Supplementary Material for An energy-based stability criterion for solitary traveling waves in Hamiltonian lattices Haitao Xu Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455 USA Center for Mathematical Science, Huanzhong University of Science and Technology, Wuhan, Hubei 430074, China Jesús Cuevas Maraver Grupo de Física No Lineal, Departamento de Física Aplicada I, Universidad de Sevilla. Escuela Politécnica Superior, C/ Virgen de África, 7, 41011-Sevilla, Spain Instituto de Matemáticas de la Universidad de Sevilla (IMUS). Edificio Celestino Mutis. Avda. Reina Mercedes s/n, 41012-Sevilla, Spain Panayotis G. Kevrekidis Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA 01003-9305, USA Anna Vainchtein Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA I. PROOFS OF SOME RESULTS IN THE MAIN TEXT In this section, we prove Propositions 4.1 and 4.2 and Lemma 5.4 in the main text. A. Proof of Proposition 4.1 Proof. Suppose S + E 2 (1, 0)Y 0 = µy 0. Letting Y (τ) = E 2 (τ, 0)Y 0, we have Y (τ + 1) = E 2 (τ + 1, 1)E 2 (1, 0)Y 0 = E 2 (τ + 1, 1) S µy 0 = S E 2 (τ, 0) S + S µy 0 = S µe 2 (τ, 0)Y 0 = S µy (τ), where we used the fact that X tw (τ + 1) = S X tw (τ) and that the operator DA in c d dτ Y (τ) = DA(X tw(τ))y (τ), τ [τ 0, τ 1 ] Y (τ 0 ) = Y 0. (1) satisfies DA( S ± x) = S ± DA(x) S. Thus Z tw (τ) = e log(µ)τ Y (τ) is a traveling wave solution of λz tw = LZ tw, L = 1 c DA(X tw) d dτ = 1 c J 2 H X 2 d X=Xtw dτ, (2) with λ = log(µ), satisfying Z tw (τ + 1) = S Z tw (τ). Conversely, if Y (τ) = Z tw (τ)e λτ solves d dτ Y (τ) = 1 c DA(X tw(τ))y (τ) = 1 c J 2 H X 2 Y (τ). (3) X=Xtw and Z tw (τ) solves (2), we set Y 0 = Y (0) = Z tw (0), so that E 2 (1, 0)Y 0 = Y (1) = e λ Z tw (1) = e λ S Z tw (0) = S µy 0, so Y 0 is an eigenfunction of S + E 2 (τ 1, τ 0 ) associated with the eigenvalue µ = e λ.
2 B. Proof of Proposition 4.2 Proof. Note that if λz 0 = LZ 0 and (L λi)z 1 = Z 0, then Ỹ1(τ) = e λτ (Z 1 (τ) + τz 0 (τ)) solves (3). Similarly, if (L λi) k Z k = (L λi) k 1 Z k 1 =... = (L λi)z 1 = Z 0, (4) then Ỹk(τ) = e λτ ( k 1 j! τ j Z k j (τ)) solves (3). Since Ỹ1(0) = Z 1 (0), then E 2 (1, 0)Ỹ1(0) = Ỹ1(1) = e λ (Z 0 (1) + Z 1 (1)) = µ S (Z 0 (0) + Z 1 (0)), where µ = e λ, and thus ( S + E 2 (1, 0) µi)ỹ1(0) = µỹ0(0). Ỹ j (0) = Z j (0) for 1 j k, then ( S + E 2 (1, 0) µi)ỹj(0) = µ( j 1 i=0 Using mathematical induction, we can show that if (j i)!ỹi(0)). 