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A Variant of Newton's Method for the Computation of Traveling Waves of Bistable Dierential-Dierence Equations Christopher E. Elmer and Erik S. Van Vlecky

June 21, 1999

Abstract. We consider a variant of Newton's method for solving nonlinear dierential-dierence equations arising from the traveling wave equations of a large class of nonlinear evolution equations. Using the Fredholm theory developed by J. Mallet-Paret we prove convergence of the method. The utility of the method is demonstrated with a series of numerical experiments.

1. Introduction. We consider a numerical method to obtain traveling wave solutions to systems of dierential-dierence equations of the form (1.1) u_ (; t) ?
u(; t) = L(u(; t); u( + r1 ; t); :::; u( + rN ; t)) ? f (u(; t); a): The left-hand side of (1.1) is the \dierential" portion of the equation where
represents the Laplacian operator and \ _ " indicates the derivative with respect to time. The operator L is a linear dierence operator that represents nonlocal diusion. The function f is a nonlinear operator of bistable type which depends on the parameter a. The ri 2 IR, i = 1; :::; N , represent \shifts" of the solution function with respect to the spatial variable. Our contribution in this paper is to derive a variant of Newton's method to nd traveling wave solutions of (1.1). Using the Fredholm theory developed by J. Mallet-Paret in [26] we show convergence of the method. Previously in [13] and [14] we derived numerical methods to obtain traveling wave solutions for speci c examples of (1.1). The method in [13] is not generally applicable to a wide range of parameter values, while the method in [14] was based upon the use of a speci c piecewise linear nonlinearity f . Our motivation here is to provide a general method based upon the use of a two-point boundary value solver such as COLNEW [2, 1] or COLMOD [8, 9] to solve at each iteration step. This is the reason for the term
u in (1.1). The method derived here can be thought of as a front end to solve boundary value problems (BVPs) with backward and forward delays using an existing BVP solver. In addition to derivation and convergence analysis of the method, we apply it to several examples, with cubic or logarithmic nonlinearities, that include the nonlinear wave equation considered in [14] and a reaction-diusion system with step function like discrete diusion. Our interest in traveling wave solutions to (1.1) follows from the extensive studies of traveling wave solutions to partial dierential equations such as the reaction-diusion equation (1.2) ut = uxx ? f (u; a); u : (x; t) ! IR for x; t 2 IR; with  > 0 and f : IR ! IR de ned as (1.3) f (u; a) = u(u ? a)(u ? 1); 0 < a < 1; where a is a detuning parameter. Solutions to (1.2) of the form (1.4) u(x; t) = (x ? ct); where c 2 IR is the unknown wave speed, are called traveling wave solutions, [19]. Substituting the traveling wave ansatz (1.4) into the reaction-diusion partial dierential equation (1.2) we obtain (1.5) ?c0 ( ) = 00 ( ) ? f (( ); a)  Material Science and Engineering Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA. y Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, CO 80401, USA. Supported by NIST contract # 43NANB714674, and by NSF grant DMS-9505049. 1

where  = x ? ct 2 IR. Equation (1.2) is de ned P for a one-dimensional spatial domain. If we want x to be in IRn , we replace uxx in (1.2) by
u = nk=1 Dxk xk and substitute the n-dimensional P traveling wave ansatz u(x; t) = (x
 ? ct), where the vector  = (1 ; 2 ; :::; n )T 2 IRn , such that nk=1 k2 = 1, is the normal to the traveling wave front. We once again obtain equation (1.5), which is independent of the direction vector  and dimension n. A spatially discrete analog of the one-dimensional partial dierential equation (1.2) is the dierentialdierence equation

ut (x; t) = [u(x + 1; t) ? 2u(x; t) + u(x ? 1; t)] ? f (u; a); where u : (x; t) ! IR for x 2 ZZ; t 2 IR, with bistable nonlinearity (1.3), which de nes a countably in nite system of ordinary dierential equations indexed by points, x, on a spatial lattice. Substituting the one-dimensional traveling wave ansatz u(x; t) = (x ? ct) into (1.6) we obtain (1.7) ?c( ) = [( + 1) ? 2( ) + ( ? 1)] ? f (u; a): Now suppose that x 2 ZZ n . Then the operator [u(x + 1; t) ? 2u(x; t) + u(x ? 1; tP )] is replaced by an operator that has shifts in each of the n spatial dimensions. A possible example is nk=1 [u(x + ek ; t) ? 2u(x; t) + u(x ? ek ; t)] with ek a unit vector of dimension n with 1 as the kth element. Substituting the n-dimensional traveling wave ansatz u(x; t) = (x
 ? ct) into the n dimensional version of (1.6), we

(1.6)

obtain

(1.8)

?c( ) = 

n X k=1

[( + ek
) ? 2( ) + ( ? ek
)] ? f (u; a);

which, unlike (1.5), does depend on the direction vector  and on the dimension n. An additional property that the traveling wave equations (1.7) and (1.8) exhibit, that their continuous counterparts do not, is propagation failure, failure of the traveling wave to propagate over a nontrivial interval of the detuning parameter a [7]. We are interested in bistable systems, systems where f (u) can be represented by a (cubic-like) nonlinear function with \stable" zeros at u? < u+ , and an \unstable" zero at the detuning parameter a 2 (u? ; u+): In this work we consider three nonlinearities f (u) that are of bistable type which include varying degrees of smoothness. By bistable type we mean that F (u) is double-welled, where F 0 (u) = f (u): The nonlinearities that we investigate are: (1.9) (1.10) and

f1 (u; a) = d1



u

u < a;

(u ? 1) u > a;

f2 (u; a) = d1 u(u ? a)(u ? 1); 

d1 2 IR+ ; d1 2 IR+ ;



(1 ? a) ]; f3(u; a) = d1 [a ? u + d2 log au(1 d1 2 IR+ ; 0 < d2 < a(1 ? a): ? u) The piecewise linear nonlinearity f1 , where u? = 0 and u+ = 1, has the advantage of being linear but has the disadvantage of possessing no spinodal region. The values of u? and of u+ for nonlinearity f2 are also 0 and 1. This is the classical bistable nonlinearity. Nonlinearity f3 is smooth like f2 but it diers from f2 in that u? and u+ depend on the value of a and the choice of d2 . In [7], [14], and [15], we used the linearized nonlinearity f (f1 from equation (1.9)) to represent the the bistable nonlinearity.

