Traveling wave oscillatory patterns in a signed Kuramoto-Sivashinsky equation with absorption
TTraveling wave oscillatory patterns in a signedKuramoto-Sivashinsky equation with absorption
Yvonne Bronsard Alama ∗ Jean-Philippe Lessard † October 8, 2019
Abstract
In this paper, a partial proof of a conjecture raised in [9] concerning existence and globaluniqueness of an asymptotically stable periodic orbit in a fourth-order piecewise linear ordinarydifferential equation is presented. The fourth-order equation comes from the study of travelingwave patterns in a signed Kuramoto-Sivashinsky equation with absorption. The proof is twofold.First, the problem of solving for the periodic orbit is transformed into a zero finding problemon R , which is solved with a computer-assisted proof based on Newton’s method and thecontraction mapping theorem. Second, the rigorous bounds about the periodic orbit in phasespace are combined with the theory of discontinuous dynamical systems to prove that the orbitis asymptotically stable. Key words.
Traveling wave patterns, Kuramoto-Sivashinsky model, discontinuous dynamical systems,periodic orbits, computer-assisted proofs
The Kuramoto-Sivashinsky equation u t + ∇ u + ∇ u − ( ∇ u ) = 0 , (1)where ∇ is the Laplace operator and ∇ is the biharmonic operator, is a fourth-order semilinearparabolic PDE which was originally introduced to model flame front propagation and later be-came a popular model to analyze weak turbulence or spatiotemporal chaos [3, 11, 12, 13, 15, 20].In an attempt to study extinction phenomena, Galaktionov and Svirshchevskii consider in [9] amodification of (1), namely the signed KS equation with absorption u t + ∇ u + sign( u ) − ( ∇ u ) = 0 . (2)Considering equation (2) on the real line (i.e. u = u ( ξ, t ), with ξ ∈ R and t ≥ u ( ξ, t ) = f ( y ) (with y def = ξ − ct ) in (2) whichleads to the problem − cf (cid:48) ( y ) + f (4) ( y ) + sign f ( y ) − ( f (cid:48)(cid:48) ( y )) = 0 , y ∈ (0 , ∞ ) and f (0) = 0 . ∗ McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, QC, H3A0B9, Canada. [email protected] † McGill University, Department of Mathematics and Statistics, 805 Sherbrooke Street West, Montreal, QC, H3A0B9, Canada. [email protected] . This author was supported by NSERC. a r X i v : . [ m a t h . D S ] O c t Following the approach of [9] (based on their experience studying the thin film equation), we onlykeep the highest derivative term in the equation and this yields f (4) ( y ) + sign f ( y ) = 0 , y ∈ (0 , ∞ ) and f (0) = 0 . (3)To capture the oscillatory component ϕ ( s ) in the solution of (3), we change coordinates ( y, f ( y )) (cid:55)→ ( s, ϕ ( s )) via f ( y ) = y ϕ ( s ) , with s def = ln y and plugging the transformation in (3) leads to the fourth-order piecewise linear ordinary differentialequation ϕ (4) ( s ) + 10 ϕ (cid:48)(cid:48)(cid:48) ( s ) + 35 ϕ (cid:48)(cid:48) ( s ) + 50 ϕ (cid:48) ( s ) + 24 ϕ ( s ) + sign( ϕ ( s )) = 0 . (4)This change of coordinates turns the search for a blowup solution into the search for a periodicsolution. The purpose of the present paper is to give a partial proof of the Conjecture 3.2 on page150 of [9], about solutions of equation [4], which we now state as a theorem. Theorem 1.1.
Equation (4) has a nontrivial asymptotically stable periodic solution.