1 We claim that there exists a (nonunique) lower triangular matrix Ω = (Ω ij ) 0 i,j k such that for Ỹ0(0) = Z 0 (0) and Y j = j i=0 Ω jiỹi(0) for 0 j k we have ( S + E 2 (1, 0) µi) k Y k = ( S + E 2 (1, 0) µi) k 1 Y k 1 =... = ( S + E 2 (1, 0) µi)y 1 = Y 0. (5) To prove the existence of such Ω, let R 1 = (R 1,j,i ) 0 j,i k and R 2 = (R 2,j,i ) 0 j,i k be such that { { 0, i j, R 1,ji = 1 (j i)!, i < j, R 1, i = j 1, 2,ji = 0, i j 1. Then we have ( S + E 2 (1, 0) µi)ỹ (0) = µr 1 Ỹ (0), ( S + E 2 (1, 0) µi)y = R 2 Y, where Y = (Y 0, Y 1,..., Y k ) T, Ỹ (0) = (Ỹ0(0), Ỹ1(0),..., Ỹk(0)) T and Y = ΩỸ (0), implying µωr 1 = R 2 Ω. By comparing the terms on both sides, one can verify that given Ω 00, one can compute the other diagonal terms Ω 11, Ω 22,..., Ω kk. Once all diagonal terms are known, given Ω 10, one can solve for Ω 21, Ω 32,..., Ω k(k 1). Assuming Ω ji are known for any j i < m and given Ω m0, one can then obtain Ω (m+1)1, Ω (m+2)2,..., Ω k(k m). Following this procedure, a matrix Ω can be found for given constants Ω j0, 0 j k. In particular, for k = 3 one choice of Ω is Ω = µ 3 0 0 0 0 µ 0 0 0 1 2 µ µ 0 1 0 3 1 1 To prove the converse, we assume that (5) holds and let Ỹj(0) = k i=0 Ω 1 ji Y i for 0 j k, with Ω 1 being the inverse matrix of Ω. Then for 1 j k, ( S + E 2 (1, 0) µi)ỹj(0) = µ( j 1 1 i=0 (j i)!ỹi(0)). Let Ỹ0(τ) = E 2 (τ, 0)Ỹ0(0). and Ỹj(τ) = E 2 (τ, 0)Ỹj(0) for 1 j k, then Ỹj(τ + 1) = S µ( j i=0 1 (j i)!ỹi(τ)). Using mathematical induction, one can show that Ỹj(τ) has the form Ỹj(τ) = e λτ ( j i=0 1 i! τ j Z j i (τ)), where Z j (τ) are traveling waves, i.e., satisfy Z j (τ + 1) = S Z j (τ), for 0 j k. Moreover, since Ỹj(τ) solves (3), one can show λz 0 = LZ 0 and (4) hold. C. Proof of Lemma 5.4. Proof. Using the arguments similar to the ones leading to Propositions 5.1 and 5.2, one can show that λ = O(ɛ 1/(n 2) ) in this case. To be more specific, using and L( τ X tw (τ; c)) = 0 (6) L(c c X tw (τ; c)) = τ X tw (τ; c), (7)
we can similarly obtain the coefficients g j and f j (as well as the analogs of h j defined in (23)). Moreover, from 0 = Q 0 g 1 + Q 1 g 0 we can derive the n-dimensional version of K 00 = 0, which eliminates the possibility of λ O(ɛ 1/n ). Using symmetries or K 10 = K 01 we rule out the case λ O(ɛ 1/(n 1) ). Below we show that 2K 20 K 11 is satisfied, so that λ = O(ɛ 1/(n 2) ) can be achieved. Substituting λ = ɛ 1/(n 2) λ 1 + ɛ 2/(n 2) λ 2 + ɛ 3/(n 2) λ 3 +... and Z = Z 0 + ɛ 1/(n 2) Z 1 + ɛ 2/(n 2) Z 2 + ɛ 3/(n 2) Z 3 +... into λz = LZ, we obtain 3 L 0 Z j = L 0 Z n 2 + L 1 Z 0 = L 0 Z n 1 + L 1 Z 1 = L 0 Z n + L 1 Z 2 = j λ k Z j k, j = 0, 1,..., n 3 n 2 λ k Z n 2 k n 1 λ k Z n 1 k n λ k Z n k. Let Z j = n 1 k=0 (C jke k ) + (Z j ) #, where (Z j ) # G n. Then the above equations are equivalent to A 0 