traveling wave solutions of parabolic systems-volpert a.i., volpert v.a. (1994)

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American Mathematical Society
Providence, Rhode Island
Volume 140
Traveling Wave Solutions
of Parabolic Systems
Aizik I. Volpert
Vitaly A. Volpert
Vladimir A. Volpert
A. I. Volpert, Vit. A. Volpert, Vl. A. Volpert
BEGUWIE VOLNY, OPISYVAEMYE
PARABOLIQESKIMI SISTEMAMI
Translated by James F. Heyda from an original Russian manuscript
2000 Mathematics Subject Classification. Primary 35K55, 80A30;
Secondary 92E10, 80A25.
Abstract. Traveling wave solutions of parabolic systems describe a wide class of phenomena in

combustion physics, chemical kinetics, biology, and other natural sciences. The book is devoted to
the general mathematical theory of such solutions. The authors describe in detail such questions as
existence and stability of solutions, properties of the spectrum, bifurcations of solutions, approach
of solutions of the Cauchy problem to waves and systems of waves. The final part of the book is
devoted to applications to combustion theory and chemical kinetics.
The book can be used by graduate students and researchers specializing in nonlinear differential
equations, as well as by specialists in other areas (engineering, chemical physics, biology), where
the theory of wave solutions of parabolic systems can be applied.
Library of Congress Cataloging-in-Publication Data
Vol

pert,A.I.(A˘ızik Isaakovich)
[Begushchie volny, opisyvaemye parabolicheskimi sistemami. English]
Traveling wave solutions of parabolic systems/Aizik I. Volpert, Vitaly A. Volpert, Vladimir A.
Volp ert.
p. cm. — (Translations of mathematical monographs, ISSN 0065-9282; v. 140)
Includes bibliographical references.
ISBN 0-8218-4609-4 (acid-free)
1. Differential equations, Parabolic. 2. Differential equations, Nonlinear. 3. Chemical
kinetics—Mathematical models. I. Volpert, Vitaly A., 1958– . II. Volpert, Vladimir A., 1954– .
III. Title. IV. Series.
QA377.V6413 1994
515

.353—dc20 94-16518
c
 1994 by the American Mathematical Society. All rights reserved.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.

Reprinted with corrections, 2000
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established to ensure permanence and durability.
Information on copying and reprinting can be found in the back of this volume.
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Contents
Preface xi
Introduction. Traveling Waves Described by Parabolic Systems 1
§1. Classification of waves 2
§2. Existence of waves 11
§3. Stability of waves 16
§4. Wave propagation speed 22
§5. Bifurcations of waves 23
§6. Traveling waves in physics, chemistry, and biology 32
Part I. Stationary Waves
Chapter 1. Scalar Equation 39
§1. Introduction 39
§2. Functionals ω
∗
and ω

∗
45
§3. Waves and systems of waves 51
§4. Properties of solutions of parabolic equations 72
§5. Approach to waves and systems of waves 85
§6. Supplement (Additions and bibliographic commentaries) 111
Chapter 2. Leray-Schauder Degree 121
§1. Introduction. Formulation of results 121
§2. Estimate of linear operators from below 128
§3. Functional c(u) and operator A(u) 134
§4. Leray-Schauder degree 138
§5. Linearized operator 141
§6. Index of a stationary point 144
§7. Supplement. Leray-Schauder degree in the multidimensional
case 149
Chapter 3. Existence of Waves 153
§1. Introduction. Formulation of results 153
§2. A priori estimates 159
§3. Existence of monotone waves 173
§4. Monotone systems 176
§5. Supplement and bibliographic commentaries 183
Chapter 4. Structure of the Spectrum 187
§1. Elliptic problems with a parameter 189
§2. Continuous spectrum 192
§3. Structure of the spectrum 198
vii
viii CONTENTS
§4. Examples 208
§5. Spectrum of monotone systems 212
Chapter 5. Stability and Approach to a Wave 217

§1. Stability with shift and its connection with the spectrum 218
§2. Stability of planar waves to spatial perturbations 225
§3. Conditions of instability 237
§4. Stability of waves for monotone systems 238
§5. On the solutions of nonstationary problems 242
§6. Approach to a monotone wave 250
§7. Minimax representation of the speed 254
Part II. Bifurcation of Waves
Chapter 6. Bifurcation of Nonstationary Modes of Wave Propagation 259
§1. Statement of the problem 259
§2. Representation of solutions in series form. Stability of
solutions 263
§3. Examples 268
Chapter 7. Mathematical Proofs 273
§1. Statement of the problem and linear analysis 273
§2. General representation of solutions of the nonlinear problem.
Existence of solutions 285
§3. Stability of branching-off solutions 295
Part III. Waves in Chemical Kinetics and Combustion
Chapter 8. Waves in Chemical Kinetics 299
§1. Equations of chemical kinetics 299
§2. Monotone systems 306
§3. Existence and stability of waves 312
§4. Branching chain reactions 316
§5. Other model systems 333
Bibliographic commentaries 335
Chapter 9. Combustion Waves with Complex Kinetics 337
§1. Introduction 337
§2. Existence of waves for kinetic systems with irreversible
reactions 338

§3. Stability of a wave in the case of equality of transport
coefficients 362
§4. Examples 366
Bibliographic commentaries 375
Chapter 10. Estimates and Asymptotics of the Speed of Combustion
Waves 377
§1. Estimates for the speed of a combustion wave in a condensed
medium 377
§2. Estimates for the speed of a gas combustion wave 392
§3. Determination of asymptotics of the speed by the method of
successive approximations 400
Bibliographic commentaries 409
CONTENTS ix
Supplement. Asymptotic and Approximate Analytical Methods in
Combustion Problems 411
§1. Narrow reaction zone method. Speed of a stationary
combustion wave 411
§2. Stability of a stationary combustion wave 415
§3. Nonadiabatic combustion 416
§4. Stage combustion 418
§5. Transformations in a combustion wave 423
§6. Application of the methods of bifurcation theory to the study
of nonstationary modes of propagation of combustion waves 426
§7. Surveys and monographs 431
Bibliography 433
Preface
The theory of traveling wave solutions of parabolic equations is one of the
fast developing areas of modern mathematics. The history of this theory begins
with the famous mathematical work by Kolmogorov, Petrovski˘ı, and Piskunov
and with works in chemical physics, the best known among them by Zel