(1.11)

This allowed us to produce both analytical (using integral transforms) and numerical solutions (using xed point methods). In this paper we concentrate on a numerical method which allows us to solve dierential-dierence equations with any bistable nonlinearity that possesses a continuous derivative. 2

Considering the equilibrium solutions of u_ (; t) = ?f (u(; t); a); we refer to u = u? and u = u+ as stable and we refer to u = a as unstable. Equilibrium solutions to (1.2) and (1.6) are the simplest solutions to obtain. The solution types that are of interest in this work are solutions, waves, that connect the two stable equilibria at u = u? and u = u+. This requires imposing the boundary conditions (?1)  !?1 lim ( ) = u? ; (1)  lim ( ) = u+ ; !1 to the traveling wave formulations. Although we are solving nonlinear dierential-dierence equations, the numerical method that we present relies on several results for linear dierential operators with an exponential dichotomy and on several results for linear dierential-dierence operators that enjoy behavior similar to an exponential dichotomy. Our method also relies on the fact that our nonlinear dierential-dierence equation can be treated as a (sometimes large) perturbation of a nonlinear dierential equation. While we present the method for a speci c subclass of the class of equations that exhibit a heteroclinic orbits parameterized by one parameter, this method appears to be applicable to a general class of parameterized dierentialdierence equations that connect two points (either in a heteroclinic or homoclinic orbit). Observe that for (1.8), if  6= 0 and c 6= 0, we have a delay equation with both forward and backward delays, a functional dierential equation of mixed type. Often in the modeling community, theory is developed based on the notion that phenomena have a local nature, that phenomena are independent of nonlocal information. Models are based on the variable and the local change of that variable. In reality, this is only a rst approximation approach to many systems. Functional dierential equations of mixed type often allow for a more realistic representation. The original foundation of studying the asymptotic behavior of delay equations which depend on energy functions can be traced back to Volterra. Recently there has been some work on general systems of such mixed functional dierential equations, most notably the work of Rustichini [31], [32] and Mallet-Paret [26], [27]. For the speci c case of mixed functional dierential equations resulting from traveling wave equations we note the seminal work of Keener [24], [25], Zinner [36], [37], and of Weinberger [35]. We also mention the work of Gao [21] in which the coecients of the dierence term were directionally dependent, and the work of Cahn, Mallet-Paret, and Van Vleck [7] on propagation failure and lattice induced anisotropy for traveling wave solutions of two dimensional spatially discrete reaction-diusion equations with a linearized bistable nonlinearity. Shen provides existence and stability results for functional traveling waves when f (u; t) is almost periodic in t [33], [34]. Other investigations into the solutions of mixed equations of the form (1.1) include [11], [16], [23], and [38]. The outline of this paper is as follows. In section 2 we introduce the class of equations we solve, we introduce the numerical method (a modi ed Newton's method) we use to solve this class of equations, and we introduce our main theorem, a convergence result for our modi ed Newton's method. The proof of this theorem relies on detailed functional analysis of the class of equations we are solving. Section 3 consists of the background, de nitions, concepts, and results that we use in the proof of the main theorem and relies heavily on the work of J. Mallet-Paret in [26] and [27]. This section includes analysis based on Fredholm operator theory and asymptotic hyperbolicity. Once these preliminary results are stated we proceed to section 4 and the proof of our convergence theorem. We conclude with a summary and remarks about the class of equations that can be solved with this method, details of our numerical implementation, and several numerical examples that illustrate the robust behaviors of these types of traveling wave equations. 2. A Variant of Newton's Method. We present a numerical method for solving the nonlinear autonomous equation (2.1) ?c'0 ( ) ? '00 ( ) = F ('( ); ) + G('( + r1 ); '( + r2 ); :::; '( + rN ); ); 3

where  2 (0; 1)  U ,  2 IR, c() : IR ! IR, and 2 IR+ [ f0g. Equation (2.1) includes the traveling wave forms of (1.1). To help illustrate the following conditions made on (2.1) we introduce the dierential-dierence equation (2.2) u_ (; t) =
u(; t) + LD u(; t) ? f2 (u(; t); a); where u(; t) maps IRn  IR ! IR, 2 IR+ [ 0, a 2 (0; 1),
is the continuous Laplacian operator P n D , and \_" denotes dierentiation with respect to t. We let L be a dierence Laplacian D i=1 i ;i operator de ned as

LD u(; t) =

n X k=1

"k [u( + ek ; t) ? 2u(; t) + u( ? ek ; t)];

where "k 2 IR+ and ek is the unit vector whose kth element equals 1, for k = 1; :::; n. The traveling wave form of (2.2) is (2.3) ?c'0 ( ) = '00 ( ) + LT '( ) ? f2 ('( ); a) with

LT '( ) =

n X k=1

"k ['( + ek
) ? 2'( ) + '( ? ek
)];

where ' : IR ! IR,  = 
 ? ct and P  is the direction normal to the plane wave front. In this case F ('( ); ) = f2 ('( ); a()) ? 2'( ) nk=1 "P k and G('( + r1 ); '( + r2 ); :::; '( + rN ); ) = nk=1 "k ['( + k ) + '( ? k )]. Let 'rj ( ) = '( + rj ) and let '( ) = ('r1 ( ); 'r2 ( ); :::; 'rN ( )) for j = 1; :::; N . The following is a list of ve conditions for F and G: (c1) F : IR  U ! IR is C 1 in IR and U , and G : IRN  U ! IR is C 1 in IRN and U . (c2) D' F : IR  U ! IR is locally Lipschitz in ' and D' G : IRN  U ! IRN is locally Lipschitz in '. (c3) For j = 1; :::; N @G('; ) > 0: (2.4) @'rj For (2.3), this means that the "k > 0. (c4) The rj , j = 1; :::; N , are the forward and backward shifts. As a matter of notation, let ri 6= rk for 1  i < k  N and rj 6= 0 for j = 1; :::; N . The rj correspond to the ek
 in (2.3) where the shifts occur in plus and minus pairs. (c5) Let '? ; '+ 2 IR such that '? < '+ , and let ? : IR  U ! IR be de ned as ?('; ) = F ('; ) + G('; '; '; :::; '; ): For some a() such that a() 2 ['? ; '+ ], ?('; ) > 0; ' 2 (?1; '? ) [ (a; '+ ); ?('; ) < 0; ' 2 ('? ; a) [ ('+ ; 1); (2.5) ?('? ; ) = ?(a; ) = ?('+ ; ) = 0; with D' ?('? ; ) < 0 if a 6= '? ; D' ?('+ ; ) < 0 if a 6= '+ ; (2.6) D'?(a; ) > 0 if a 2 ('? ; '+ ): 4

For equation (2.3), ?('; ) = f2 ('; a()), '? = 0 and '+ = 1. Assumptions (2.5) and (2.6) imply a bistable nature to (2.1) along with the fact that we have exactly the three equilibrium solutions ' = '? ; a; '+ when a 2 ('? ; '+ ). Assumptions (2.5) and (2.6) also imply that ' = '+ is stable for positive increasing  and that ' = '? is stable for negative decreasing  . In all the examples in this paper the operators F and G can be expressed in the form

F ('( ); ) + G('( + r1 ); '( + r2 ); :::; '( + rN ); ) =

N X i=1

i ['( + ri ) ? '( )] ? f ('( ); )

where f ('( ); ) is one of our bistable nonlinearities, f1 , f2 , or f3 . We are interested in solutions of (2.1), ('; c); that connect '? and '+ . In other words, for each  2 (0; 1), we want to nd G ('; c; ) = 0 for (2.7)

G ('; c; )( ) = ?c'0 ( ) ? '00 ( ) ? F ('( ); ) ? G('( ); );

with boundary conditions

'(+1) = '+ : As we have done in previous works, we de ne the phase condition '(0) = a() to select a unique translate, where a is a monotone increasing function. An advantage gained from (2.5) and (2.6) is that we do not need to consider solutions which join '? and a(), and a() and '+ . (2.8)

'(?1) = '?

and

Let

(2.9)

F ('; c; )( ) = ?c'0 ( ) ? '00 ( ) ? F ('( ); ) ? G('( ); );

where  2 [0; 1) is a relaxation parameter. Since F and G do not depend on c and since F and G are in C 1 , F and G are also C 1 Frechet-dierentiable. Taking the derivative of F with respect to the rst two variables we obtain N