C j = A 0 C n 2 + M 1 Q 1 C 0 = A 0 C n 1 + M 1 Q 1 C 1 = A 0 C n + M 1 Q 1 C 2 = j λ k C j k, j = 0, 1,..., n 3 (8) n 2 λ k C n 2 k (9) n 1 λ k C n 1 k (10) n λ k C n k, (11) where C j and n-dimensional vectors with components C jk and M = ( J 1 e j, e k ), Q 0 = ( J 1 e j, L 0 e k ) and Q 1 = K = ( J 1 e j, L 1 e k ) are n n matrices, with j, k = 0,..., n 1. As before, we have Q 0 = MA 0, where A 0 is the n n matrix whose only nonzero entries are ones along the superdiagonal (generalization of the previous 4-by-4 matrix A 0 ), Q 0 and Q 1 are symmetric, and M is a skew-symmetric invertible matrix (recall that n is even). Setting g 0 = (1, 0, 0,..., 0) T, we obtain C j = C n 2 = C n 1 = C n = j λ k A T 0 C j k + C j,0 g 0, j = 0, 1,..., n 3 (12) n 2 λ k A T 0 C n 2 k A T 0 M 1 Q 1 C 0 + C n 2,0 g 0, (13) n 1 λ k A T 0 C n 1 k A T 0 M 1 Q 1 C 1 + C n 1,0 g 0, (14) n λ k A T 0 C n k A T 0 M 1 Q 1 C 2 + C n,0 g 0. (15) With C 00 = 1, we get C j,j = λ j 1 and C j,k = 0 for 0 j (n 3) and j < k. The last row of (13) is K 00 = 0; the last row of (14) can be directly verified by using symmetries. The last row of (11) is the same as the last row of M 1 Q 1 C 2 = λ 1 C n 1, which yields λ n 1 + (2K 20 K 11 )λ 2 1 = 0. (16)
Similar to (5.32) in the main text, we can show that the coefficient in front of λ 2 1 equals H (c 0 ). Indeed, using H and 2 H to simplify notation and recalling the symmetry of K, we have H (c 0 ) = 2 H, U 2 + U 1, 2 HU 1 = 2c 0 J 1 e 0, U 2 + c 0 U 1, J 1 L 0 U 1 + c 0 U 1, J 1 τ U 1 4 = e 1, J 1 ( τ U 1 c 0 L 1 U 1 ) + U 1, J 1 e 0 + c 0 (f 10 e 0 + 1 c 0 e 1 ), J 1 τ U 1 = e 2, J 1 L 1 e 0 e 1, J 1 L 1 e 1 + 0 c 0 f 10 e 1, J 1 L 1 e 0 + (c 0 f 10 e 1 + e 2 ), J 1 L 1 e 0 (17) Thus, nonzero roots of (16) satisfy = K 20 + K 11 + f 10 c 0 K 10 f 10 c 0 K 10 K 20 = K 11 2K 20. λ n 2 1 = K 11 2K 20 = H (c 0 ). II. PROJECTION COEFFICIENTS In this section, we provide some additional expressions for the projection coefficients arising in and 3 τ U j = (g jk e k ) + ( τ U j ) #, ( τ U j ) # G 4 (18) k=0 3 U j = (f jk e k ) + (U j ) #, (U j ) # G 4. (19) k=0 We start with (18). Recall that g 0j = δ 1,0 and that g 1j coefficients are given by (5.21). Using 0 = Q 0 g 2 +Q 1 g 1 +Q 2 g 0, we obtain 3 j=1 g 1jK 0j + L 00 = 0, and, recalling g 1 = A T 0 M 1 Q 1 g 0 + g 10 g 0, we have g 2 = A T 0 M 1 Q 1 g 1 A T 0 M 1 Q 2 g 0 + g 20 g 0 = [( A T 0 M 1 Q 1 ) 2 + ( A T 0 M 1 Q 2 )]g 0 + ( A T 0 M 1 Q 1 )g 10 g 0 + g 20 g 0. (20) This yields g 21 = α2 2K 10 K 11 α 2 ( K 12 K 20 + K 11 K 30 + 2K 10 K 31 ) α 4 1 + K 30K 31 K 20 K 32 + K 10 K 33 + α 