The periodic solution of Theorem 1.1 is portrayed in Figure 1 and the corresponding travelingwave pattern u ( ξ, t ) = f ( ξ − ct ) = ( ξ − ct ) ϕ (ln( ξ − ct )) is plotted in Figure 2. Note that we set f ( ξ − ct ) = 0 for ξ − ct ≤
0, and that we did not solve for the wave speed c . X 2 X , R +
Figure 2: Different snapshots of u ( ξ, t ) = f ( ξ − ct ) = ( ξ − ct ) ϕ (ln( ξ − ct )).The proof of Theorem 1.1 has two parts. The first part of the proof (existence) is presented inSection 2, where the problem of finding the periodic solution ϕ ( s ) of (4) is transformed (via thesymmetry argument of Lemma 2.1) into a zero finding problem F ( a ) = 0 where F : R → R isdefined in (12). Proving the existence of ˜ a ∈ R such that F (˜ a ) = 0 is done with a computer-assistedproof based on a Newton-Kantorovich type theorem (Theorem 2.2). The second part of the proofis presented in Section 3, where the rigorous enclosure of the periodic solution is combined with thetheory of discontinuous dynamical systems to prove that the orbit is asymptotically stable. Thesetwo parts conclude the proof of Theorem 1.1. In this section, we prove the existence of a periodic solution ϕ ( s ) of (4). To achieve this goal, wereformulate this into a zero finding problem F ( a ) = 0 defined on R . Proving the existence of asolution is done by verifying the hypotheses of Theorem 2.2 with the help of the digital computerand interval arithmetic (e.g. see [17, 19]).We begin by making the change of variables x def = ϕ , x def = ϕ (cid:48) , x def = ϕ (cid:48)(cid:48) and x def = ϕ (cid:48)(cid:48)(cid:48) to rewritethe fourth-order equation (4) as the system˙ x = M x + g ( x ) def = − − − − x x x x + − sign( x ) . (5)Equation (5) is a piecewise smooth dynamical system and changes rule as x = ( x , x , x , x )goes through the switching manifold defined byΣ def = { x = ( x , x , x , x ) ∈ R : x = 0 } . The switching manifold Σ separates the phase space R into the two regions R + and R − definedby R + def = { x ∈ R : x ≥ } and R − def = { x ∈ R : x ≤ } . Denoting b def = (0 , , , − x = (cid:40) f + ( x ) def = M x + b, x ∈ R + f − ( x ) def = M x − b, x ∈ R − . (6)Given b ∈ {± b } , the unique solution of ˙ x = M x + b , x (0) = x ∈ R is given by t (cid:55)→ e Mt x + e Mt (cid:90) t e − Ms b ds = e Mt (cid:0) x + M − b (cid:1) − M − b . (7)Note that M − = − − − − Moreover, M = P DP − , e Mt = P e Dt P − where e Dt = diag (cid:0) e − t e − t e − t e − t (cid:1) and P = − − − −
116 9 4 1 − − − − and P − = − − − −
72 12 − − − −
133 32 16 . (8)We now introduce a result which exploits the symmetry of the problem and establishes a mech-anism to obtain a periodic solution of (6). Lemma 2.1.
If there exist
L > and a solution φ : [0 , L ] → R of ˙ x = f + ( x ) with φ ( L ) = − φ (0) (9) and φ ([0 , L ]) ⊂ R + , (10) then φ (0) , φ ( L ) ∈ Σ and Γ : [0 , L ] → R defined by Γ( t ) def = (cid:40) φ ( t ) , t ∈ [0 , L ] − φ ( t − L ) , t ∈ [ L, L ] (11) is a L -periodic solution of (6) .Proof. First, φ ([0 , L ]) ⊂ R + implies that ( φ (0)) , ( φ ( L )) ≥ ≤ ( φ ( L )) = − ( φ (0)) ≤