X D1;2 F ('; c; )( ; b; )( ) = ?c 0 ( ) ? 00 ( ) ? D'F ('; ) ( ) ?  [D'rj G('; )] ( + rj ) ? b'0 ( ): j =1

(2.10)

Notice that D1;2 F ('; c; ) is a linear (possibly nonautonomous) operator. Definition 2.1. Using operator (2.10), we de ne, for xed , the modi ed Newton's iteration (2.11)

D1;2 F ('n ; cn ; )('n+1 ; cn+1 ; ) = D1;2 F ('n ; cn ; )('n ; cn ; ) ? G ('n ; cn ; ):

Unless otherwise stated,  is xed throughout this paper. In sections 3 and 4 we construct a proof that shows the modi ed Newton's method in (2.11) converges for ('0 ; c0 ) in an open neighborhood about the solution of (2.1). Let

L1  L1 (IR); let

W 1;1 = ff 2 L1 jf is absolutely continuous and f 0 2 L1 g; and let

W01;1 = ff 2 W 1;1 jf (0) = a()g: 5

Definition 2.2. The pair ('; c) is a point of attraction of the iteration de ned in (2.11) if there is an open neighborhood, S , of ('; c) such that S  W01;1  IR and, for any ('0 ; c0 ) 2 S , the iterates de ned by (2.11) all lie in W01;1  IR and converge to ('; c).

We now present our main result.

Theorem 2.1. (Convergence of Modi ed Newton's Method) Let ('; c) be a pair of functions such

that G ('; c) = 0, with G ('; c) as de ned in (2.7) and boundary conditions (2.8). Also assume that the operators F and G of G ('; c) satisfy conditions (c1) through (c5). Then ('; c) is a point of attraction for the modi ed Newton iteration (2.11).

3. Preliminary Results. The following results on asymptotic hyperbolicity and Fredholm theory follow directly from the work of W.-J. Beyn [5], K.J. Palmer [29], and J. Mallet-Paret [26], [27]. The wave speed c 6= 0 throughout this section. 3.1. De nitions. In this subsection, we de ne what we mean when we call an operator hyperbolic, asymptotically autonomous, or asymptotically hyperbolic. Let c : W01;1 ! L1 be a nonautonomous bounded linear operator of the form (3.1)

N

X (c )( ) = ?c 0 ( ) ? 00 ( ) ? A0 ( ) ( ) ? Aj ( ) ( + rj ): j =1

Definition 3.1. Suppose that the Ai (i = 0; :::; N ) in (3.1) can be written in the form

Ai ( ) = Ai + Bi ( ) where Ai , i = 0; :::; N , are constant coecient operators, such that

Ai+ = lim A ( ); !1 i

Ai? = !?1 lim Ai ( );

i = 0; :::; N:

Then the operator (3.1) is called asymptotically autonomous.

If c is asymptotically autonomous then

c = c ? M ;

where c : W01;1 ! L1 is the autonomous bounded linear operator (3.2)

N

X (c )( ) = ?c 0 ( ) ? 00 ( ) ? A0; ( ) ? Aj ( + rj ) j =1

and M : W01;1 ! L1 such that (3.3)

(M )( ) = B0 ( ) ( ) +

N X j =1

Bj ( ) ( + rj ):

where lim kM+k = 0;

!1

lim kM?k = 0:

6

!?1

A particular case of (3.1) is 0c : W01;1 ! L1 , the autonomous bounded linear operator (3.4)

N

X (0c )( ) = ?c 0 ( ) ? 00 ( ) ? A0;0 ( ) ? Aj;0 ( + rj ) j =1

where Ai;0 , i = 0; :::; N , are constant coecient operators. Consider solutions to 0c = 0 of the form = e v, v 2 IR, v 6= 0. Then for each root s =  of the characteristic equation det
0c (s) = 0, where
0c (s) = ?s c ? s2 ? A0;0 ?

(3.5)

N X j =1

Aj;0 esrj ;

there corresponds a set of eigensolutions to 0c = 0. Definition 3.2. The constant coecient operator 0c (3.4) is called hyperbolic if its characteristic equation (3.5) has no roots on the imaginary axis; i.e, if det
0c (i) 6= 0 for  2 IR, then (3.4) is called

hyperbolic.

We de ne hyperbolicity solely based on the characteristic equation det
0c (s) = 0 and not on the dynamics of the operator or of the solution. Having a constant coecient operator be hyperbolic is an essential element to having an exponential dichotomy. We desire to imply that an asymptotically autonomous operator whose asymptotic limit is an hyperbolic operator that has an exponential dichotomy also exhibits an exponential dichotomy. Definition 3.3. If (3.1) is asymptotically autonomous and (3.2) is hyperbolic, then we call (3.1) asymptotically hyperbolic.

Let

An asymptotically hyperbolic operator allows us to set up a linear Fredholm theory for the operator.

Rc = R(c ) = f 2 L1 j = c for some  2 W

;1 g;

1 0

the range c (3.1), and let

Kc = K(c ) = f 2 W

;1 j

1 0

c  = 0g;

the kernel of c (3.1). The operator c is a Fredholm operator if 1) the kernel Kc  W01;1 is nite dimensional, 2) the range Rc  L1 is closed, and 3) Rc has nite codimension in L1 .

3.2. Results. The following results depend on c being a Fredholm operator which includes knowing speci c information about the kernel and range of c. Theorem 3.1. Let c be de ned as in (3.1) and assume c is asymptotically hyperbolic. Then c is a Fredholm operator. Proof. The result follows from Theorem A in [26] and from Lemma 4.2 in [29]. 7

The formal adjoint operator c of c is de ned N

X (c )( ) = c 0 ( ) ? 00 ( ) ? A0 ( ) ( ) ? Aj ( ? rj ) ( ? rj ); j =1

where Aj is the conjugate transpose of Aj , j = 1; :::; N . Let Rc be the range of adjoint of c and Kc be the kernel of the adjoint of c . Then we have from Theorem A in [26] that 

Rc = h 2 L1 j

Z1

?1

  y( )h( )d = 0 for all y 2 Kc :

We now present two hypotheses which are used in the lemma below. (H1) Let c be de ned as in (3.1) and assume there exist quantities

j ; j 2 IR; 0  j  N; with j > 0; 1  j  N; such that

j  Aj ( )  j 8 2IR; 8 j 2 f1; 2; :::; N g; and

0  A0 ( )  0 8 2 IR; In addition assume N X i=0

Ai < 0:

(H2) Let c be de ned as in (3.1) and assume there exist quantities

j ; j 2 IR; 0  j  N; with j  0; 1  j  N; such that

?j  Aj ( )  j 8 2IR; 8 j 2 f1; 2; :::; N g; and

0  A0 ( )  0 8 2 IR; In addition assume N X i=1

A(0) l < 0:

Lemma 3.1. Let c be de ned as in (3.1) and assume c is asymptotically autonomous. Also assume that there exists a solution to c ( ) = 0 which is nonnegative and bounded. If c satis es either (H1) or (H2), then 1) c is asymptotically hyperbolic, (is a Fredholm operator) 2) there exists an element p in the kernel of c, such that p > 0, 8