2 (g 10 K 10 + L 00 ) α 2 1 g 22 = α 2 K 10 K 21 α 2 1( K 20 K 22 + K 10 K 23 + K 21 K 30 ) α 4 1 g 10K 30 + L 30 + g 10K 20 + L 20 g 23 = α 2( K 10 K 11 ) + (K 10 K 13 K 12 K 20 + K 11 K 30 ) α1(g 2 10 K 10 + L 10 ) α1 3. (21) Condition 3 j=1 g 2jK 0j + 3 g 1jL 0j + B 00 = 0 follows from 0 = Q 0 g 3 + Q 1 g 2 + Q 2 g 1 + Q 3 g 0, which together with g 1 and g 2 also yields g 3 = A T 0 M 1 Q 1 g 2 A T 0 M 1 Q 2 g 1 A T 0 M 1 Q 3 g 0 + g 30 g 0 = [( A T 0 M 1 Q 1 ) 3 + ( A T 0 M 1 Q 1 )( A T 0 M 1 Q 2 ) + ( A T 0 M 1 Q 2 )( A T 0 M 1 Q 1 ) + ( A T 0 M 1 Q 3 )]g 0 + [( A T 0 M 1 Q 1 ) 2 + ( A T 0 M 1 Q 2 )]g 10 g 0 + ( A T 0 M 1 Q 1 )g 20 g 0 + g 30 g 0,
5 resulting in 3 g 31 = g 2jK 3j + 3 g 1jL 3j + B 30 α 2 g 33 3 g 32 = g 2jK 2j + 3 g 1jL 2j + B 20 3 g 33 = g 2jK 1j + 3 g 1jL 1j + B 10 To summarize, (5.21), (21) and (22) determine the projection coefficients g ij, i, j = 1, 2, 3, in (18). Considering now the coefficients in (19), (5.26)-(5.28) yield (22) f 1 = 1 c 0 A T 0 g 0 + f 10 g 0, f 2 = 1 2c 0 [A T 0 g 1 A T 0 M 1 (Q 0 + c 0 Q 1 )f 1 ] + f 20 g 0 = 1 2c 0 [A T 0 g 1 A T 0 M 1 Q 1 A T 0 g 0 c 0 A T 0 M 1 Q 1 f 10 g 0 1 c 0 A T 0 g 0 ] + f 20 g 0, f 3 = 1 [A T 0 g 2 A T 0 M 1 (Q 1 + c 0 Q 2 )f 1 A T 0 M 1 (Q 0 + c 0 Q 1 )2f 2 ] + f 30 g 0 = 1 [A T 0 g 2 A T 0 M 1 (Q 1 + c 0 Q 2 )( 1 c 0 A T 0 g 0 + f 10 g 0 ) A T 0 M 1 (Q 0 + c 0 Q 1 )( 1 c 0 A T 0 g 1 1 c 0 A T 0 M 1 Q 1 A T 0 g 0 A T 0 M 1 Q 1 f 10 g 0 1 c 2 0 A T 0 g 0 + f 20 g 0 )] + f 30 g 0. Using (5.26), we find that f 1 = (f 10, 1 c 0, 0, 0) T, which together with (5.27) yields f 21 = α2 1(c 0 g 10 1) + α 2 c 0 (K 11 + c 0 f 10 K 10 ) c 0 (c 0 f 10 K 30 + K 31 ) 2α 2 1 c2 0 f 22 = c 0f 10 K 20 + K 21 K 30 2 c 0. and f 23 in (5.30). Using (5.28) we obtain f 31 = 1 [g 20 f 32 = 1 [g 21 3 1 (K 3j + c 0 L 3j )f 1j 3 3 ( α 2 (K 0j + c 0 L 0j ) 1 (K 2j + c 0 L 2j ))f 1j f 33 = 1 [g 22 1 α 2 1 c 0 1 K 3j 2f 2j 2f 21 2 α 2 f 23 ] α 2 f 33 + g 22 + f 30 3 3 (K 1j + c 0 L 1j )f 1j 1 3 c 0 K 1j 2f 2j 2f 23 ] c 0 ( α 2 α1 2 K 0j 1 K 2j )2f 2j 2f 22 ] To simplify some calculations, it is convenient to define h j = c 0 jf j + (j 1)f j 1 for j = 1, 2, 3, so that h 1 = A T 0 g 0 + h 10 g 0 h 2 = A T 0 g 1 A T 0 M 1 Q 1 h 1 + h 20 g 0 (23) h 3 = A T 0 g 2 A T 0 M 1 Q 2 h 1 A T 0 M 1 Q 1 h 2 + h 30 g 0 We note that the last rows of the equations (5.36)-(5.38) are closely connected to the constraints (6) (7), one of the reasons for which is as follows: Remark 1. If the constant term in C 2k is given by g k with C 2k,0 = g k,0, then the constant term in C 2k+2 will be g k+1 with C 2k+2,0 = g k+1,0.