0. Hence, ( φ (0)) = ( φ ( L )) = 0, that is φ (0) , φ ( L ) ∈ Σ.Now, for t ∈ [ L, L ], ψ ( t ) def = − φ ( t − L ) solves ˙ x = f − ( x ), as ψ (cid:48) ( t ) = − φ (cid:48) ( t − L ) = M ( − φ ( t − L )) − b = M ψ ( t ) − b = f − ( ψ ( t )). Also φ ([0 , L ]) ⊂ R + implies that ψ ([ L, L ]) ⊂ R − . Moreover, ψ ( L ) = − φ (0) = φ ( L ) = − ψ (2 L ).By definition of Γ( t ) in (11), Γ([0 , L ]) ⊂ R + , Γ([ L, L ]) ⊂ R − and φ ( L ) = − φ ( L − L ) = − φ (0).Hence Γ is continuous at t = L . Finally, since Γ(0) = φ (0) = − φ ( L ) = − φ (2 L − L ) = Γ(2 L ), weconclude that Γ( t ) is a 2 L -periodic orbit of (6).To find the segment of the orbit φ : [0 , L ] → R solving ˙ x = f + ( x ) = M x + b as in Lemma 2.1we use formula (7), impose that the segment begins in the switching manifold (i.e. φ (0) ∈ Σ) andthat φ ( L ) = − φ (0). Note that if φ (0) ∈ Σ, then φ (0) = (0 , a , a , a ) for some a , a , a ∈ R . Using(7), the condition φ ( L ) = − φ (0) reduces to solving0 = φ (0) + φ ( L )= φ (0) + e ML φ (0) + e ML (cid:90) L e − Ms b ds = P ( I + e DL ) P − a a a + (cid:90) L P e D ( L − s ) P − b ds. Denote a def = ( L, a , a , a ) and let F ( a ) = F La a a def = P (cid:0) e DL + I (cid:1) P − a a a + (cid:90) L P e D ( L − s ) P − b ds. (12)To prove the existence of a periodic solution ϕ ( s ) of (4), it is sufficient to prove the existenceof a zero of F : R → R defined in (12) and then to verify the extra condition (10). While therigorous verification of (10) is done a-posteriori using interval arithmetics, the existence of a zeroof F is done using the radii polynomial approach (e.g. see [4, 10, 16]) which is essentially theNewton-Kantorovich theorem (e.g. see [18]). We now introduce this approach for a general C map defined on R n . Endow R n with the supremum norm (cid:107) a (cid:107) ∞ = max i =1 ,...,n | a i | and denote by B r ( b ) def = { a ∈ R n | (cid:107) a − b (cid:107) ∞ ≤ r } ⊂ R n the closed ball of radius r and centered at b . Theorem 2.2.
Let F : R n → R n be a C map. Consider ¯ a ∈ R n (typically a numerical approxima-tion with F (¯ a ) ≈ ). Assume that the Jacobian matrix DF (¯ a ) is invertible and let A def = DF (¯ a ) − .Let Y ≥ be any number satisfying (cid:107) AF (¯ a ) (cid:107) ∞ ≤ Y . (13) Given a positive radius r ∗ > , let Z = Z ( r ∗ ) be any number satisfying sup a ∈ B r ∗ (¯ a ) (cid:18) max ≤ i ≤ n (cid:88) ≤ k,m ≤ n (cid:12)(cid:12)(cid:12) (cid:88) ≤ j ≤ n A ij D km F j (cid:0) a (cid:1)(cid:12)(cid:12)(cid:12)(cid:19) ≤ Z . (14) Define the radii polynomial by p ( r ) def = Z r − r + Y . (15) If there exists r ∈ (0 , r ∗ ] with p ( r ) < , then there exists a unique ˜ a ∈ B r (¯ a ) such that F (˜ a ) = 0 .Proof. Let r ≤ r ∗ and consider c ∈ B r (¯ a ). Applying the Mean Value Inequality and using (14), ||| A ( DF ( c ) − DF (¯ a )) ||| ∞ ≤ sup (cid:107) h (cid:107) ∞ =1 sup a ∈ B r (¯ a ) ||| AD F ( a ) h ||| ∞ (cid:107) w − ¯ a (cid:107) ∞ ≤ Z r, (16)where ||| · ||| ∞ denotes matrix norm. Define the Newton-like operator T : R n → R n by T ( a ) = a − AF ( a ). Since A is invertible, F (˜ a ) = 0 if and only if T (˜ a ) = ˜ a . Let r > p ( r ) <
0. Hence Z r + Y < r and Z r + Y r <
1. Since Y , Z ≥
0, one gets that Z r < . (17)For any a ∈ B r (¯ a ), apply (16) to get ||| DT ( a ) ||| ∞ = ||| I − ADF ( a ) ||| ∞ = ||| A [ DF (¯ a ) − DF ( a )] ||| ∞ ≤ Z r < . Hence, (cid:107) T ( a ) − ¯ a (cid:107) ∞ ≤ (cid:107) T ( a ) − T (¯ a ) (cid:107) ∞ + (cid:107) T (¯ a ) − ¯ a (cid:107) ∞ ≤ sup c ∈ B r (¯ a ) ||| DT ( c ) ||| ∞ (cid:107) a − ¯ a (cid:107) ∞ + (cid:107) AF (¯ a ) (cid:107) ∞ ≤ ( Z r ) r + Y < r . Then T maps B r (¯ a ) into itself. Finally, given a , a ∈ B r (¯ a ) combine (17) with the Mean ValueInequality to get (cid:107) T ( a ) − T ( a ) (cid:107) ∞ ≤ sup c ∈ B r (¯ a ) ||| DT ( c ) ||| ∞ (cid:107) a − a (cid:107) ∞ ≤ ( Z r ) (cid:107) a − a (cid:107) ∞ ≤ κ (cid:107) a − a (cid:107) ∞ , where κ def = Z r <
