Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation

Transcription

1 Can a traeling wae connect two nstable states? The case of the nonlocal Fisher eqation Grégoire Nadin, Benoît Perthame, Min Tang To cite this ersion: Grégoire Nadin, Benoît Perthame, Min Tang. Can a traeling wae connect two nstable states? The case of the nonlocal Fisher eqation.. <hal-57986> HAL Id: hal Sbmitted on 9 No HAL is a mlti-disciplinary open access archie for the deposit and dissemination of scientific research docments, whether they are pblished or not. The docments may come from teaching and research instittions in France or abroad, or from pblic or priate research centers. L archie oerte plridisciplinaire HAL, est destinée a dépôt et à la diffsion de docments scientifiqes de niea recherche, pbliés o non, émanant des établissements d enseignement et de recherche français o étrangers, des laboratoires pblics o priés.

2 Can a traeling wae connect two nstable states? The case of the nonlocal Fisher eqation Grégoire Nadin Benoît Perthame Min Tang Noember, Abstract This note inestigates the properties of the traeling waes soltions of the nonlocal Fisher eqation. The eistence of sch soltions has been proed recently in [] bt their asymptotic behaior was still nclear. We se here a new nmerical approimation of these traeling waes which shows that some traeling waes connect the two homogeneos steady states and, which is a striking fact since is dynamically nstable and is nstable in the sense of Tring. Key-words Nonlocal Fisher eqation, Tring instability, traeling waes. AMS Sbject Classification: 5B6, 9B5, 7M99. Introdction For the semilinear eqation t = f(), f() = f() =, () atraelingwaecanconnectthetwosteadystates and inariossitations. This is the case when is nstable and is stable (Fisher/monostable). This is also the case when and are both stable (Allen-Cahn/bistable). It is known that these waes are attractie and are obtained as the long time limit of the dynamics () associated with compactly spported initial data (see [6]). When and are nstable, in between there is a stable steady state of f which preents any traeling wae to eists. For systems and non-local eqations, the classification becomes more complicated becase a steady state can be Tring nstable, which means that is nstable with respect to some periodic pertrbations in a bonded range of periods. In this note we consider the nonlocal Fisher eqation (), the simplest to prodce Tring instability. t = +µ( φ ), φ () = ( y)φ(y)dy, () UPMC, CNRS, UMR 7598, Laboratoire Jacqes-Lois Lions, F-755, Paris. nadin@ann.jssie.fr INRIA Paris-Rocqencort, Eqipe BANG. benoit.perthame@pmc.fr mintang@ann.jssie.fr R

3 t t t t m= m= a) m= m= b) Figre : Nmerical simlations of the time eoltion for the nonlocal Fisher/monostable eqation () with kernel (5). The comptational domain is [ 8, 8]. Left: the isoales show that it is not a traeling wae bt a more complicated strctre; the bottom sbplots are zooms of the top sbplots. Right: the fnction L connects the (dynamically) nstable state with a periodic tail at =. Two ales of µ are sed a) µ = ; b) µ =. with µ >, φ(), φ() >, R φ()d =. () The steady state can be Tring nstable when the Forier transform of φ changes sign and µ is largeenogh (see [, 4, ]). This creates seeral differences with the monostable or bistable eqations. First of all, one can obsere that the soltion of () associated with a compactly spported initial datm does not conerge, for large times, toward the traeling wae bt it conerges towards a more complicated strctre (see [], [4] and [5]). The nmerical simlation presented in b) of Figre 4 shows that the soltion of the eoltion eqation conerges to a plsating front, that is a fnction ( σt,) which is periodic in its second ariable. These types of fronts typically arise in the framework of reaction-diffsion eqations with periodic coefficients ([], [7]). In a) of Figre 4, the soltion seems to conerge to the sperposition of a traeling wae and a plsating front, with two different speeds. These periodic patterns are a symptom of Tring instability on the fll line R. Secondly, it is proed in [] that a traeling wae (,t) = ( σt) always