6 Remark 2. If the constant term in C 2k 2 is given by g k 1 with C 2k 2,0 = g k 1,0 and the coefficient for λ 1 in C 2k 1 is given by h k with h k,0 = 0, then the coefficient for λ 1 in C 2k+1 will be h k+1 with h k+1,0 = 0. By these two remarks, the constant terms in the last rows of (5.36)-(5.38) are always zero. A more involved calculation, omitted here, shows that the coefficients for λ 1 in the last rows of these equations also vanish. Similar techniques can be used in other cases, including the one considered in Sec. 5, to show the constant terms and λ 1 coefficients vanish in the last rows of the equations. III. THE DEGENERATE CASE H (c 0) = H (c 0) = 0 In this section, we briefly discuss the degenerate case when H (c 0 ) = H (c 0 ) = 0 but H (c 0 ) 0. For simplicity, we assume that dim(gker(l 0 )) = 4. Following the arguments in Proposition 5.1 and Proposition 5.2, one can show that the near-zero eigenvalues are at most O(ɛ). Proceeding as before, one obtains the following equation for λ 1 : where We omit the details. This yields λ 4 1 bλ 2 1 = 0, b = 1 [ 2K01 K 32 + 2K 02 K 22 K 2 03 + K 11 K 31 K 2 12 + α 2(2K 21 K 10 + K 2 20 K 2 11) α 2 1 + 2L 02 L 11 ]. Lemma III.1. Suppose all the assumptions in Theorem 5.3 hold except that 3 c (X tw,j (τ; c)) D 0 ([0, 1]) (D 0 a([0, 1]) if weighted spaces are used), H (c 0 ) = 0 and b defined in (24) is nonzero. Then there exist c 1 < c 0 and c 2 > c 0 such that for c (c 1, c 2 ), the leading-order terms of nonzero eigenvalues of L will be λ = ± b(c c 0 ) + o( c c 0 ). It is not clear whether the coefficient b is related to H (c 0 ), which is given by [ 1 H (c 0 ) = 2 ( 3K 13 K 20 + K12 2 4K 20 K 22 + 4K 10 K 23 + 3K 21 K 30 + 2c 0 f 10 (K 10 K 13 K 12 K 20 + K 11 K 30 )) + α 2 α1 2 (K11 2 4K 10 K 21 4c 0 f 10 K 10 K 11 ) 2g 10 K 20 K 20 + L 11 4L 20 c 0 ] + c 0 (2f 20 K 10 3f 10 g 10 K 10 2f 10 L 10 ). We note that it is possible to make f 10 = f 20 = 0 with a careful choice of {e 0, e 1, e 2, e 3 }. While this case has an intrinsic theoretical value, we have not yet identified any Hamiltonian lattice models for which this degeneracy condition is satisfied. (24) IV. NUMERICAL PROCEDURES In this section we describe the numerical procedures we used to obtain the solitary traveling waves and analyze their stability. More specifically, to identify the solitary wave structures, we use the procedure followed in [1]. To this end, we seek solutions of (6.3) in the co-traveling frame corresponding to velocity c: obtaining the advance-delay partial differential equation r n (t) = Φ(ξ, t), ξ = n ct, (25) Φ tt + c 2 Φ ξξ 2cΦ ξt + 2V (Φ(ξ, t)) V (Φ(ξ + 1, t)) V (Φ(ξ 1, t)) + Λ(m)(2Φ(ξ, t) Φ(ξ + m, t) Φ(ξ m, t)) = 0. m=1 (26)
Solitary traveling waves φ(ξ) are stationary solutions of (26). They satisfy the advance-delay differential equation (6.4) and (6.5). Following the approach in [2], we assume that φ(ξ) = o(1/ξ) and φ (ξ) = o(1/ξ 2 ) as ξ, multiply Eq. (6.4) by ξ 2 and integrate by parts to derive the identity [ ] c 2 m 2 Λ(m) φ(ξ)dξ V (φ(ξ))dξ = 0, (27) m=1 which imposes the constraint (6.5) on the traveling wave solutions. Here we assume that Λ(m) decays faster than 1/m 3 at infinity, so that the