dovich
and Frank-Kamenetski˘ı in combustion theory and by Semenov, who discovered
branching chain flames.
Traveling wave solutions are solutions of special type. They can be usually
characterized as solutions invariant with respect to translation in space. The
existence of traveling waves appears to be very common in nonlinear equations,
and, in addition, they often determine the behavior of the solutions of Cauchy-type
problems.
From the physical point of view, traveling waves usually describe transition
processes. Transition from one equilibrium to another is a typical case, although
more complicated situations can arise. These transition processes usually “forget”
their initial conditions and reflect the properties of the medium itself.
Among the basic questions in the theory of traveling waves we mention the
problem of wave existence, stability of waves with respect to small perturbations
and global stability, bifurcations of waves, determination of wave speed, and systems
of waves (or wave trains). The case of a scalar equation has been rather well studied,
basically due to applicability of comparison theorems of a special kind for parabolic
equations and of phase space analysis for the ordinary differential equations. For
systems of equations, comparison theorems of this kind are, in general, not appliΓ
cable, and the phase space analysis becomes much more complicated. This is why
systems of equations are much less understood and require new approaches. In
this book, some of these approaches are presented, together with more traditional
approaches adapted for specific classes of systems of equations and for a more
complete analysis of scalar equations. From our point of view, it is very important
that these mathematical results find numerous applications, first and foremost in
chemical kinetics and combustion. The authors understand that the theory of
traveling waves is far from being complete and hope that this book will help in its
development.
This book was basically written when the authors worked at the Institute of

Chemical Physics of the Soviet Academy of Sciences. This scientific school, created
by N. N. Semenov, Director of the Institute for a long time, by Ya. B. Zeldovich,
who worked there, and by other outstanding personalities, has a strong tradition
xi
xii PREFACE
of collaboration among physicists, chemists, and mathematicians. This special
atmosphere had a strong influence on the scientific interests of the authors and was
very useful to us. We would like to thank all our colleagues with whom we worked
for many years and without whom this book could not have been written.
Aizik Volpert
Department of Mathematics, Technion, Haifa, 32000, Israel
Vitaly Volpert
Universite Lyon 1, CNRS, Villeurbanne Cedex, 69622 France
Vladimir Volpert
Northwestern University, Evanston, Illinois 60208
June 1993
INTRODUCTION
Traveling Waves Described by Parabolic Systems
Propagation of waves, described by nonlinear parabolic equations, was first
considered in a paper by A. N. Kolmogorov, I. G. Petrovski˘ı, and N. S. Piskunov
[Kolm 1]. These mathematical investigations arose in connection with a model for
the propagation of dominant genes, a topic also considered by R. A. Fisher [Fis 1].
Moreover, when [Kolm 1] appeared in 1937, the fact that waves can be described
not only by hyperbolic equations, but also by parabolic equations, did not receive
the proper attention of mathematicians. This is indicated by the fact that sub-
sequent mathematical papers in this direction (Ya. I. Kanel

[Kan1,2,3]) did
not appear until more than twenty years later, although mathematical models,
which form a basis for these papers, models of combustion, were formulated by

Ya.B.Zel

dovich somewhat earlier (see, for example, [Zel 4, 5]). Itwasnotuntil
the seventies, under the influence of a great number of the most diverse problems of
physics, chemistry, and biology, that an intensive development of this theme began.
At the present time a large number of papers is devoted to wave solutions of
parabolic systems and this number continues to increase. In recent years, along
with the study of one-dimensional waves, an interest in multi-dimensional waves
has developed. This interest was stimulated by observation of spinning waves in
combustion, spiral waves in chemical kinetics, etc.
The overwhelming number of natural science problems mentioned above leads
to wave solutions of the parabolic system of equations
(0.1)
∂u
∂t
= A∆u + F(u),
where u =(u
1
, ,u
m
) is a vector-valued function, A is a symmetric nonnegative-
definite matrix, ∆ is the Laplace operator, and F (u) is a given vector-valued
function, which we will sometimes refer to as a source. System (0.1) is considered
in a domain Ω of space R
n
on whose boundary, assuming Ω does not coincide with
R
n
, boundary conditions are specified.
We attempt in the present introduction to give a general picture of current

results concerning wave solutions of system (0.1) (see also [Vol 47]). Later on in
the text we present in detail results of a general character, i.e., results connected
with general methods of analysis and with sufficiently general classes of systems.
In the remaining cases we limit ourselves to a brief exposition or to references to
original papers. However, in selecting material for a detailed exposition interests of
the authors are dominant.
Numbering of formulas and various propositions are carried out according to
sections, the first digit indicating the section number. If in references the chapter is
not indicated, it may be assumed that reference is being made to a section within
the current chapter.
1
2 INTRODUCTION. TRAVELING WAVES DESCRIBED BY PARABOLIC SYSTEMS
§1. Classification of waves
Waves described by parabolic systems can be divided into several classes. The
most conventional is the class of waves referred to as stationary. By a stationary
wave we mean a solution u(x, t) of system (0.1) of the form
(1.1) u(x, t)=w(x
1
− ct, x

),
where w(x) is a function of n variables, x =(x
1
, ,x
n
), x

=(x
2
, ,x

n
), and c
is a constant (speed of the wave). We assume here that Ω is a cylinder and that
the system of coordinates is chosen so that axis x
1
is directed along the axis of the
cylinder.
In recent years a large body of experimental material has accumulated and, in
addition, a number of mathematical models connected with it have been studied
in which not just stationary waves can be observed. In particular, we can observe
periodic waves , defined as solutions u(x, t) of system (0.1) of the form
(1.2) u(x, t)=w(x
1
− ct, x

,t),
where the function w(x, t)isperiodicint; Ω, as defined above, is a cylinder; and
x
1
is directed along the axis of the cylinder.
Other forms of waves also occur, some of which we indicate below.
1.1. Stationary waves. We present a classification of stationary waves cur-
rently being studied. Part I of the present text is devoted to stationary waves.
1.1.1. One-dimensional planar waves. We consider system (0.1) with the fol-
lowing boundary condition on the surface of cylinder Ω:
(1.3)
∂u
∂ν
=0,
where ν is the normal to the surface. We refer to a solution of the form