1. Then, by the Contraction Mapping Theorem, T has a unique fixed point˜ a ∈ B r (¯ a ). It follows from the invertibility of A that ˜ a is the unique zero of F in B r (¯ a ).We now apply Theorem 2.2 to prove the existence of a zero of F defined in (12). This beginsby computing an approximate solution. Applying Newton’s method, we find an approximate zeroof F given by ¯ a = . . × − . × − − . × − . (18)Then, using INTLAB (see [19]) we compute rigorous enclosures of DF (¯ a ) and A def = DF (¯ a ) − .We then verify rigorously that Y def = 7 . × − (cid:62) (cid:107) AF (¯ a ) (cid:107) ∞ , which settles the computation ofthe bound (13).The next bound to compute is Z satisfying (14). The only non zero second partial derivativesare the terms ∂ F j ∂a ∂a k for j, k ∈ { , . . . , } , where we note that by Clairaut’s theorem ∂ F j ∂a ∂a k = ∂ F j ∂a k ∂a for k ∈ { , , } , j ∈ { , . . . , } . Hence, we can write the bound (14) as Z ( r ∗ ) ≥ max ≤ i ≤ sup c ∈ B r ∗ (¯ a ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) ≤ j ≤ A i,j ∂ F j ∂ a ( c ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) ≤ j ≤ A i,j ∂ F j ∂a ∂a ( c ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) ≤ j ≤ A i,j ∂ F j ∂a ∂a ( c ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) ≤ j ≤ A i,j ∂ F j ∂a ∂a ( c ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (19)Choosing r ∗ = 0 .
01 we use interval arithmetic to obtain that Z def = 41 satisfies (19).Therefore, for r (cid:54) r ∗ the radii polynomial is given by p ( r ) = 41 r − r + 7 . × − . Using interval arithmetic, we show that for every r ∈ [7 . × − , . p ( r ) <
0. By Theorem 2.2,there exists a unique zero ˜ a of F in B . × − (¯ a ). Denote ˜ a = ( ˜ L, ˜ a , ˜ a , ˜ a ). Then since | ˜ L − ¯ a | ≤(cid:107) ˜ a − ¯ a (cid:107) ∞ ≤ . × − and ¯ a = 1 . L >
0. By construction,˜ φ ( t ) def = e Mt (cid:16) ˜ φ (0) + M − b (cid:17) − M − b = e Mt a ˜ a ˜ a + M − b − M − b (20)defines a solution ˜ φ : [0 , ˜ L ] → R of ˙ x = f + ( x ) with ˜ φ ( ˜ L ) = − ˜ φ (0). The last hypothesis which needsto be verified to apply Lemma 2.1 is the condition (10), that is ˜ φ ([0 , ˜ L ]) ⊂ R + . Using a MATLABprogram using INTLAB, we consider a uniform time mesh (of size 300) of the time interval [0 , ˜ L ],that is 0 = t < t < · · · < t = ˜ L . For each mesh interval I k = [ t k − , t k ] ( k = 1 , . . . , φ ( I k ) using formula (20). Then the code verifies that ˜ φ ( t ) > t ∈ I k and k = k , . . . , k for some 1 < k < k < φ ([ t k − , t k ]) ⊂ R + .Afterward, it verifies that ˜ φ (cid:48) ( t ) = ˜ φ ( t ) > t ∈ I k and k = 1 , . . . , k −
1. Hence ˜ φ ( t ) isstrictly increasing over the interval [0 , t k − ], and since ˜ φ (0) = 0, it follows that ˜ φ ( t ) > t ∈ (0 , t k − ], that is ˜ φ ([0 , t k − ]) ⊂ R + . Similarly, the code verifies that ˜ φ (cid:48) ( t ) = ˜ φ ( t ) < t ∈ I k and k = k + 1 , . . . , φ ( t ) is strictly decreasing over the interval [ t k , ˜ L ], and since˜ φ ( ˜ L ) = 0, it follows that ˜ φ ( t ) > t ∈ [ t k , ˜ L ), that is ˜ φ ([ t k , ˜ L ]) ⊂ R + . We conclude that˜ φ ([0 , ˜ L ]) = ˜ φ ([0 , t k − ]) ∪ ˜ φ ([ t k − , t k ]) ∪ ˜ φ ([ t k , ˜ L ]) ⊂ R + . Hence ˜ φ : [0 , ˜ L ] → R verifies the hypotheses of Lemma 2.1. We conclude that ˜ φ (0) , ˜ φ ( ˜ L ) ∈ Σ andthat ˜Γ( t ) def = (cid:40) ˜ φ ( t ) , t ∈ [0 , ˜ L ] − ˜ φ ( t − ˜ L ) , t ∈ [ ˜ L, L ] (21)is a 2 ˜ L -periodic solution of (6). All the computational steps described in this section are carriedout in the MATLAB program Proof.m available at [1].