series on the left hand side converges. To solve Eq. (6.4) numerically, we introduce a discrete mesh with step ξ, where 1/ ξ is an integer, so that the advance and delay terms φ(ξ ± m) are well defined on the mesh. We then use a Fourier spectral collocation method for the resulting system with periodic boundary conditions [3] with large period l. Implementation of this method requires an even number N of collocation points ξ j j ξ, with j = N /2 + 1,..., N /2, yielding a system for ξ in the domain ( l/2, l/2], with l = N ξ being an even number, and the long-range interactions are appropriately truncated. To ensure that the solutions satisfy (6.5), we additionally impose a trapezoidal approximation of (27) on the truncated interval at the collocation points. This procedure is independent of the potential and the interaction range. However, the choices of ξ and l depend on the nature of the problem. To investigate spectral stability of a traveling wave φ(ξ), we substitute Φ(ξ, t) = φ(ξ) + ɛa(ξ) exp(λct), (28) into (26) and consider O(ɛ) terms resulting from this perturbation. This yields the following quadratic eigenvalue problem: 7 c 2 λ 2 a(ξ) = c 2 a (ξ) + 2λc 2 a (ξ) 2V (φ(ξ))a(ξ) + V (φ(ξ + 1))a(ξ + 1) + V (φ(ξ 1))a(ξ 1) Λ(m)(2a(ξ) a(ξ + m) a(ξ m)). By defining b(ξ) = cλa(ξ), we transform this equation into the regular eigenvalue problem ( ) ( ) a(ξ) a(ξ) cλ = M b(ξ) b(ξ) m=1 (29) (30) for the corresponding linear advance-delay differential operator M. Note that this problem is equivalent to the eigenvalue problem (3.5) via the transformation ξ = τ and (a(ξ), b(ξ)) = (Γ a ( τ), Γ b ( τ) cγ a( τ)). Spectral stability can be determined by analyzing the spectrum of the operator M after discretizing the eigenvalue problem the same way as the nonlinear Eq. (6.4) and again using periodic boundary conditions. A traveling wave solution is spectrally stable when the spectrum contains no real eigenvalues. To analyze the dependence of the eigenvalues spectrum obtained using this procedure on the grid size ξ and the size l of the lattice, we consider as an example an α-fpu lattice with V (r) = r 2 /2 r 3 /3 and without long-range interaction (Λ(m) 0). In this case, H (c) > 0 for every c, and one expects all waves to be stable (see, e.g., [4]). However, the numerically obtained spectra show mild oscillatory instabilities, which are associated with complexvalued eigenvalues with nonzero imaginary parts and small but nonzero real parts. As one can see in the example shown in Fig. 1, corresponding to c = 1.3, the real part of the eigenvalues does not change when ξ is varied (only their imaginary parts are affected, given that smaller ξ enables accessing higher wavenumbers); however, when the system s length l is increased, the real parts of the eigenvalues decrease. This suggests that the oscillatory instabilities are a numerical artifact due to the finite length of the lattice that can, in principle, be expected to disappear in the infinite lattice limit. An alternative method for determining the spectral stability of the traveling waves is to use Floquet analysis. To this end, we cast traveling waves φ(ξ) as fixed points of the map [ ] [ ] {rn+1 (T )} {rn (0)}, (31) {ṙ n+1 (T )} {ṙ n (0)} which is periodic modulo shift by one lattice point, with period T = 1/c. Indeed, one easily checks that ˆr n (t) = φ(n ct) = φ(n t/t ) satisfies ˆr n+1 (T ) = ˆr n (0) = φ(n) and ˆr n+1 (T ) = ˆr n (0) = cφ (n). To apply the Floquet