(1.4) u(x, t)=w(x
1
− ct)
as a planar wave. This, obviously, corresponds to the definition given above of a
stationary wave, one-dimensional in space, i.e., a solution of the system
(1.5)
∂u
∂t
= A
∂
2
u
∂x
2
1
+ F (u).
Function w of the variable ξ = x
1
− ct is a solution of the following system of
ordinary differential equations over the whole axis:
(1.6) Aw

+ cw

+ F (w)=0.
Obviously, the system of equations (1.6) can be reduced to the system of first order
equations
(1.7) w

= p, Ap


= −cp − F(w).
Thus, the problem of classifying planar waves can be reduced to the study of
the trajectories of system (1.7). Apparently, however, not all trajectories are of
interest. Solutions of system (1.6) are stationary solutions of system (1.5), written
in coordinates connected with the front of a wave; of most interest are those waves
which are stable stationary solutions.
We present a classification of planar waves encountered in applications.
§1. CLASSIFICATION OF WAVES 3
ξ
c
w
Figure 1.1. A monotone wave front
By wave fronts we mean solutions w(ξ) of system (1.6), having limits as ξ →
±∞,
(1.8) lim
ξ→±∞
w(ξ)=w
±
,
where
(1.9) w
+
= w
−
.
Typical representatives of such waves are waves of combustion and waves in chemical
kinetics, in particular, frontal polymerization, concentrational waves in Belousov-
Zhabotinsky reactions, cold flames, etc. A characteristic form of a monotone wave
front for each component of the vector-valued function w is shown in Figure 1.1. If

we return to the initial coordinate x
1
, the wave front is then the profile shown in
this figure moving along the x
1
-axis at constant speed c.
It is readily seen that we have the equalities
(1.10) F (w
+
)=0,F(w
−
)=0
if the function w(ξ), together with its first derivative, is bounded on the whole axis
and if the limits (1.8) exist. Actually, in this case it is easy to show that
w

(ξ) → 0andw

(ξ) → 0as|ξ|→∞,
and, passing to the limit in (1.6), we obtain (1.10).
Thus, w
+
and w
−
are stationary points of the nondistributed system
(1.11)
du
dt
= F (u),
corresponding to system (1.5). It turns out to be the case that in studying wave

fronts connecting points w
+
and w
−
(i.e., solutions of system (1.6) satisfying
conditions (1.8)) it is very important to have information concerning stability of the
stationary points w
+
and w
−
. Obviously, only the following three types of sources
F (u) are possible:
A. Both points w
+
and w
−
are stable stationary points of equation (1.11).
B. One of the points w
+
or w
−
is stable, the other is unstable.
C. Both points w
+
and w
−
are unstable.
As we shall show below, answers to questions concerning the existence of waves,
their uniqueness, and a number of other questions, depend on the source type for
F (u).

4 INTRODUCTION. TRAVELING WAVES DESCRIBED BY PARABOLIC SYSTEMS
u
w
−
w
+
F
Figure 1.2. A Type A source (bistable case)
F
w
+
w
−
u
Figure 1.3. A Type A source (bistable case)
F
w
+
w
−
u
Figure 1.4. A Type B source (monostable case)
Sources of various types are shown in Figures 1.2–1.5 in the case of a scalar
equation (1.5). Figures 1.2 and 1.3 display sources of Type A; Figures 1.4 and 1.5
display sources of types B and C, respectively. Sources shown in Figures 1.2, 1.4,
and 1.5 are encountered in problems concerned with the propagation of dominant
genes (see [Kolm 1]and[Aro 1]); the source shown in Figure 1.3 appears in
problems of combustion (see [Zel 5]).
InthecaseofaTypeAsourceweshallalsosaythatwehaveabistablecase,
for Type B sources a monostable case, and for Type C sources an unstable case.

Figure 1.6 depicts a Type A source of more complex form. It has a stable
intermediate stationary point w
0
, so that one can speak of two waves: one joining
point w
+
with w
0
, and one joining w
0
with w
−
; one can also speak of the wave
§1. CLASSIFICATION OF WAVES 5
F
w
+
w
−
u
Figure 1.5. A Type C source (unstable case)
F
w
+
w
−
w
0
Figure 1.6. A Type A source with a stable intermediate station-
ary point

joining w
+
with w
−
. We shall concern ourselves with the question of which waves
may be realized in actuality when, in what follows, we discuss systems of waves.
Pulses differ from wave fronts only by the fact that, instead of (1.9), we have
the equality
(1.12) w
+
= w
−
.
Currently, the most studied equations describing pulses (as well as periodic waves,
see below) are the equations for propagation of nerve impulses, namely, the Hodgkin-
Huxley equations and the simpler Fitz-Hugh-Nagumo equations, which are special
cases of system (1.5). The characteristic form of the pulses described by the
equations mentioned is shown in Figures 1.7 and 1.8 on the next page.
Waves periodic in space are solutions of system (1.6) for which the function
w(ξ) is periodic. Periodic waves were discovered in problems of propagation of
nerve impulses and in problems of chemical kinetics. Corresponding to them in the
phase plane are the limit cycles of system (1.7).
1.1.2. Multi-dimensional waves are solutions of the form (1.1) which cannot be
written in the form (1.4). Experimentally such waves may be observed as a uniform
displacement of a “curved” front along the axis of the cylinder. Obviously, if instead
of boundary condition (1.3) we consider a different boundary value problem, for
example, of the first or the third kind, then a stationary wave, if it exists, is multi-
dimensional. However, even in the case of condition (1.3) multi-dimensional waves
can also be realized.
6 INTRODUCTION. TRAVELING WAVES DESCRIBED BY PARABOLIC SYSTEMS

w
ξ
Figure 1.7. A wave solution in the form of a pulse
w
ξ
Figure 1.8. A wave solution in the form of a pulse
Let us assume that the following limits exist:
(1.13) lim
ξ→±∞
w(ξ, x