3) there exists an element p in the kernel of the adjoint such that each element p ( ) > 0 for all  2 IR, and 4) the range of c contains no elements, h( ), such that h( ) < 0 or h( ) > 0, for all  2 IR. Proof. In this proof, we combine the results of Mallet-Paret, [26], [27], for rst order dierentialdierence operators of the form N

X (H 1 )( ) = ?c 0 ( ) ? A0 ( ) ( ) ? Aj ( ) ( + rj ); j =1

and the results of Fife and McLeod [19] and Beyn [5] for reaction-diusion equations whose linearized form can be represented by operators of the form (H 2 )( ) = ?c 0 ( ) ? 00 ? B0 ( ) ( ):

Consider hypothesis (H1). Then the operator c can be treated as a perturbation of the operator H 1 . For the conditions set forth in the lemma and in (H1), Theorem 4.1 in [27] implies conclusions 1), 2), and 3). Now consider hypothesis (H2). Then the operator c can be treated as a perturbation of the operator H 2 . Results 1), 2), and 3) follow from the fact that this operator has an exponential dichotomy. As for result 4), with either (H1) or (H2), by result 3) there exists an element, p, in Rc such that  p ( ) > 0 for all  2 IR. Since 

Rc = h 2 L1 j any h( ) 2 Rc must satisfy

Z1

?1

  y( )h( )d = 0 for every y 2 Kc ;

Z1

?1

p ( )h( )d = 0:

Hence h( ) 2 Rc cannot be have h( ) < 0 for all  2 IR, or h( ) > 0 for all  2 IR.

4. Proof of Theorem 2.1. We now present the proof of Theorem 2.1. First we establish that

D1;2 F (; b) is invertible in a neighborhood of the solution ('; c). Then we rewrite the Newton iteration

(2.11) as

('n+1 ; cn+1 ) = H ('n ; cn ) where

H (; b) = (; b) ? [D1;2F (; b)]?1 G (; b); which is well-de ned. We next show that H (4.1) is Frechet dierentiable in the neighborhood in which it

(4.1)

is well-de ned. Last, we show that the Newton iteration converges to the solution in our neighborhood, that the solution is a point of attraction. This includes showing that the spectral radius of D1;2 H is less than 1. Let  be xed and assume c 6= 0. Let ('; c) 2 W01;1  IR be such that G ('; c) = 0. And let F and G satisfy conditions (c1) through (c5) throughout this section. Lemma 4.1. There exists a ball of radius  de ned as

S';c; = f(' ; c ) 2 W01;1  IR j k('; c) ? (' ; c )k   for  > 0g  W01;1  IR such that D1;2 F (' ; c ) is invertible for all (' ; c ) 2 S';c; . 9

Proof. It follows from the work of Beyn [5] and Mallet-Paret [26] [27] that D ; F ('; c) is an isomorphism from W ;1  IR onto L1 , and hence is invertible. Set  = k[D ; F ('; c)]? k and let 0 <  < 1=(2). Since D ; F ('; c) is continuous at ('; c), we can choose a  > 0 such that when (' ; c ) 2 S';c; , kD ; F ('; c) ? D ; F (' ; c )k < : Let I be the identity operator from W ;1  IR to W ;1  IR. Since 12

1 0

1

12

12

12

12

1 0

1 0

kI ? [D ; F ('; c)]? D ; F (' ; c )k = k[D ; F ('; c)]? (D ; F ('; c) ? D ; F (' ; c ))k    21 < 1; 1

12

12

1

12

12

12

Neumann's Lemma implies that [D1;2 F ('; c)]?1 D1;2 F (' ; c ), and hence D1;2 F (' ; c ), is invertible. Also

k[D ; F (' ; c )]? k = k[I ? (I ? [D ; F ('; c)]? D ; F (' ; c ))]? [D ; F ('; c)]? k 1 X   ()i = 1 ?  1

12

1

12

1

12

12

1

i=1

for all (' ; c ) 2 S';c; .

Remark 4.1. Thus the operator H : S';c; ! W01;1  IR given by (4.2) H (' ; c ) = (' ; c ) ? [D1;2 F (' ; c )]?1 G (' ; c )

is well de ned.

c is

Lemma 4.2. The operator H (4.2) is Frechet-dierentiable and the derivative with respect to ' and

D1;2 H ('; c) = I ? [D1;2 F ('; c)]?1 D1;2 G ('; c):

Proof. Since G is Frechet-dierentiable at ('; c), kG (' ; c ) ? G ('; c) ? D ; G ('; c)[(' ; c ) ? ('; c)]k  k(' ; c ) ? ('; c)k for all (' ; c ) 2 S';c; . Using the fact that ('; c) = H ('; c), we obtain kH (' ; c ) ? H ('; c) ? [I ? [D ; F ('; c)]? D ; G ('; c)][(' ; c ) ? ('; c)]k 12

12

1

12

= k[D1;2 F ('; c)]?1 D1;2G ('; c)[(' ; c ) ? ('; c)] ? [D1;2 F (' ; c )]?1 G (' ; c )k

 k ? [D ; F (' ; c )]? [G (' ; c ) ? G ('; c) ? D ; G ('; c)((' ; c ) ? ('; c))]k +k[D ; F (' ; c )]? [F ('; c) ? F (' ; c )][D ; F ('; c)]? D ; G ('; c)((' ; c ) ? ('; c))k 1

12

12

12

1

12

1

12

(4.3)  (2 + 22 kD1;2G ('; c)k)k(' ; c ) ? ('; c)k; for all (' ; c ) 2 S';c; . This implies H is Frechet-dierentiable and the derivative of (4.2) about the solution ('; c) is (4.4)

D1;2 H ('; c) = I ? [D1;2 F ('; c)]?1 D1;2 G ('; c) 10

where N X

D1;2 G ('; c)( ; b)( ) = D1;2 F ('; c)( ; b)( ) ? (1 ? )

j =1

[D'rj G(')] ( + rj ):

D'rj G(') is a linear, possibly nonautonomous, operator. Lemma 4.3. Let ^ be the spectral radius of D1;2 H ('; c). Then ^ < 1. Proof. Writing the eigenproblem for D1;2H ('; c) we obtain the equation (1 ? )[D1;2 F ('; c)]?1

N X j =1

[D'rj G(')]( + rj ) ? I (; b) = (0; 0)

where  is the eigenvalue, and (; b) are the eigenfunctions. Rewriting this relation we obtain N X 1 ?  [D'rj G(')]( + rj ) = 0; ?D1;2F ('; c)(; b) +  j =1

) [c0 ( ) + 00 ( ) + D'F (')( )] + (1 ? ) ) [c0 ( ) + 00 ( ) + D'F (')( )] + (1 ? ) (4.5)

N X j =1 N X j =1

[D'rj G(')]( + rj ) + b'0 ( ) = 0;

[D'rj G(')]( + rj ) = ?b'0 ( ):

The right-hand side of (4.5), contains the derivative of the solution to (2.1), '0 ( ), which we know from Lemma 3.1 with (H1), is strictly positive. We now analyze the left-hand side of (4.5) using the results presented in the previous section. Let J (; b) be

J (; b) = [c0 ( ) + 00 ( ) + D' F (')( )] + (1 ? )

N X i=1

[D'ri G(')]( + ri ):