4 eists for all σ µ, with a generalized formlation { σ + +µ( φ ) =, R, liminf () >, (+ ) =. (4) The athors were only able to obtain, de to the nonlocal effect, the weak bondary condition at = rather than the epected condition ( ) =. When µ is small enogh or when the Forier transform of the kernel F(φ) is positie eerywhere, the traeling wae connects to. Bt this leaes open the qestion to know whether for µ large, the traeling wae soltion of (5) connects the (dynamically) nstable state to a stable periodic state or to the Tring nstable state. Also, when F(φ) can take negatie ales, it is proed in [] that for µ large enogh monotonic traeling waes cannot eist. In the seqel, we will firstly focs on the case σ = µ and φ() = [,](). (5) Then we also test two other cases ( φ() = +β ) [ β,β], β = ( ) (6a) 4 φ() = ( 4 + ) [,] () (6b) In all these cases, F(φ)(ξ) takes negatie ales. For instance, for(5), F(φ)(ξ) = sin ξ/ξ. Finally, to complete the tests, the traeling waes for the Gassian kernel φ() = ) ep (, (7) π which has positie Forier transform, are displayed. In section, we deelop a specific algorithm which allows s to bild the traelingwae(4) and not the plsating front. Then, in section in, we perform nmerical simlations to decide which alternatie holds tre. The algorithm Nmerically one can only sole the problem on a bonded domain of length L = r l (this is also the analytical constrction in []) { σ L L + L +µ L ( φ L ) =, l < < r, L ( l ) =, L ( r ) =, L (8) () = ǫ, The conoltion is compted by etending L by on (, l ) and on ( r,+ ). The parameter ǫ, small enogh, is needed for technical reasons bt intitiely its ale has to be below the oscillations obsered in Figre. Or algorithm for soling (8) is to diide the comptational domain into two parts: I = [ l,] and I = [, r ]. Being gien σ, in each interal an elliptic eqation with Dirichlet bondary conditions is soled σ i = i +µ i ( φ ), i () = ǫ, i =,, ( l ) =, ( r ) =. (9)

5 The conoltion term is compted by defining as on I, and I, on (, l ) and on ( r,+ ). Bt the eqation does not necessarily hold tre at =. We define σ L so as to impose that the jmp of deriaties at zero anishes σ L = [ ( r ) ( l )]+ r l µ( φ )d. () Lemma.. When (σ L, L,L ) satisfies (9) and () simltaneosly, then L is C on ( l, r ) and satisfies (8). We can write abstractly this problem as a fied point for a system of two eqations (σ, ) = (P(), T(σ)). It is straightforward to make it discrete sing finite differences. The most efficient way to sole it is to se Newton iterations. The nmerical reslts. Conergence of the scheme The nmerical reslts we present in this section are obtained with the hat fnction φ in (5), ǫ =. and we stdy the effect of the bifrcation parameter µ. The diffsion term is treated implicitly by centered three point finite difference while the reaction term is pt eplicit. In order to erify that the iteratie scheme described in section conerges to the right soltion, a crcial qantity to look at is the trncation errors at zero E() = L L +L +σ LL L + L ( (φ L ) ). Here L,L,L are the ales of L at the grid points,,. The conergence reslts are displayed in Table. One can see that E() conerges to zero as, which shows that or nmerical reslts is a good approimation of (8) on the whole comptational domain. Note that better accracy of E() can be obtained if we se higher order nmerical integration methods for the conoltion term. L E() σ L L E() σ L L E() σ L Table : Conergence of the trncation error at zero E() and of the traeling elocity σ L for arios ales of and L, with kernel (5) and µ = 64. For L = the speed is σ = µ = 6. 4