)=w
±
(x

).
We can then expect that the function w
±
(x

) will satisfy the equation
(1.14) Φ(w
±
)=0.
Here Φ is the nonlinear operator
(1.15) Φ(u)=A∆

u + F(u),
where ∆


is the Laplace operator in the variables x

=(x
2
, ,x
n
). The operator
Φ(u) is considered on functions given in a cross-section of the cylinder Ω, satisfying
the same boundary conditions as in the initial problem (for simplicity we assume
that in these conditions there is no dependence on x
1
).
Here, as we did in the one-dimensional formulation, we can speak of three
cases A, B, and C, except that here, instead of equation (1.11), we must consider
the operator equation
(1.16)
du
dt
=Φ(u).
1.2. Periodic waves. Periodic waves, determined by equation (1.2), describe
various processes that were observed in combustion (see the supplement to Part III)
and other physico-chemical processes (see, for example, [Beg 1]). Currently, the
§1. CLASSIFICATION OF WAVES 7
Figure 1.9. Two-spot mode of wave propagation in a strip
basic general methods for studying periodic waves are the methods of bifurcation
theory (see below). We now present some forms of periodic waves encountered in
applications.
It is convenient to describe the character of wave propagation by considering the
motion of certain characteristic points, for example, maximum points of solutions.
We shall refer to these maximum points as hot spots. This type of terminology

arose in experiments dealing with combustion, where luminous spots, corresponding
to maximum points of the temperature, were observed propagating along the
specimen.
1.2.1. One-dimensional waves are solutions of system (1.5) of the form w(x
1
−
ct, t), where w(ξ,t) is a periodic function of t. As an example of a physical model
we cite the oscillatory mode of combustion (see [Shk 3]) in which a planar front of
combustion performs periodic oscillations relative to a uniformly moving coordinate
frame. We remark that the character of the oscillations can be fairly complex. In
particular, in numerical modeling of combustion problems, bifurcations have been
observed leading to the successive doubling of the period, and then to irregular
oscillations [Ald 6, Dim 1, Bay 4].
1.2.2. Two-dimensional waves are solutions of the form (1.2) of system (0.1),
considered in an infinite strip of width l: −∞ <x
1
< +∞,0 x
2
 l.We
shall assume that condition (1.3) is satisfied. The nature of the wave propagation
manifests itself by motion of the hot spots. A planar wave propagates along the
strip when the width of the strip is sufficiently small. As l increases, a critical
value l = l
1
is attained and a one-spot mode of wave propagation then arises: the
spot moves along the line x
2
= 0, then moves onto the line x
2
= l,afterwhich

the motion of the spot again takes place along the line x
2
=0,andsoon,ina
periodic fashion. With further widening of the strip a second critical value l = l
2
appears, giving rise to a two-spot mode. The spots move simultaneously along the
lines x
2
=0andx
2
= l, after which, moving towards one another, they merge onto
the line x
2
= l/2 and move along this line: they then diverge and, once again, two
spots appear on the lines x
2
=0andx
2
= l, after which the motion described
is repeated. Depending on how much the width of the strip is increased, at some
l = l
3
a three-spot mode appears, and so forth. Such modes for combustion of a
plate were obtained numerically in [Ivl 3]. A two-spot mode is shown in Figure 1.9.
1.2.3. Spinning waves. We consider the case of a three-dimensional space
(n = 3) and a circular cylinder Ω, along whose axis a wave is propagating. We
introduce polar coordinates r and ϕ in a disk cross-section of the cylinder. By a
8 INTRODUCTION. TRAVELING WAVES DESCRIBED BY PARABOLIC SYSTEMS
Figure 1.10. One-spot spinning mode of wave propagation in a
circular cylinder

Figure 1.11. Two-spot spinning mode of wave propagation in a
circular cylinder
spinning wave we mean a solution of system (0.1) of the form
u(x, t)=w(x
1
− ct,r,ϕ−σt),
where c and σ are constants; c is the speed of propagation along the axis of the
cylinder, σ is the angular rate.
Spinning waves were observed in the combustion of condensed systems [Mer7]
as the motion of luminous spots along a spiral on the surface of the cylindrical
specimen. Spinning wave propagation patterns were studied experimentally, in
particular, their dependence on the radius of the cylinder. It was established that
for small radii the spinning mode is not present; with an increase in the radius
a one-spot spinning mode appears (Figure 1.10); next, a two-spot mode appears
(Figure 1.11) when two spots move simultaneously along a spiral, and so forth.
1.2.4. Symmetric waves. In studies made using methods of bifurcation theory
[Vol 13, 21, 30] waves coexisting with spinning waves were observed in a circular
cylinder. These were called symmetric waves (they are also called standing waves).
An analysis of the stability of these waves (see Chapter 6) showed that spinning
and symmetric waves cannot be simultaneously stable at their birth occurring as
the result of a loss of stability of a plane wave. Symmetric waves, just like spinning
waves, can have one spot, two spots, etc. In a one-spot symmetric mode motion
of the spots proceeds as follows: a spot moves along the surface of the cylinder
parallel to its axis; it then bifurcates and two spots appear, moving along the
surface and meeting on the diametrically opposite side of the cylinder surface, etc.
in a periodic mode. A one-spot symmetric mode is depicted in Figure 1.9 if we
§1. CLASSIFICATION OF WAVES 9
Figure 1.12. One-spot symmetric mode of wave propagation on
the surface of a cylinder
Figure 1.13. An end-view of a one-spot symmetric mode

Figure 1.14. An end-view of a two-spot symmetric mode
identify lines bounding the strip from its sides. A direct representation of a one-
spot symmetric mode on the surface of a cylinder is shown in Figure 1.12. An
end-view is shown in Figure 1.13. Motion of the spots in the case of a two-spot
symmetric mode is completely analogous, except that now two spots, located at
diametrically opposite points of the surface, move simultaneously; each of the spots
bifurcates and they meet at points shifted with respect to the initial points by an
angle π/2. Figure 1.14 depicts an end-view of a two-spot symmetric mode.
The question as to whether a symmetric mode in combustion has been observed
experimentally is an open one. In experiments modes have been observed in
which the spots move along the surface of a circular cylinder towards each other;
unfortunately, however, there is no detailed description of these modes. It can be
assumed that these modes are symmetric.
1.2.5. Radial waves. We have in mind periodic waves in a circular cylinder,
i.e., waves of the form (1.2) in which there is no dependence on angle, so that
motion takes place along the axis of the cylinder and in the direction of the radius.
Obviously, such waves can be considered in a section of the cylinder by plane
x
1
= 0 and a corresponding body of revolution about the axis of the cylinder can
be obtained. As bifurcation analysis shows, such waves can exist with a various
number of spots. In the case of a one-spot mode, motion of the spots takes place
10 INTRODUCTION. TRAVELING WAVES DESCRIBED BY PARABOLIC SYSTEMS
Figure 1.15. One-spot radial wave
Figure 1.16. A mode of wave propagation in a cylinder of rect-
angular cross-section
Figure 1.17. A mode of wave propagation in a cylinder of rect-
angular cross-section
as follows: a spot moves along the axis of the cylinder, then goes over onto the
surface, fills in a complete circle, then again falls onto the axis, and so forth (see