Let A0 ( ) = D' F (') and Aj ( ) = (1 ? )D'rj G('), j = 1; :::; N . Then J , the left-side of (4.5) can be written as c as de ned in (3.1). We now show that c = J satis es (H1) and the assumptions of Lemma 3.1 for   1 where 1 2 (0; 1). By (2.4), 0 < j  Aj ( )  j for all  2 IR, for all j = 1; :::; n. From the conditions on ?('; ) (2.5), (2.6), along P with the boundary conditions (2.8) we have that there exists a 0 < 1 < 1 such that, for  > 1 , Ni=0 Ai < 0 and that J is asymptotically autonomous. Let 1 be the smallest such 1 . Hence by Lemma 3.1, for   1, the range of J contains no strictly positive nor negative sets of elements. Now notice that the right-hand side of (4.5), a constant times the derivative of the solution ', is a set of strictly positive or negative elements. Thus (4.5) has no nontrivial solution for b 6= 0 and   1 . Next we consider  < 0,

jj[c0 ( ) + 00 ( ) + D'F (')( )] ? (1 ? )

N X j =1

[D'rj G(')]( + rj );

and Lemma 3.1 with (H2). For  < 0, the conditions of Lemma 3.1 with (H2) are met for the same reasons the conditions of Lemma 3.1 with (H1) are met for   1 . Thus the right-hand side of (4.5) is 11

not in the range of J , the left-hand side, when  < 0. Hence   1 , for 0 < 1 < 1, and  < 0 are not in the spectral radius of (4.4). In section 4 of [27], Mallet-Paret shows that when  > 1 , the real eigenvalues of (4.5) lie in both the intervals (?1; 0) and (0; 1), and as we saw above, when  > 1 ,  is not in the spectrum of (4.4). He also showed that for 0 <  < 1 , the real eigenvalues of the determinant of the characteristic equation of (4.5) all lie in either (?1; 0) or (0; 1). Suppose 0 <  < 1 , and consider the formal adjoint of the left-hand side of (4.5). This adjoint operator has a kernel, Kc , of dimension zero. Since the operator represented by the left-hand side of (4.5) is a Fredholm operator, the codimension of the range of (4.5) equals the dimension of Kc which is zero. Hence the right-hand side of (4.5) is in the range of the left-hand side. Thus 0 <  < 1 is in the spectrum of (4.4). Remark 4.2. The value 1 , which is both direction and dimension dependent, can be thought of as the contraction factor for our method. Lemma 4.4. For D1;2 H : W01;1 ! W01;1 the limit

lim k[D1;2H ]m k1=m = ^

m!1

exists, where ^ is the spectral radius. Proof. The result is a statement of Theorem 10.13 in Rudin [30]. Lemma 4.5. The solution ('; c) is a point of attraction of (2.11). Proof. By Lemma 4.4, for any  > 0 there exists an integer N^ such that for all m  N^ k[D1;2 H ('; c)]m k1=m  ^ + ;

which implies

k[D ; H ('; c)]m k  (^ + )m : Let H p ( ; b) = H [H (H f:::[H ( ; b)]:::g)]; the operator H applied p times. Observe that H p : W ;1 ! W ;1 and that H p ('; c) = ('; c). Choose  > 0 such that ^ +  < 1, choose p 2 ZZ such that 12

1 0

1

+

1 0

(^ + )p1 +  < 1, and let p > max(N^ ; p1 ). Since H is Frechet-dierentiable at ('; c), H p is also. Thus there exists a  > 0 such that, for all (' ; c ) 2 S';c; ,

kH p (' ; c ) ? H p ('; c) ? D ; H p ('; c)((' ; c ) ? ('; c))k  k(' ; c ) ? ('; c)k: 12

The Frechet derivative of H p ,

D1;2 H p ('; c) = [D1;2 H ('; c)]p ; through application of the chain rule. Thus,

kH p (' ; c ) ? ('; c)k  kH p(' ; c ) ? H p ('; c) ? D ; H p ('; c)((' ; c ) ? ('; c))k + k[D ; H ('; c)]p kk(' ; c ) ? ('; c)k 12

 [(^ + )p + ]k(' ; c ) ? ('; c)k for all (' ; c ) 2 S';c; :

12

12

Therefore, if ('0 ; c0 ) 2 S';c; , k('p ; cp ) ? ('; c)k = kH p('0 ; c0 ) ? ('; c)k  [(^ + )p + ]k('0 ; c0 ) ? ('; c)k; which implies that ('p ; cp ) 2 S';c; . By an induction argument, (4.6) k('n
p ; cn
p ) ? ('; c)k  [(^ + )p + ]n k('0 ; c0 ) ? ('; c)k and ('n
p ; cn
p ) 2 S';c; for all n 2 ZZ + [ 0: Since (^ + )p +  < 1, (4.6) implies that lim (' ; c ) = ('; c): n!1 n
p n
p Since H is a bounded operator, and since (4.6) holds for all p > max(N^ ; p1), lim (' ; c ) = ('; c): n!1 n n This concludes the proof of Theorem 2.1. Remark 4.3. In section 2, we assumed that F and G were C 1 . In proving Theorem 2.1, we only need that G , D1;2G (' ; c ), and D1;2 F (' ; c ) are continuous in a neighborhood of the solution ('; c). Thus we only need (' ; c ) 2 S';c; , D' F , and D'ri G to be continuous. If we de ne f1 (a) = 0 and D' f1 (a) = d1 , then (2.1) with the piecewise linear nonlinearity f1 , appears to be included in the class of

equations that can be solved with this numerical method. But recall that for this problem, with nonlinearity f1 , '00 ( ) is discontinuous at a and the spectral radius argument holds for continuous operators. It is not clear at this time if the proof can be made to hold for nonlinearities such as f1 . The relaxation presented here allows us to solve problems without the use of continuation, by just Newton iteration. In the examples here, and those presented in [15], we always started the iteration with ' equal to the hyperbolic tangent, the wave speed c equal to 1, and  = 0. We also note that these results easily extent to coupled systems. The single equation case was presented for ease of notation. Remark 4.4. To solve at each iteration we use the boundary value problem solver COLNEW. This imposes the requirement that 6= 0, that we always include the '00 term. If we used a functional dierential equation solver, as opposed to a BVP solver, then we could solve for the case when = 0.