8 200 150 100 ξ=0.025 ξ=0.05 ξ=0.1 50 l=400 l=200 l=100 50 Im(λ) 0 Im(λ) 0 50 100 150 200 5 0 5 Re(λ) x 10 4 50 5 0 5 Re(λ) x 10 4 FIG. 1: Dependence of the linearization operator spectrum with the discretization parameter (left panel, with fixed l = 100) and the system length (right panel, with fixed ξ = 0.1) for a stable solitary traveling wave solution with c = 1.3 of the problem with the α-fpu potential governing nearest-neighbor interactions and no long-range interaction. analysis, we trace time evolution of a small perturbation ɛw n (t) of the periodic-modulo-shift (traveling wave) solution ˆr n (t). This perturbation is introduced in (6.3) via r n (t) = ˆr n (t) + ɛw n (t). The resulting O(ɛ) equation reads ẅ n + 2V (ˆr n )w n V (ˆr n+1 )w n+1 V (ˆr n 1 )w n 1 + Λ(m)(2w n w n+m w n m ) = 0. m=1 Then, in the framework of Floquet analysis, the stability properties of periodic orbits are resolved by diagonalizing the monodromy matrix F (representation of the Floquet operator at finite systems), which is defined as: [ ] [ ] {wn+1 (T )} {wn (0)} = F. (33) {ẇ n+1 (T )} {ẇ n (0)} For the symplectic Hamiltonian systems we consider in this work, the linear stability of the solutions requires that the monodromy eigenvalues µ (also called Floquet multipliers) lie on the unit circle. The Floquet multipliers can thus written as µ = exp(iθ), with Floquet exponent θ (note that it coincides with iλ). The results were complemented by numerical simulations of the ODE system (6.3), performed using the fourth-order explicit and symplectic Runge-Kutta-Nyström method developed in [5], with time step equal to 10 3. In particular, to explore the evolution of unstable waves, we used initial conditions with such waves perturbed along the direction of the Floquet mode causing the instability. (32) V. DISCUSSION: CONNECTION TO EXISTING ENERGY CRITERIA In this section, we discuss connections between the energy-based stability criterion we derived and analyzed for solitary traveling waves in Hamiltonian lattices and related criteria for such waves in continuum systems and for discrete breathers. A. Energy criteria for discrete breathers in Hamiltonian lattices As we discuseed in Section 4, solitary traveling wave solutions in lattices can be viewed as time-periodic solutions (or discrete breathers, if they are exponentially localized) modulo an integer shift. In particular, suppose (2.1) has a family of time-periodic solutions x per (t; ω) parametrized by the frequency ω and satisfying x per (t + 1 ω ; ω) = x per(t; ω). Setting τ = ωt and X per (τ; ω) = x per (t; ω), one can similarly define the linear operator L = 1 c DA(X per) d dτ = 1 c J 2 H X 2 X=X per d dτ. With ω playing the role of c for traveling waves, almost all of the results in this work, including Theorem 5.1, Lemma 5.1 and Lemma III.1, can also be applied to discrete breathers; see [6] for details and a number of examples.