Figure 1.15). A mode of this kind has been observed in combustion [Mak 3].
Remark. As follows from a bifurcation analysis (see below), with a loss of
stability of a planar wave such that a pair of complex conjugate eigenvalues goes
across the imaginary axis, there generically arise, in the circular cylinder, periodic
waves of four and only four types: one-dimensional, spinning, symmetric, and radial.
1.2.6. Waves in a cylinder of rectangular cross-section. In this case various
modes of propagation of the spots are possible. They have been studied by the
methods of bifurcation theory (see §5). Two of these are shown in Figures 1.16
and 1.17. Modes of this kind were obtained experimentally in combustion [Vol 32].
We restrict the discussion here to cylinders with circular and rectangular cross-
sections. Cylinder of arbitrary cross-section will be discussed in §5.
§2. EXISTENCE OF WAVES 11
1.2.7. Waves of more complex structure. We have enumerated in some sense
the simplest forms of periodic waves. All of them can be obtained as bifurcations
in the vicinity of a planar wave. More complex waves are also possible, being
obtained through interaction of modes already described. Such waves emerge
during computer calculations, for example, as secondary bifurcations. As examples,
we can point to the birth of symmetric waves from developed one-dimensional
oscillations [Meg 1, 2], and also, waves which appear as the rotation of “curved”
fronts [Buc 3].
1.3. Other forms of waves. Along with the waves presented above, other
forms of waves are encountered.
1.3.1. Rotating and spiral waves. Rotating waves are similar to spin waves,
differing only in that propagation is with respect to an angular coordinate. The
pertinent domain Ω is a body of revolution about an axis (or about a point for
n =2).
Spiral waves have been observed experimentally in chemical kinetics, wherein
the spot of a chemical reaction moves along a spiral. In this case Ω is taken to be a
plane, and rotation of the wave is described in polar coordinates with simultaneous
propagation along a radius. Three-dimensional spiral waves have also been studied.

Rotating and spiral waves have been discussed in a large number of papers (see
[Ale1,Ang1,Auc1,2,Bark1,Bern1,Brazh1,Coh1,Duf1,Ern1,2,
Gom 1, Gre 1–4, Grin 1, Hag 4, Kee 1, 2, Koga 1, Kop 6, Kri 1–3, Nan 1,
Ort 1, Pelc 1, Ren 1, Win 1–3]).
1.3.2. Target type waves. Waves of this kind are observed experimentally in
chemical kinetics as concentric waves diverging from a center with simultaneous
generation of new waves at the center. References concerned with these waves
are [Erm4, Fife4,5, Hag2, Kop1,2,5,6, Tys1].
1.4. Systems of waves. A study of the behavior of solutions of a Cauchy
problem for system (0.1) for large values of t showsthatitisnotalwayssingle
waves that are involved. We arrive at this conclusion already in the study of a
scalar equation in the one-dimensional case. This has already been mentioned for
the source shown in Figure 1.6. Possible waves here are [w
+
,w
0
]-, [w
0
,w
−
]-, and
[w
+
,w
−
]-waves,whereweusethenotation[w
±
,w
0
] to show that the waves connect

the points w
±
and w
0
. In Figure 1.18 the possible cases are shown schematically:
[w
0
,w
−
]-wave with a speed c
−
,[w
+
,w
0
]-wave with a speed c
+
. Existence of a
[w
+
,w
−
]-wave depends on the relationship between the speeds c
+
and c
−
.If
c
−
>c

+
, then the [w
0
,w
−
]-wave overtakes the [w
+
,w
0
]-wave; the waves then merge
and the wave [w
+
,w
−
] emerges as a solitary wave with an intermediate speed. But
if c
−
 c
+
, then the two waves coexist and we have a system of waves. Studies
presented in Chapter 1 show that in precisely this way an asymptotic solution of
system (0.1) is obtained as t →∞.
Systems of waves, or in other terminology wave trains, or minimal decompo-
sition of waves, were studied first from the physical point of view in combustion
theory [Kha 3, Mer 6] and mathematically in [Fife 7].
§2. Existence of waves
2.1. Methods of proof for the existence of waves. At the present time
there is a large number of papers concerned with the existence of waves in which
12 INTRODUCTION. TRAVELING WAVES DESCRIBED BY PARABOLIC SYSTEMS
w

w
−
w
0
w
+
c
+
c
−
ξ
Figure 1.18. A system of waves (wave train)
various methods of analysis are employed. It appears, however, that we can single
out three basic approaches:
1. Topological methods, in particular, the Leray-Schauder method.
2. Reduction of a system of equations of the second order to a system of first
order ordinary equations and various methods of analyzing the trajectories
of this system (for one-dimensional waves).
3. Methods of bifurcation theory.
Other methods are also in use. In this section, and in the supplement to
Chapter 3, we attempt briefly to characterize the known methods and results on
the existence of waves. A more detailed discussion of the Leray-Schauder method
will be given; we develop this method in the text in connection with wave solutions
of parabolic systems and, as it appears to us, it is a very promising method. We
remark that in the overwhelming majority of papers the existence of waves for
systems of equations is discussed in the one-dimensional case.
2.1.1. Leray-Schauder method. As is well known, the Leray-Schauder method
consists in constructing a continuous deformation of an initial system to a model
system for which it is known that solutions exist and possess the required properties.
For these systems we consider the vector field generated by them in a functional