5. Numerical Results. This section focuses on several numerical examples which exhibit behavior that we are only beginning to investigate analytically. The algorithms presented in [14] and in sections 2, 3, and 4 provide us with the numerical tools for investigating a large class of both linear and nonlinear traveling wave dierential-dierence equations. In the rst example, Example 1, we revisit the reaction-diusion, wave, and damped wave dierentialdierence equations of [14]. In addition to the nonlinearity f1 , we also consider the bistable nonlinearities f2(') = d1 '(' ? a)(' ? 1); 



? 1) ): f3 (') = d1 (a ? ' + 0:1 log a'(('a ? 1)

We discover that the phenomena of propagation failure and step-like solution pro les occur for all three nonlinearities and are not artifacts of the piecewise continuous nonlinearity f1. In the second example we present wave speed c() plots, where  is the direction through the two dimensional integer lattice, for the ve point star and the nine point box versions of the spatially discrete diusion operator. This allows us to explicitly show the lattice anisotropy inherent in the discrete operator, and to show how the anisotropy changes with the detuning parameter a. We nish with an example which demonstrates several approximations to the one dimensional spatially discrete problem where the spatial diusion has two values, one on each half of the real line. 13

5.1. Example 1: Nonlinear Dierential-Dierence Equations of Reaction-Diusion, Wave, and Damped Wave Type. In this example, we consider the traveling wave equation (2.2),

which is of the form (5.1) where

?c'0 ( ) ? c'00 ( ) = LD '( ) ? f ('( ); a); LD '( ) =

P

n X k=1

"k ['( ? k ) ? 2'( ) + '( + k )];

with nk=1 k2 = 1 and k 6= 0, k = 1; :::n. Comparing Equation (5.1) with the general dierential-dierence form (2.1), the operator

F ('; ) = ?2 the operator

G('; ) =

n X k=1

n X

k=1

"k '( ) ? f ('( ); a);

"k ['( ? k ) + '( + k )]

and is linear, and a() =  = a. The condition that F 2 C 1 requires that we use a C 1 nonlinearity f when solving with the Newton's method. We have experimented with using this method to solve (5.1) with f1 with limited success. There are ve conditions imposed on F and G for convergence of the Newton's method. Equation (5.1) satis es conditions (1) through (4) with condition (5) satis ed by (5.1) with nonlinearities f2 and f3 . Since in [14] we presented several numerical experiments for this problem with nonlinearity f1 , we concentrate here on the comparison of these results with results obtained with the cubic nonlinearity f2 and the logarithmic nonlinearity f3 . The numerical solutions with the nonlinearities f1 , f2 , and f3 , presented in this section, are obtained using the Newton's method presented in sections 2, 3, and 4. 5.1.1. Step-Like Wave Pro les. We begin our presentation with Figures 5.1 and 5.2. The problems solved to produce the solution curves in these two gures are identical except for the scaling of the nonlinearities. In Figure 5.1(a), we plot the three nonlinearities f1 , f2 , and f3, all of which have maximum and minimum values that are approximately the same (the same order of magnitude). In addition, f2 and f3 have approximately the same slope at a. Figure 5.1(b) shows the solution curves, '( ), for (5.1) for each of the three nonlinearities with a = 1=2. Remark 5.1. In the equation at the bottom of Figure 5.1 we claim that c = 0. In actuality, because of numerical error, c  10?7. Remark 5.2. In the solution plot of for Equation (5.1), Figure 5.1(b), the step phenomenon appears regardless of which nonlinearity is used. This implies that the step-like solution pro le is not generated by the jump discontinuity in the nonlinearity f1 , but is generated by the nonlocal nature of the spatially discrete diusion term. In Figure 5.2(a), we have the same relationships between f1 , f2, and f3 as in Figure 5.1(a), only now the magnitude is about a factor of 12 smaller. Remark 5.3. In the solution plot of Figure 5.2(b) we see that the solution curves for f2 and f3 no longer have a step like behavior. Comparing Figures 5.1 and 5.2 gives us our rst indication that the slope of the nonlinearity at the detuning parameter a eects the solution pro le. 5.1.2. The Cubic Nonlinearity. In Figure 5.3 we solve (5.1) with the cubic nonlinearity f2. A set of solution curves, for various values of the detuning parameter a, is plotted in plot(a) and the a(c) curve for this problem is plotted in plot (b). Remark 5.4.

14

f (') 1 f1 f2 f3

0.8 0.6 0.4 0.2

(a)

0 -0.2 -0.4 -0.6 -0.8

1

'

10



-1 0

0.1

0.2

0.3

0.4

0.5



' < 0:5 f1 ('; 0:5) = 2 ' '; ? 1; ' > 0:5

0.6

0.7

0.8

0.9

f2 ('; 0:5) = 12'(' ? 0:5)(' ? 1)

  f3 ('; 0:0) = 4[0:5 ? ' + 0:1 log 1?'' ]

'( ) 1 Curve 1 Curve 2 Curve 3

0.9 0.8 0.7 0.6

(b)

0.5 0.4 0.3 0.2 0.1 0 -10

-8

-6

-4

-2

0

2

4

6

8

. Example 1: Step-like solution pro les and nonlinearity plots for nonlinearities f1 , f2 , and f3 .

Fig. 5.1

i) We present only solution curves for a  0:5 in Figure 5.3 because of the symmetry these solutions possess with the solutions for a  0:5. ii) In [14], for the nonlinearity f1 , we presented a(c) plots which admitted a nontrivial interval of a in which c = 0. We also learned that this propagation failure only occurs when = 0, but we saw that for small, a(c) curves possess an interval of a (approximately the same interval as for propagation failure) for which c is small. In the a(c) plot, Figure 5.3(b), we once again see this 15

f (') 0.08 f1 f2 f3

0.06

0.04

0.02

(a)

0

-0.02

-0.04

-0.06

1

'

10



-0.08 0

0.1

0.2

0.3

0.4

0.5



' < 0:5 f1('; 0:5) = 0:16 ' '; ? 1; ' > 0:5

0.6

0.7

0.8

0.9

f2 ('; 0:5) = '(' ? 0:5)(' ? 1)

  f3 ('; 0:5) = 0:4[0:5 ? ' + 0:1 log 1?'' ]

'( ) 1 Curve 1 Curve 2 Curve 3

0.9 0.8 0.7 0.6

(b)

0.5 0.4 0.3 0.2 0.1 0 -10

-8

-6

-4

-2

0

2

4

6

8

. Example 1: A plot of nonlinearities and a plot of solutions.

Fig. 5.2

behavior, where for = 10?4, there exists an interval of a for which jcj < 10?3. This suggests that we may also have propagation failure for = 0 when solving with the cubic nonlinearity f2 . This also suggests that propagation failure is due to the discrete diusion term, not the discontinuity in the nonlinearity f1 . Referring to both the solution '( ) plot and the a(c) plot in Figure 5.3, we point out three main types of solution pro les that can be classi ed by ja ? 1=2j. The rst type (solutions C6 and C7 in Figure 16

'( ) 1 C1 C2 C3 C4 C5 C6 C7

0.8

: : : : : : :

a a a a a a a

= = = = = = =

0.50 0.54 0.55 0.65 0.75 0.85 0.95

0.6

(a) 0.4

0.2



0 -4

-2

0

2

4

?c'0 ( ) ? 10? '00 ( ) = ['( + 1) ? 2'( ) + '( ? 1)] ? 10'(' ? a)(' ? 1) 4

a(c) 1 0.9 0.8 0.7 0.6

(b)

0.5 0.4

a(c) a(c) a(c) a(c) a(c) a(c) a(c) a(c)

0.3 0.2

Curve for C1 for C2 for C3 for C4 for C5 for C6 for C7

0.1 0 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

c

Fig. 5.3. Example 1: The spatially discrete/continuous reaction-diusion equation with the cubic nonlinearity and various values of the detuning parameter a.