9 B. Energy criteria for solitary waves in continuum Hamiltonian systems Stability of solitary waves in continuum Hamiltonian systems with symmetry (e.g U(1)-invariance) was analyzed in [7, 8]. Solitary traveling waves in such systems possess translational invariance in space, which breaks down in the case of lattice dynamical systems. However, in what follows, we rewrite the energy-based stability criteria obtained in [7, 8] to identify the similarities with our criterion for solitary traveling waves in lattices. Following the formulation in [7], we consider a continuum Hamiltonian system u t = JE (u), u(x, t) = ( q(x, t) p(x, t) ), J = ( 0 I I 0 ). (34) where E is the energy functional, while q(x, t) and p(x, t) are the displacement and momentum fields, respectively. Let T (s) be a one-parameter unitary operator group with the infinitesimal generator T (0), i.e. T (s) = exp(st (0)). We assume that E is invariant under T (s) for any s, i.e., E(u) = E(T (s)u). This invariance corresponds to the symmetry of the system. In particular, T (s)u(x) = e s x u(x) = u(x + s) corresponds to the translational symmetry. Assuming that T (s)jt (s) = J and differentiating it with respect to s at s = 0 gives T (0)J = J(T (0)), so that J 1 T (0) is self-adjoint. As a result, there exists a self-adjoint bounded linear operator B such that JB extends T (0). This implies that Q(u) = 1 2 u, Bu is also invariant under T (s), i.e., Q(u) = Q(T (s)u). Observe that if φ ω (x) satisfies E (φ ω ) = ωq (φ ω ), then u(x, t) = T (ωt)φ ω (x) is a solitary solution of (34). In fact, the converse also holds. Indeed, if u(x, t) = T (ωt)φ ω (x) solves (34), then JE (φ ω ) = JT (ωt) E (T (ωt)φ ω ) = JT (ωt) J 1 t (T (ωt)φ ω) = JT (ωt) J 1 t (T (ωt)φ ω) = T ( ωt)ωt (0)T (ωt)φ ω (35) = ωt (0)φ ω = JωQ (φ ω ). Here we used the fact that T (0)T (ωt) = T (ωt)t (0). We now define the free energy d(ω) = E(φ ω ) ωq(φ ω ) and the associated Hessian H ω = E (φ ω ) ωq (φ ω ). Observe that they satisfy H ω (T (0)φ ω ) = 0, H ω ( ω φ ω ) = Q (φ ω ) = J 1 T (0)φ ω, d (ω) = Q(φ ω ), d (ω) = ω φ ω, Q (φ ω ) = ω φ ω, H ω ( ω φ ω ) = d dω Q(φ ω) = 1 d ω dω E(φ ω). As stated in [7], if H ω has at most one negative eigenvalue, ker(h ω ) is spanned by T (0)φ ω and the rest of its spectrum is bounded below from zero, then d (ω) > 0 implies that φ ω -orbit {T (ωt)φ ω, t R} is stable. On the other hand, d (ω) < 0 implies instability. In other words, d (ω) = 0 = 1 ω d dω E(φ ω) implies the change of stability, which is similar to our energy-based criterion H (c 0 ) = 0 (and to the corresponding one for discrete breathers). On the one hand, the solutions in our lattice problems are discrete in space and the translation operator T (s) = e s x cannot be directly applied except in the quasi-continuum, advance-delay variant of the problem. However, the nature of the traveling waves implies that the time dynamics on all sites is connected to a continuous profile function and one can easily replace the translation in space by translation in time by going to the co-moving frame. In fact, replacing T (s) = e s x by T (s) = e 1 ω s t, we can similarly define (36) Q(u) = 1 2 u, Bu = 1 2 u, 1 ω J 1 t u, H ω = E (φ ω ) ωq (φ ω ) = E (φ ω ) J 1 t and derive (36) where u(t) = T (ωt)φ ω is a solitary traveling wave solution of the lattice system with E (φ ω ) = ωq (φ ω ). Note also the relation L = ωjh ω, with ω = c and x tw (c) = φ ω. A more complete understanding of the relationship between the energy-based criteria in discrete and continuum systems remains a challenging problem to be considered in the future work. [1] Yasuda H, Chong C, Charalampidis EG, Kevrekidis PG, Yang J. 2016 Formation of rarefaction waves in origami-based metamaterials. Phys. Rev. E 93, 043004.
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