space, and we assume that a homotopic invariant is defined, namely, rotation of
the vector field, or, in other terminology, the Leray-Schauder degree, satisfying the
following properties:
1. Principle of nonzero rotation.
If on the boundary of a domain in a functional space the degree is defined and
different from zero, then in this domain there are stationary points.
2. Homotopic invariance.
If during a continuous deformation of a system the solution does not reach the
boundary of a domain, then the degree does not vary on this boundary.
Thus, if we have a priori estimates of solutions, i.e., in a homotopy process
solutions are found in some ball in functional space, and if for a model system the
degree on the ball boundary is different from zero, then it is also different from zero
for the initial system. Consequently, solutions also exist for the initial system.
Thus, to apply the Leray-Schauder method it is necessary to define the degree
with the indicated properties; to construct a model system for which the degree
is different from zero on the boundary of a ball of sufficiently large radius; and to
construct a continuous deformation of the initial system to the model system such
that there are a priori estimates of solutions.
§2. EXISTENCE OF WAVES 13
Rotation of a vector field for completely continuous vector fields is well defined
and widely applied, in particular, in proving existence of solutions by the Leray-
Schauder method. Systems of equations of the type (1.6) can be reduced to
completely continuous vector fields, but only in case they are considered in bounded
domains (see §1 of Chapter 2). For waves, i.e., for solutions considered in unbounded
domains, one cannot make use of an existing theory for completely continuous vector
fields, and this is actually the case. Essentially the situation is the following.
To construct the degree it is necessary to select, in an appropriate way, a
functional space and to define an operator A vanishing on solutions of system (1.6),
i.e., on waves. A vector field will thereby be determined. Operator A can be
approximated in various ways by operators A

n
, which correspond to completely
continuous vector fields and for which the degree can be defined in the usual way.
Moreover, this can be done so that the degree for operators A
n
is independent of
n if n is sufficiently large, and we can take this quantity as the degree of operator
A.Ifγ(A, D), the degree of operator A on the boundary of domain D, is different
from zero, then γ(A
n
,D) will also be different from zero; consequently, there exists
a sequence of functions u
n
, belonging to domain D,forwhichA
n
(u
n
)=0. This
sequence is bounded (domain D is assumed to be bounded) and, consequently, some
subsequence converges weakly. The main difficulty here is that the weak limit of
this sequence may not belong to domain D, and, as a consequence, the principle
of nonzero rotation can be violated. To avoid this situation we need to show,
for the class of operators considered, that weak convergence of solutions implies
strong convergence (precise statements appear in Chapter 2). To proceed we need
to obtain estimates from below for operator A. These estimates for operators
corresponding to the system of equations (1.6) were obtained, thereby making it
possible to define the degree by Skrypnik’s method [Skr 1]. It should be noted that
rotation of a vector field possessing the usual properties cannot be constructed in an
arbitrary functional space. Even in the case of a scalar equation it is easy to give an
example whereby, in the space of continuous functions C, a wave under deformation

disappears with no violation of a priori estimates. This is connected with the fact
that during motion with respect to a parameter a wave can be attracted to an
intermediate stationary point and, instead of a wave satisfying conditions (1.8), we
will have a system of waves. In constructing the degree for operators describing
traveling waves, it is convenient to use weighted Sobolev spaces.
Yet another difficulty arising here is that solutions of equation (1.6) are invari-
ant with respect to a translation in the spatial variable. In addition, the speed
c of the wave is an unknown and must also be found in solving the problem. It
is therefore convenient in the study of waves to introduce a functionalization of
the parameter. This means that the speed of the wave is considered not as an
unknown constant, but as a given functional defined on the same space as operator
A. Here the value of the functional depends on the magnitude of the translation
of the stationary solution, making it possible to single out one wave from a family
of waves. Thus functionalization of a parameter allows us to consider an isolated
stationary point in a given space instead of a whole line of stationary points.
We remark that the degree for operators describing traveling waves is defined
without any assumptions as to the form of the nonlinearity of F(u), except, natu-
rally, for smoothness and stability of the points w
±
. This result, apparently, can be
rather easily generalized to the multi-dimensional case. As for the monostable case,
there arise here additional complexities associated with the facts that waves exist
14 INTRODUCTION. TRAVELING WAVES DESCRIBED BY PARABOLIC SYSTEMS
for a whole half-interval (half-axis) of speeds and form an entire family of solutions.
It can be expected that the degree can be successfully introduced with a proper
selection of weighted norms, identifying a single wave (a single speed) of a family
of waves.
Having defined the degree, we see that the possibility of obtaining a priori
estimates of solutions also determines the class of systems for which we can suc-
cessfully apply the Leray-Schauder method to prove the existence of waves. (The

construction of a model system can be carried out rather easily and is, in the main,
of a technical nature.) Hitherto it has been possible to do this for locally-monotone
systems (see §2.2), yet one may expect that there exist other types of nonlinearities
also for which a priori estimates of solutions can be obtained. It should be noted
that the problem of obtaining a priori estimates in one form or another also arises
for other methods of proving the existence of waves; one should therefore not assume
that this restricts the application of the Leray-Schauder method in comparison with
other approaches.
2.1.2. Other methods for proving the existence of waves. A widely used me-
thod is one based on phase portrait analysis. This means that the system of
equations (1.6) is placed into correspondence with the first order system of equa-
tions (1.7). As has already been noted, its trajectories correspond to waves. In
particular, if the question concerns waves satisfying conditions (1.8), i.e., wave
fronts or pulses, we then have in mind trajectories of system (1.7) joining the
stationary points (w
+
, 0) and (w
−
, 0) in the phase space (w, p). To periodic waves
there correspond limit cycles.
Thus the problem of proving existence of waves reduces to proving existence of
corresponding trajectories of system (1.7).
This method is very suitable when applied to a scalar equation. Actually, in
this case (1.7) is a system of two equations and the analysis is carried out in the
two-dimensional phase plane, a situation rather well studied. To prove existence of
wave fronts a trajectory is drawn off from one of the stationary points (w
+
, 0) or
(w
−