5.3) are solutions that have a hyperbolic tangent shape. The values of a are the farthest from 1=2 for this type. Remark 5.5. As ja ? 1=2j increases towards 1=2, the magnitude of the wave speed jcj becomes large. This rst type of solution pro le, the large wave speed case, is typically what is studied when traveling wave solutions are considered. The third type of pro le (solutions C1, C2, and C3) consists of solutions whose pro le exhibit step-like behavior. Away from the wave front (the internal layer), the tails of these solutions decay asymptotically. The values of a for this type de ne an interval about 1=2. 17

Remark 5.6.

i) The interval for a for which c is small corresponds to the solutions that are step-like. ii) Existing work on propagation failure considers solutions of this type. The second type of solution pro le (solutions C4 and C5) consists of \transition" solutions, solutions that contain a mix of elements from the rst and third types of solution pro les. The distance from 1=2, ja ? 1=2j, for the values of a for this type of solution pro le is greater than ja ? 1=2j for the values of a of the third type and is less than ja ? 1=2j for the values of a of the rst type. Remark 5.7. The second type of solution pro les is least often considered when traveling wave solutions are sought. 5.1.3. The Logarithmic Nonlinearity. Continuing with our rst example, in Figure 5.4 we solve (5.1) with the same parameter values as for the cubic case above, Figure 5.3, only now we use the logarithmic nonlinearity f3 . Remark 5.8. The a(c) plot (bottom plot) of Figure 5.4, has an interval of a values for which c is small, a region that may imply propagation failure as in the cubic case. These values of a correspond to the step-like solution pro les, Figure 5.4(a). Remark 5.9. The main dierence between the solutions which appear in Figure 5.4 and the solutions which appear in Figure 5.3 is that the stable equilibria of (5.1) with nonlinearity f3 depend on the value of the unstable equilibrium a while the stable equilibria, 0 and 1, for (5.1) with nonlinearity f2 do not depend on a. 5.2. Example 2: Comparison Five Point and Nine Point Stencils. In this example we numerically solve the equation ?c'0 ( ) = '00 ( ) + L '( ) ? f2('; a); with d1 = 10 and L = L5 or L9 , to compare the directional dependence of the wave speed for the ve point star discrete Laplacian L5 '( ) = ['( + 1 ) + '( ? 1 ) + '( + 2 ) + '( ? 2 ) ? 4'( )] and the nine point box L9'( ) = 6 [4'( + 1 ) + 4'( ? 1 ) + 4'( + 2 ) + 4'( ? 2 ) +'( + 1 + 2 ) + '( ? 1 + 2 ) + '( + 1 ? 2 ) + '( ? 1 ? 2 ) ? 20'( )]: In these discretizations 1 = cos() and 2 = sin(), where  is the direction that the traveling wave is

owing through the lattice. Figure 5.5(i){(vi) is a series of wave speed c versus direction angle  plots, each plot representing a dierent value of the detuning parameter a. We see that as a increases from 1=2, the average wave speed c increases and c becomes more isotropic for both L5 and L9. All six plots also show that c for L5 is always greater than c for L9 . These plots are meant to give an indication about the shape of a crystal growing with a primitive square lattice. The value of the detuning parameter a can be thought of as the free energy available for growth. Figure 5.5(iii) shows us that for a = 0:55, we might expect the crystals to grow into squares. For a = 0:54, with L5 we again would expect a square, but with L9 we should expect an octagon. 5.3. Example 3: Step Function Diusion Coecient. In this next example we consider the lattice dierential equation (5.2) u_ j (t) = j+1 [uj+1 (t) ? uj (t)] + j [uj?1 (t) ? uj (t)] ? f (uj (t); a) where uj : IR ! IR, f (uj (t); a) = 10uj (uj ? a)(uj ? 1), and 8 j  ?1; < b1 j = 0; j = : b2 b3 j  1; 18

'( ) 1 0.9 0.8 0.7

C1 C2 C3 C4 C5 C6 C7

0.6

(a)

0.5

: : : : : : :

a a a a a a a

= = = = = = =

0.50 0.53 0.535 0.54 0.65 0.75 0.85

0.4 0.3 0.2 0.1 0 -4

-2

0

2

4

?c'0 ( ) ? 10? '00 ( ) = ['( + 1) ? 2'( ) + '( ? 1)] ? 4(a ? ' + 0:1 log 4

h



i a?1)' ) a('?1)

(

a(c) 0.9

0.8

0.7

0.6

(b)

0.5

a(c) a(c) a(c) a(c) a(c) a(c) a(c) a(c)

0.4

0.3

Curve for C7 for C6 for C5 for C4 for C3 for C2 for C1

0.2

0.1 -1.5

-1

-0.5

0

0.5

1

1.5

c

Fig. 5.4. Example 1: The spatially discrete/continuous reaction-diusion equation with the logarithmic nonlinearity and various values of the detuning parameter a.

with b1 ; b2 ; and b3 2 IR: This problem describes a reaction-diusion system where the diusion coecient has one value for the left half of the lattice, another value for the right half of the lattice, and a third value to describe the diusion between the two half lattices. Substituting the traveling wave ansatz R uj (t) = '(j ? tt0 cj (s)ds)

(5.3) into (5.2) we obtain (5.4)

?cj (t)'0 (j ) = j ['(j ) ? '(j )] + j ['(j? ) ? '(j )] ? f ('(j ; a)); +1

+1

19

1

0.15 0.1

0.2 5 star 9 box

0.15

5 star 9 box

0.1 0.05 0.05

(i)

(ii)

0 0.15

0 0.05

0.05 0.1 0.1 0.15

0.2

0.15

a = 0:53

0.2

5 star 9 box

0.4

0.1

(iii)

5 star 9 box

0.2

(iv)

0

0 0.2

0.1

0.2

a = 0:54

0.4

a = 0:55

a = 0:6 2

1

5 star 9 box

1.5

5 star 9 box

1

0.5

0.5

(v)

(vi)

0

0 0.5

0.5

1

1 1.5

a = 0:7

2

a = 0:8

Fig. 5.5. Example 2: c( ) polar plots: A comparison of the wave speeds for the spatially discrete reaction-diusion equation with the ve point star and the nine point box discretizations for various values of the detuning parameter a.

R

where j = j ? tt0 cj (s)ds. This traveling wave ansatz allows each point in the lattice to have its own wave speed cj (t). Fixing t0 = t = 0, de ne (5.5) j (x) = '(x ? j ); (for a more detailed explanation of this traveling wave ansatz (5.3) and function substitution (5.5) see [15]). We now have an in nite system of equations indexed by j 2 ZZ . Since we wish to demonstrate 20

(x) 1 0.9 0.8 0.7 0.6 0.5 0.4

j j j j

0.3

< = = >

0, 0, 1, 1,

c c c c

= = = =

0.2002 0.1440 0.2049 0.3310

0.2 0.1 0 -3

-2

-1

0

1

2

3

x

.

Fig. 5.6

(x) 1 0.9 0.8 0.7 0.6 0.5 0.4

j j j j j j

0.3

< = = = = >

-1, -1, 0, 1, 2, 2,

c c c c c c

= = = = = =

0.2002 0.2004 0.1428 0.2191 0.3700 0.3310

0.2 0.1 0 -3

-2

-1

0

.