6 5 5 m= m= m= Figre : The traeling wae soltion for the nonlocal Fisher eqation (4) with kernel (5). Left: the nmerical reslts for µ = and µ =. Right: the reslts for µ = 5, the top sbplot depicts while the bottom sbplot is a zoom of the tail.. Conergence of the traeling waes to The traeling wae shapes for µ =, µ = and µ = 5 for kernel (5) are depicted in Figre. When µ =, we obsere a monotone traeling wae which connects to, as for the local Fisher eqation. When µ grows, some oscillations appear. Nmerically, when we increase L, the amplitdes of the tail decrease and the bigger µ is, the slower the amplitdes decrease. The shapes of sggest that thogh is Tring nstable with the kernel (5) when µ = and µ = 5, the traeling waes will still connect to. We do not obtain the same type of strctre than when we compte the soltion of the eoltion eqation depicted in Figre 4. This means that there eist some traeling waes that connect to, bt that these waes do not attract the soltion of the Cachy problem associated with compactly spported initial data.. Monotonicity of the traeling waes Lastly, we consider the critical ale of µ for which the monotonicity of the traeling waes is broken. Since the monotone traeling waes always connect to, we perform the linearization, close to =, by assming e λ withλarealpositienmber. Afterinsertingthisforminto(4),singσ = µ and the smallness of e λ, λ can be determined by µ throgh the eqation e λ e λ µ λ µ λ =, () λ a qadratic eqation for µ, which gies µ = λ +λ λ +λsinhλ sinhλ. Ths the ( critical µ that makes λ no longer eist is µ c = sp λ +λ ) λ +λsinhλ λ> sinhλ 8.9. We hae checked that this threshold µ c 8.9 is correct on the nmerical ales. The maimm of L is when µ = 9, bt eceeds slightly when We do not know if sch a linearization is legitimate, bt the technical argments sed in [] to proe non-monotonicity for large µ are close to a linearization 5

7 m= m= Figre : The traeling wae soltion for the nonlocal Fisher eqation (4) with µ = and kernels in (6). Left: the nmerical reslts with kernel (6a). Right: the reslts for (6b). In all these pictres, the top sbplot depicts while the bottom sbplot is a zoom of the tail..5 m= Figre 4: The shapes of traeling wae soltion for the nonlocal Fisher/KPP eqation with Gassian kernel (7) when µ =. µ =. In Figre with µ =, the wae is nearly monotonic, bt, checking the nmerical ales, the maimm of is.8. This indicates that actally might be not monotone een for L finite. In [] the athors hae proed that, when µ > µ c, the traeling wae is not monotone. One open qestion is to knowif the traelingwaeis monotonewhen µ < µ c. Or simlation answers positiely to this open qestion nmerically..4 Traeling wae shapes for kernels in (6) The nmerical reslts with kernels in (6) are depicted in Figre. With large µ =, we can see similar phenomena sch that some oscillations appear and the waes become nonmonotone. For the two kernels in (6), the state is Tring nstable as for (6) and Figre sggests again that the traeling waes will connect to. Finally, to complete the tests, we show the traeling wae for the Gassian kernel (7) for µ = in Figre 4. 6

8 References [] H. Berestycki, F. Hamel, Front propagation in periodic ecitable media, Comm. Pre Appl. Math. 55, 949,. [] H. Berestycki, G. Nadin, B. Perthame and L. Ryzhik, The non-local Fisher- Kpp eqation: traeling waes and steady states. Nonlinearity (), 8 844, 9. [] M. A. Fentes, M. N. Kperman and V. M. Kenkre, Nonlocal interaction effects on pattern formation in poplation dynamics, Phys. Re. Lett. 9(5), ,. [4] S. Genieys, V. Volpert and P. Ager, Patternand waesfor amodel in poplation dynamics with nonlocal consmption of resorces, Math. Modelling Nat. Phenom., 65 8, 6. [5] S. A. Gorley, Traelling front soltions of a nonlocal Fisher eqation, J. Math. Biol. 4(),. [6] A.N. Kolmogoro, I.G. Petrosky and N.S. Piskno, Etde de l éqation de la diffsion aec croissance de la qantité de matière et son application à n problème biologiqe, Blletin Uniersité d Etat à Mosco (Bjl. Moskowskogo Gos. Uni.), 6, 97. [7] N. Shigesada, K. Kawasaki, E. Teramoto, Traeling periodic waes in heterogeneos enironments, Theor. Poplation Biol., 4 6,