, 0) and it is proved that the constant c can be selected so that this trajectory
reaches the other of these stationary points. Precisely this method was used for the
first time in [Kolm 1] to prove the existence of a wave.
The situation is far more involved for the system of equations (1.6). Here it
is necessary to consider a phase space of dimensionality greater than or equal to
4; application of the method indicated entails essential difficulties. To successfully
apply this method one must deal with a system of special form, possessing specific
properties. It is of interest to note that many systems arising in various physical
problems possess the required properties. Therefore, it is this approach that was
successfully used to prove the existence of wave fronts in various mathematical
models of physics, chemistry, and biology (see Chapter 3, §5).
To prove the existence of a pulse, it is obviously sufficient to establish in the
phase plane (w, p) the existence of a trajectory of system (1.7) leaving and entering
the stationary point (w
+
, 0). In the general case only results obtained with the aid
of bifurcation theory are available: under certain conditions birth of a separatrix
loop from the stationary points may be proved.
One of the general approaches to proving the existence of periodic waves also
yields a theory of bifurcations. The question concerns birth of periodic waves of
small amplitude from constant stationary solutions under a change of parameters
(see below). Another approach to proving the existence of periodic waves is
§2. EXISTENCE OF WAVES 15
connected with an assumption concerning the existence of stable limit cycles for
the system (1.11). Use is made of the small parameter method: for large speeds c
a small parameter can be introduced into system (1.6) so that as c →∞we obtain
system (1.11).
Results concerning the existence of multi-dimensional waves for scalar equations
are presented in the supplement to Chapter 1. Multi-dimensional waves close to
planar waves have been studied by methods of bifurcation theory (see §5).

2.2. Locally-monotone and monotone systems. Scalar equation. In
this subsection we present basic results on the existence of waves of wavefront type
for a class of systems of equations. We assume that the matrix A is a diagonal
matrix and that the vector-valued function F(u) satisfies the conditions
(2.1)
∂F
i
∂u
j
 0,i,j=1, ,n, i= j,
for all u ∈ R
n
. In this case the systems of equations (0.1) and (1.6) are called
monotone systems. Such systems of equations are often encountered in applications
(see §6 and Chapters 8 and 9).
The simplest particular case of such systems is the scalar equation (n =1). If
conditions (2.1) with strong inequalities are satisfied only on the surfaces F
i
(u)=0
(i =1, ,n), the system of equations (0.1) is then said to be locally monotone (a
generalization of this concept is given in §2 of Chapter 3).
As we have already remarked, for wave fronts we assume existence of the lim-
its (1.8) and F (w
±
) = 0. We seek monotone waves (for definiteness, monotonically
decreasing) with the same direction of monotonicity for all components w
i
(x)of
the vector w(x). (Nonmonotone waves for monotone systems are unstable; see
Chapter 5, §6.) Obviously, then, w

+
<w
−
and it is sufficient to require that
inequality (2.1) be satisfied in the interval [w
+
,w
−
], i.e., for w
+
 w  w
−
.
We formulate a theorem for the existence of a wave in the case of a source of
Type A.
Theorem 2.1. Let system (0.1) be monotone. Further, let the vector-valued
function F (u) vanish in a finite number of points u
k
, w
+
 u
k
 w
−
(k =1, ,m).
Let us assume that all the eigenvalues of the matrices F

(w
+
) and F


(w
−
) lie in the
left half-plane, and that the matrices F

(u
k
)(k =1, ,m) are irreducible and have
at least one eigenvalue in the right half-plane. Then there exists a unique monotone
traveling wave, i.e., a constant c and a twice continuously differentiable monotone
vector-valued function w(x) satisfying system (1.6) and the conditions (1.8).
For Type B sources we have the following theorem for the existence of a wave.
Theorem 2.2. Let system (0.1) be monotone. Assume further that the vector-
valued function F (u) vanishes at a finite number of points u
k
, w
+
 u
k
 w
−
(k =1, ,m). Suppose that all eigenvalues of the matrix F

(w
−
) lie in the left
half-plane and that the matrices F

(w

+
), F

(u
k
)(k =1, ,m) have eigenvalues in
the right half-plane. There exists a positive constant c
∗
such that for all c  c
∗
there
exist monotone waves, i.e., solutions of system (1.6) satisfying conditions (1.8).
When c<c
∗
, such waves do not exist. The constant c
∗
is determined with the aid
of a minimax representation (see §4).
Finally, we have the following result for Type C sources (where the system is
not assumed to be monotone).
16 INTRODUCTION. TRAVELING WAVES DESCRIBED BY PARABOLIC SYSTEMS
Theorem 2.3. Let the matrices F

(w
+
) and F

(w
−
) have eigenvalues in the

right half-plane. Then a monotone wave does not exist, i.e., no monotone solution
of system (1.6) exists satisfying conditions (1.8).
For simplicity of exposition these theorems are stated here under conditions
more stringent than necessary (see Chapter 3).
Theorem 2.1 is proved by the Leray-Schauder method and is generalized to a
locally-monotone system (with no assertion regarding uniqueness of the wave). In
Theorem 2.2 existence of solutions is first proved on a semi-axis x  N,andthen
we pass to the limit as N tends to infinity. The last theorem may be proved rather
easily based on an analysis of the sign of the speed for a wave tending towards an
unstable stationary point of system (1.11).
We remark that the existence and the number of waves for monotone systems
is determined by the type of source. For a Type A source a wave exists for a unique
value of the speed; for a Type B source it exists for speed values belonging to a
half-axis; for a Type C source it does not exist.
This generalizes known results for a scalar equation (see Chapter 1), which
are readily obtained from an analysis of trajectories in the phase plane. Moreover,
Theorem 2.3 for a scalar equation is a consequence of a necessary condition for the
existence of waves, a condition which may be formulated in the following way.
For the existence of a solution (c, w) of scalar equation (1.6) with condi-
tions (1.8) and (1.9) it is necessary that one of the following inequalities be satisfied:
w

w
+
F (u) du < 0,
w
−

w
F (u) du > 0,

for all w ∈ (w
+
,w
−
), where it is assumed that w
+
<w
−
. For the case in which
the first of these inequalities is satisfied, the speed c  0; in the case of the second
inequality, c  0. Simultaneous satisfaction of both inequalities for all w ∈ (w
+
,w
−
)
is a necessary and sufficient condition for existence of a wave with zero speed.
A proof of this simple theorem is given in Chapter 1.
As examples we can consider sources shown in Figures 1.2–1.5. For the first
three of these the necessary condition for existence is satisfied; for the fourth it
is not satisfied and, consequently, the wave does not exist. As we shall see later,
this necessary condition for existence of a wave is not a sufficient condition. For
example, for a Type A source (Figure 1.6) it can be satisfied, while a wave with
the limits (1.8), under certain conditions, does not exist. Instead of a wave there
appears a system of waves. Sufficient conditions for the existence of waves for a
scalar equation, which are not encompassed by Theorems 2.1 and 2.2, are discussed
in Chapter 1.
Fairly detailed studies have been made of wave systems for a scalar equation.
We introduce here the concept of a minimal system of waves, which describes the
asymptotic behavior of solutions of a Cauchy problem and which, as will be shown
later, exists for arbitrary sources.