In this section, we demonstrate that the 2 ˜ L -periodic orbit ˜Γ( t ) defined in (21) is asymptoticallystable using the theory of discontinuous dynamical systems (e.g. see [5]). We do this by computingthe monodromy matrix X (2 ˜ L ) of ˜Γ and show that all its nontrivial Floquet multipliers have modulusless than one.Define h : R → R by h ( x ) def = x so that the switching manifold is given byΣ = { x ∈ R : h ( x ) = 0 } . Denote by ˜ a (1) def = ˜ φ ( ˜ L ) = (0 , − ˜ a , − ˜ a , − ˜ a ) T the point at which ˜Γ crosses Σ coming from R + andentering in R − . Similarly, denote by ˜ a (2) def = (0 , ˜ a , ˜ a , ˜ a ) T the point at which ˜Γ crosses Σ comingfrom R − and entering in R + . For that reason, ˜Γ is called a crossing periodic orbit (e.g. see [6, 14]).Denote by ˜ T def = 2 ˜ L the period of ˜Γ, ¯ t def = ˜ L , and denote by Φ( t, x ) the solution of (6) at time t withinitial condition x . Then (i.e. see [7]) the monodromy matrix is given by X ( ˜ T ) = X ( ˜ T , ¯ t ) S − + (˜ a (2) ) X (¯ t, S + − (˜ a (1) ) (22)where S + − (˜ a (1) ) def = I + (cid:18) ( f − − f + ) ∇ h T · f + ∇ h T (cid:19) (˜ a (1) ) S − + (˜ a (2) ) def = I + (cid:18) ( f + − f − ) ∇ h T · f − ∇ h T (cid:19) (˜ a (2) )are called the saltation matrices , and where the fundamental matrix solutions X ( t,
0) and X ( t, ¯ t )satisfy ˙ X ( t,
0) = Df − (Φ( t, ˜ a (1) )) X ( t,
0) =
M X ( t, , for t ∈ [0 , ˜ L ] , with X (0 ,
0) = I ˙ X ( t, ˜ L ) = Df + (Φ( t, ˜ a (1) )) X ( t, L ) = M X ( t, ˜ L ) , for t ∈ [ ˜ L, L ] , with X ( ˜ L, ˜ L ) = I. This implies that X ( t,
0) = e Mt and therefore X (¯ t,
0) = X ( ˜ L,
0) = e M ˜ L . Similarly, X ( t, ¯ t ) = e M ( t − ˜ L ) and then X ( ˜ T , ¯ t ) = X (2 ˜ L, ˜ L ) = e M ˜ L .Since ∇ h = (1 , , , T we obtain that ∇ h T · f + (˜ a (1) ) = − ˜ a and ∇ h T · f − (˜ a (2) ) = ˜ a . Simplecomputations yield S + − (˜ a (1) ) = − a = S − + (˜ a (2) ) . Hence, the monodromy matrix is given by X (2 ˜ L ) = e M ˜ L − a e M ˜ L − a . (23)Using interval arithmetics and that | ˜ L − ¯ a | , | ˜ a − ¯ a | ≤ . × − , we compute rigorously aninterval enclosure of (23) and using the rigorous computational method from [2] we prove that thespectrum σ ( X (2 ˜ L )) of X (2 ˜ L ) satisfies σ ( X (2 ˜ L )) ⊂ (cid:91) i =1 B i where B def = [0 . , . B def = [0 . , . B def = [0 . , . B def = [0 . , . . This rigorous computation is carried out in the MATLAB program
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