1

2

3

x

Fig. 5.7

our numerical algorithm we make the assumption that if we are far enough away from the change in the diusion coecient that the wave speed and in fact the solution are not in uenced by the change in diusion. Figures 5.6{5.9 are sets of solutions to (5.4) for j = 0:75; j < 0; 0 = 0:85, and j = 1; j > 0. The dierence in the solutions presented in the four gures is what we assume to be far enough. In Figure 5.6 we assume that we have four possible values of ck (t), k = ?1; 0; 1; 2. We assume that the solution pairs (?1 ; c?1 ) and (2 ; c2 ) are independent of the other solution pairs and each other. In Figure 5.7 we assume that we have six possible values of ck (t), k = ?2; ?1; 0; 1; 2; 3. We assume the solution pairs (?2 ; c?2 ) and (3 ; c3 ) are independent. In Figure 5.8 we assume that we have eight possible values of 21

(x) 1 0.9 0.8 0.7 0.6 0.5 0.4

j j j j j j j j

0.3 0.2

< = = = = = = >

-2, -2, -1, 0, 1, 2, 3, 3,

c c c c c c c c

= = = = = = = =

0.2002 0.2002 0.2004 0.1428 0.2194 0.3774 0.3021 0.3310

0.1 0 -3

-2

-1

0

1

2

3

x

.

Fig. 5.8

(x) 1 0.9 0.8 0.7 0.6 0.5 0.4

j j j j j j j j j j

0.3 0.2

< = = = = = = = = >

-3, -3, -2, -1, 0, 1, 2, 3, 4, 4,

c c c c c c c c c c

= = = = = = = = = =

0.2002 0.2002 0.2002 0.2004 0.1428 0.2194 0.3778 0.2973 0.3509 0.3310

0.1 0 -3

-2

-1

0

.

1

2

3

x

Fig. 5.9

ck (t), k = ?3; ?2; ?1; 0; 1; 2; 3; 4. We assume the solution pairs (?3 ; c?3 ) and (4 ; c4 ) are independent. And in Figure 5.9 we assume that we have ten possible values of ck (t), k = ?4; ?3; ?2; ?1; 0; 1; 2; 3; 4; 5. We assume the solution pairs (?4 ; c?4 ) and (5 ; c5 ) are independent.

6. Conclusions. We presented a numerical method for nonlinear systems of bistable traveling wave reaction-diusion dierential-dierence equations. The key to this method is treating the forward and backward delay terms as \known" information, and then solving the problem by iterating. The iterative method used is Newton's since we wished to work with the linear variation of the nonlinear system. Convergence of this iterative method was proved through the use of linear Fredholm operator theory which is a way to formulate the exponential-dichotomy-like behavior that the systems exhibit. 22

The primary limitation of our iterative technique is that the second derivative term '00 is needed so that we can use a standard BVP solver. This limitation, among others, indicates the need to develop a dierential-dierence solver. The primary advantages of our relaxation method are that 1) we can use \o the shelf" boundary value problem solvers, 2) we do not need to use continuation from a known solution, and 3) we can solve two boundary dierential-dierence equations without the '00 term with a functional dierential equation solver. We have provided a sampling of numerical examples, showing some of the versatility of the method. The proof of Newton's method helped in producing numerical results by providing guidance in applying the method. Through these examples, we have seen that propagation failure is not an artifact of a discontinuous nonlinearity and that the slope at a aects the solution pro le. The step-like solution behavior rst exhibited in [14], when solving with nonlinearity f1 , also appears when solving with smooth nonlinearities. We have also seen directional anisotropy based on direction and on the detuning parameter a for two possible discrete Laplacian operators, the ve point star and the nine point box, which agree with existing theory of crystal growth for primitive cubic lattice materials. We nish the numerical study by looking at approximate solutions of waves traveling in a domain consisting of two adjacent lattices with diering diusion coecients. REFERENCES [1] U. M. Ascher and G. Bader, Stability of Collocation at Gaussian Points, SIAM J. Numer. Anal. 23 (1986), 412{422. [2] U. Ascher, J. Christiansen, and R.D. Russell, Collocation Software for Boundary-Value Odes, Acm Trans. Math Software 7 (1981), 209-222. [3] J. Bell, Some Threshold Results for Models of Myelinated Nerves, Math. Biosciences 54 (1981) 181{190. [4] J. Bell and C. Cosner, Threshold Behavior and Propagation for Nonlinear Dierential-Dierence Systems Motivated by Modeling Myelinated Axons, Quart. Appl. Math. 42 (1984) 1{114. [5] W.J. Beyn, The Numerical Computation of Connecting Orbits in Dynamical Systems, IMA J. Numer. Anal. 9 (1990) 379{405. [6] J.W. Cahn, S.-N. Chow, and E.S. Van Vleck, Spatially Discrete Nonlinear Diusion Equations, Rocky Mountain J. Math. 25 (1995) 87{117. [7] J.W. Cahn, J. Mallet-Paret, and E.S. Van Vleck, Traveling Wave Solutions for Systems of ODE's on a Two-Dimensional Spatial Lattice, SIAM J. Appld. Math. 59 (1999), 455{493. [8] J.R. Cash, G. Moore, and R.W. Wright, An Automatic Continuation Strategy for the Solution of Singularly Perturbed Linear Two-Point Boundary Value Problems, J. Comp. Phys. 122 (1995), 266{279. [9] J.R. Cash, G. Moore, and R.W. Wright, An Automatic Continuation Strategy for the Solution of Singularly Perturbed Nonlinear Two-Point Boundary Value Problems, to appear. [10] H. Chi, J. Bell, and B. Hassard, Numerical Solution of a Nonlinear Advance-Delay-Dierential Equation from Nerve Conduction Theory, J. Math. Biol. 24 (1986) 583-601. [11] S.-N. Chow and W. Shen, Stability and Bifurcation of Traveling Wave Solutions in Coupled Map Lattices, Dyn. Syst. Appl. 4 (1995) 1{26. [12] E.J. Doedel and M.J. Friedman, Numerical Computation of Heteroclinic Orbits, J. Comp. Appl. Math. 25 (1989) 155{171. [13] C.E. Elmer and E.S. Van Vleck, Computation of Traveling Wave Solutions for Spatially Discrete Bistable ReactionDiusion Equations, Appld. Numer. Math. 20 (1996) 157{169. [14] C.E. Elmer and E.S. Van Vleck, Analysis and Computation of Traveling Wave Solutions of Bistable DierentialDierence Equations, Nonlinearity 12 (1999) 771{798. [15] C.E. Elmer and E.S. Van Vleck, Traveling Wave Solutions of Bistable Dierential-Dierence Equations with Periodic Diusion, submitted (1999). [16] G. Fath, Propagation Failure of Traveling Waves in a Discrete Bistable Medium, Physica D 116 (1998) 176{190. [17] P. C. Fife, \Diusive Waves in Inhomogeneous Media," Proc. Edinburgh Math. Soc. 32, (1989), pp. 291{315. [18] P.C. Fife and L. Hsiao, The Generation and Propagation of Internal Layers, Num. Anal. TMA 12 (1988) 19{41. [19] P. Fife and J. McLeod, The Approach of Solutions of Nonlinear Diusion Equations to Traveling Front Solutions, Arch. Rat. Mech. Anal. 65 (1977) 333{361. [20] M.J. Friedman and E.J. Doedel, Numerical Computation and Continuation of Invariant Manifolds Connecting Fixed Points, SIAM J. Numer. Anal. 28 (1991) 789{808. [21] W.-Z. Gao, Threshold Behavior and Propagation for a Dierential-Dierence System, SIAM J. Math. Anal. 24 (1993) 89{115. [22] J.K. Hale, Theory of Functional Dierential Equations (Springer-Verlag, New York, NY, 1977). 23

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