§3. Stability of waves
3.1. Stability and spectrum. One of the most widely used methods for
studying the stability of stationary solutions of nonlinear evolutionary systems is
the method of infinitely small permutations of a stationary solution. As a result of
§3. STABILITY OF WAVES 17
linearizing nonlinear equations we arrive at the problem concerning the spectrum
of a differential operator (call it L) and, therefore, a need to solve two problems:
first, to find the structure of the spectrum of operator L; second, what can be said
concerning stability or instability of a stationary solution, knowing the spectrum
structure. For the case in which the domain of variation of the spatial variables
is bounded (and the system itself satisfies certain conditions, ordinarily met in
applied problems), the spectrum of operator L is a discrete set of eigenvalues, and
a stationary solution is stable if all the eigenvalues have negative real parts (i.e.,
lie in the left half of the complex plane) and unstable if at least one of them has a
positive real part.
A substantially more involved situation arises when we consider stability of
traveling waves. In this case, owing to the unboundedness of the domain of
variation of the spatial variables, the spectrum of operator L includes not only
discrete eigenvalues, but also a continuous spectrum. Moreover, operator L can
have a zero eigenvalue (this is connected with invariance of a traveling wave with
respect to translations). Nevertheless, it proves to be the case that a linear analysis
allows us to make deductions, not only concerning stability or instability of a
traveling wave, but also concerning the form of stability: in some problems we
have ordinary asymptotic stability (with an exponential estimate for the decrease
of perturbations), and in others we have stability with shift.
Stability with shift means that if the initial condition for a Cauchy problem for
the system of equations
(3.1)
∂u
∂t

= A
∂
2
u
∂x
2
+ c
∂u
∂x
+ F (u)
is close to a wave w(x) in some norm, then the solution tends towards the wave
w(x + h)inthisnorm,whereh is a number whose value depends on the choice of the
initial conditions. Stability with shift arises because of the invariance of solutions
with respect to translation and the presence of a zero eigenvalue. These questions
are considered in Chapter 5, where a conditional theorem is proved concerning
stability of traveling waves for the case in which the entire spectrum of the linearized
problem, except for a simple zero eigenvalue, lies in the left half of the complex
plane.
If operator L has eigenvalues in the right half-plane, the wave is unstable.
Let us suppose now that there are points of the continuous spectrum in the right
half-plane. Such a situation is characteristic of the monostable case. A transition
to weighted norms makes it possible to shift the continuous spectrum, something
that was first done in [Sat 1,2]. If in a weighted space the continuous spectrum
and eigenvalues lie in the left half-plane, the wave is then asymptotically stable
with weight. Here stability can be both with shift and without shift, depending on
whether the derivative w

(x) belongs to the weighted space considered.
Thus, there arises a problem concerning the study of the spectrum of an
operator linearized on a wave, which for one-dimensional waves, described by the

system of equations (1.6), has the form
(3.2) Lu = Au

+ cu

+ F

(w(x))u,
where w(x) is a wave. (In Chapter 4 these problems are studied in a somewhat
more general setting.) Here we consider a question concerning structure of the
18 INTRODUCTION. TRAVELING WAVES DESCRIBED BY PARABOLIC SYSTEMS
spectrum of this operator, being restricted to waves having limits at infinity (wave
fronts, pulses):
w
±
= lim
x→±∞
w(x).
The spectrum of operator L consists of a continuous spectrum and eigenvalues.
The continuous spectrum is given by the equations
det(−Aξ
2
+ icξ + F

(w
±
) − λ)=0,
and is located in the half-plane Re λ  b,whereb is some number. It is obvious
that if A is a scalar matrix, then the continuous spectrum is determined by the set
of parabolas

(3.3) −Aξ
2
+ icξ + w
±
k
− λ =0,
where w
±
k
are eigenvalues of the matrix F

(w
±
)(k =1, ,n). From (3.3) it is easy
to see that if all the eigenvalues of the matrices F

(w
±
) lie in the left half-plane,
then the continuous spectrum also lies in the left half-plane. If these matrices have
eigenvalues in the right half-plane, then there are also points of the continuous
spectrum there. But if A is not a scalar matrix, it is then possible to have points
of the continuous spectrum in the right half-plane even when all the eigenvalues of
the matrix F

(w
±
) lie in the left half-plane.
Besides a continuous spectrum, operator L has eigenvalues, also distributed in
some half-plane Re λ  b

1
, with only a finite number of eigenvalues appearing in
the right half-plane. It is easy to see that λ = 0 is an eigenvalue of operator L with
eigenfunction w

(x). Presence of a zero eigenvalue leads, as will become clear later,
to certain peculiarities in the stability of waves.
Here we present briefly basic facts concerning the spectral distribution of
an operator linearized on a wave and its connection with the stability of waves.
It should be noted that if the distribution of the continuous spectrum can be
obtained fairly simply, then determination of the eigenvalues, or conditions for
their determination, in the left half-plane is coupled with great difficulties. Specific
results are available for the study of individual classes of systems. A complete
study has been made of the problem concerning stability of waves for monotone
systems and for the scalar equation, in particular. These results are discussed in
the following subsection.
Papers have appeared in which stability is proved, in the case of scalar equa-
tions, for waves propagating at large speeds [Bel 1]. These approaches readily carry
over to certain classes of systems. In the monostable case, in which wave speeds can
occupy an entire half-axis, this yields stability of waves for speeds larger than some
value. In the bistable case the speed of a wave, only in individual cases, satisfies
the conditions imposed on it connected with properties of the matrix F

.
Yet another approach to the study of the stability of waves is based on asymp-
totic methods. The methods most developed, apparently, are those in combustion
theory [Zel 5]. For many combustion problems presence of a small parameter
is typical; this makes it possible to find, approximately, both a stationary wave
and the boundary of its stability in parametric domains (see the supplement to
Part III). We remark that the propagation of waves of combustion of gases, under

specified conditions, is described by monotone systems. This makes it possible to
apply the theory, developed for such systems, to the description of these processes