c
,
FI[#’@]
3 +
F2[
vx
-
--X--
w
0
(3.1)
(z;0,c)]
and
In addition, differentia-
(3.7) yields
(0;X,c)
c
G2[#’] c 0,
,
,3Yl
3Y2
both
(z-0;c),
satisfy the same differential equation.
tion of the initial condition
]
3 +
Therefore
793
EQUATIONS
(0;c),
c
in particular,
x (0;0
c)
w
vl (z;0,c)
w
v
(0;c)
C
and hence the equality
<
(z;c), 0
C
<
z
(3.12)
p(c).
The equalities (3.8) and (3.12) together give
,
v
w
(z;,c)
0
Knowing
also
z
_v* (z;k,c)
(z;c) +
Z
<
z
X (z;k,c)
<
,
p(c),
v_*
(z;c) +
w
C
O(X2),
(3.13)
O.
as
(O;k,c),
by equation (3.13) for
z
p(c) and
O, we get
w
Z
"ptc);c) + k w C (p(c);c) + 0
X (p(c);l,c) [w z (o;e) + I
We
(),2)
(3.14)
(0;e)].
Substitution of the equation (3.6) containing p’(c) into the left hand side of equation
(3.14) and periodicity lead to
X (p(c);X,c) [w
z
(O;c) + X
[I -I p’(c)][w z (O;c) + X
Hence the eigenvalue
l(X,c)
satisfies
,(X,c)
-p’(c).
,=0
The proof is now complete.
(O;c)]
We
Wc
(0;c)] + 0
(X2).
794
R. LING
On the other hand,
under certain conditions,
two eigenvalues have modulus less
than one and one has modulus greater than one.
(u,w) and
THEORFa 3.3. Suppose (a) F (u,w) is a non-zero constant and (b)
2
are constant, then for
sufficiently large, two eigenvalues of X (p(c);k,c)
(u,w)
2
G!
G
<
have modulus
>
and one has modulus
I.
A(z;,c)
Decompose the matrix
PROOF:
as follows
B (k,c) + E (z;c)
A (z;k,c)
X
c
-F
G
0
G2-X
Let s. (k,c)
0
0
0
0
+
2
-F
0
2 3 be the eigenvalues of B (k,c)
i
-s
3 +
0
0
The characteristic equation of
eigenvectors.
(z;c)]
l[(z;c),
G2-I
(-----+c)s
2
+
(2k-G2)s
0
B(l,c)
+ X
and
the corresponding
qi
is
X-G2
(----)
F2GI
c
0.
It follows that as
sl(X,c)
s2(l,c)
s
The vectors
=-
+ 0(1)
/[ + 0(1)
3(x,c)
+ o(1).
qi (X’c) are
qi(k,c)
s.
s
2.1
c s.
-F 2
and let
(3.15)
Q(X,c)
be the non-singular matrix
Q(,,c)
-
1,2,3,
i
(3.16)
ql(,,c), q2(),,c), q3(),,c)],
then
Q-1BQ
Sl(X,c)
o
0
0
s2(),,v)
0
0
0
s3(, ,c
UNSTABLE PERIODIC WAVE SOLUTIONS OF NERVE AXON DIFFUSION
EQUATIONS
795
Now consider the matrix
-I
Q
X(z;,c) Q
Y(z;l,c)
X(z;l,c),
which has the same eigenvalues as
in particular with z
p(c), and satisfies
the differential equation
dz
Y(z;%,c)
Q
A(z;l,c) Q Y(z;,c)
[Q -I B(%,c)Q + Q -I E(z;c)Q] Y(z;%,c),
d
zX
since
-IBQ
But
Q
and
(3.16)
(z;l,c)
A (z;k,c) X (z;k,c).
is the diagonal matrix from before and it can be shown easily using (3.15)
that
eigenvalues of
all
exp
It follows from
<
elements
Y( p( c) l ,c)
Q-IEQ
of
and hence of
[si(l,c)
p(c)], i
(3.15) that as
> I.
%
X(p(c) ;l ,c)
1,2,3
,
o(I)
are
,
as
therefore
the
approach
.
as
X( p( c) k ,c)
two eigenvalues of
have modulus
and one has modulus
To summarize, under the assumptions of both Theorem (3.2) and Theorem (3.3), at
X( p( c) % ,c)
least two eigenvalues of
values of
X( p( c) k ,c)
must have modulus
solution
have modulus
<
have modulus
for some
(#(z;c), (z;c))
>
0
as
>
==.
as
%
0+,
and two eigen-
Hence one of the eigenvalues
and under Theorem (3.1), the travelling wave
is unstable.
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1.
FITZHG}{, R.
2.
NAGUMO, J, ARIMOTO, S., and TOSHIZAWA, S. An Active Pulse Transmission Line Simulating Nerve Axon, Proc. I.R.E. 50 (1962), 2061.
ALLEN, Linda J.S. Persistence and Extinction in Lotka-Volterra Reaction-Diffusion
Equations, Mathematical BioSciences 65 (1983), I.
3.
Mathematical Models of Excitation and
Biological Engineering, McGraw-Hill (1969), 1-85.
4.
COSNKR, C.
5.
GARD-KR, R.A.
Propagation in Nerve, in
Pointwise a Priori Bounds for Strongly Coupled Semilinear Systems of
Parabolic Partial Differential Equations, Indiana University Mathematics
Journal 30 (1981), 607.
Asymptotic
Boundary
(1980), 161.
Dirichlet
Behavior of Semilinear Reaction-Diffusion Systems with
Conditions, Indiana University Mathematics Journal 2__9
GRTIN, M.E. and MAC CAM, R.C.
Product Solutions and Asymptotic Behavior for
Age-Dependent, Dispersing Populations, Mathematical Biosciences 62 (1982),
157.
JA(]Q]F.Z, J.A.
The Physiological Role of Myoglobin: More than a Problem in Reaction-Diffusion Kinetics, Mathematical Biosciences 68 (1984), 57.
796
R. LING
A Semiliner Reaction- Diffusion Prey-Predator System with Nonlinear
Coupled Boundary Conditions: Equilibrium and Stability, Indiana University
Math. J. 31 (1982), 223.
LEUNG, A.
Stationary Solutions and Asymptotic Behavior of a Quasilinear Degenerate Parabolic Equation, Indiana University Math. J. 33 (1984), I.
SCHATZMAN, M.
EWANS, J.W.
Nerve Axon Equations: IV. The Stable and The Unstable Impulse,
Indiana University Math. J. 24 (1975), I[69.
The Existence of Periodic Solutions to Nagumo’s Equation, .Quart. J.
Math. 25 (1974), 369.
HASTINGS, S.
RINZKL, J. and KELLER, J.B.
Travelling Wave
Equation, Biophy s. J. 13 (1973), t3t3.
DAI, L.S.
Solutions
of
a
Nerve Conduction
A Stable Travelling-Wave Solution of a Model of Cellular Control Processes with Positive Feedback, Mathematical Biosciences 61 (1982), 267.