A lower bound for the power of periodic solutions of the defocusing discrete nonlinear Schrodinger equation

A lower bound for the power of periodic solutions of the defocusing Discrete Nonlinear Schr¨odinger equation∗ J. Cuevas, Departamento de Fisica Aplicada I, Escuela Universitaria Polit´enica, C/ Virgen de Africa, 7, University of Sevilla, 41011 Sevilla, Spain J. C. Eilbeck, Maxwell Institute and Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland N. I. Karachalios, Department of Mathematics, University of the Aegean, Karlovassi, 83200 Samos, Greece Abstract We derive a lower bound on the power of time periodic solutions of the defocusing Discrete Nonlinear Schr¨odinger Equation with power nonlinearity, supplemented with Dirichlet boundary conditions. The lower bound depends not only on the dimension of the lattice, the lattice spacing, and the frequency of the periodic solution, but also on the excitation threshold of time periodic and spatially localized solutions of the focusing DNLS, proved in M. Weinstein, Nonlinearity 12, 673–691, 1999. The simple proof via a direct variational method, makes use of the interpolation inequality proved by M. Weinstein, and its optimal constant related to the excitation threshold. A numerical study is performed to test the efficiency of the lower bound. 1 Introduction In [6], M. Weinstein considered the focusing Discrete Nonlinear Schr¨odinger Equation (DNLS) [2, 4] i˙ ψn+²(∆dψ)n+|ψn|2σψn= 0, σ > 0, n ∈ZN,(1.1) and resolved the hypothesis suggested by S. Flach, K. Kladko & R. MacKay [5] for this equation, on the existence of excitation thresholds for the existence of nonlinear localized modes for Hamiltonian dynamical systems defined on multidimensional lattices. More precisely, the numerical studies and heuristic arguments of [5], suggested that there is a lower bound on the energy of a breather (time periodic and spatially localized standing wave solutions), if the lattice dimension is greater than or equal to a certain critical value. For (1.1), where (∆dψ)n, stands for the N-dimensional discrete Laplacian and ²is a discretization parameter ²∼h−2 with hbeing the lattice spacing, the hypothesis of [5] was resolved by Theorem 1.1 (M. Weinstein [6, Theorem 3.1,pg. 678]). Let σ≥2 N. Then there exists a ground state excitation threshold Rthresh >0. A minimizer of the variational problem IR= inf {H[φ] : P[φ] = R} .(1.2) ∗Keywords and Phrases: Discrete Nonlinear Schr¨odinger Equation, periodic solutions, excitation thresholds, variational methods AMS Subject Classification: 37L60, 35Q55, 47J30. 1 Periodic solutions of the defocusing DNLS 2 is called a a ground state [6, Definition, pg. 676]. Here H[φ] and P[φ] are the fundamental conserved quantities H[φ] = ²(−∆dφ, φ)2−1 σ+ 1 X n∈ZN |φn|2σ+2,(1.3) P[φ] = X n∈ZN |φn|2,(1.4) the Hamiltonian and the power, respectively. By (·,·)2we denote the `2-scalar product. Theorem 1.1, states that if 0 < σ < 2 N, then IR<0 for all R>0. That is, the variational problem (1.2) has a solution for all R>0 and there is no excitation threshold. However when σ≥2 N, there exists an excitation threshold Rthresh such that (a) if R>Rthresh then IR<0, and a ground state exists and (b) if R<Rthresh then IR= 0, and there is no ground state minimizer of (1.2). Theorem 1.1, justifies the existence of an excitation threshold for spatially localized and time periodic solutions of the form ψn(t) = eiωtφn, ω > 0, n ∈ZN, t ∈R,(1.5) φn∈`2(ZN). The threshold value, Rthresh, is related to the best constant of an interpolation inequality which is a discrete analogue of the Sobolev-Gagliardo-Nirenberg inequality. Theorem 1.2 (M. Weinstein [6, Theorem 4.1,pg. 682]) Assume that σ≥2 N. Then there exists C > 0, such that for all φ∈`2, the following interpolation inequality holds X n∈ZN |φn|2σ+2 ≤CÃX n∈ZN |φn|2!σ (−∆dφ, φ)2.(1.6) If C∗is the infimum over all such constants for which inequality (1.6) holds, then the excitation threshold Rthresh is defined by [6, pg. 680, Eqn. (4.2)] (σ+ 1)²(Rthresh)−σ=C∗,(1.7) and the optimal constant C∗has the variational characterization 1 C∗ = inf φ∈`2 φ6= 0 ¡Pn∈ZN|φn|2¢σ(−∆dφ, φ)2 Pn∈ZN|φn|2σ+2 . In this short note we shall derive and perform a numerical study, on a lower bound on the power of timeperiodic solutions of the defocusing DNLS supplemented with Dirichlet boundary conditions i˙ ψn+²(∆dψ)n−Λ|ψn|2σψn= 0,Λ>0,|n| ≤ K, (1.8) ψn= 0,|n|> K. (1.9) Problem (1.8)-(1.9) has time periodic solutions of the form ψn(t) = e−iΩtφn,Ω>0.(1.10) The existence of solutions (1.10) of prescribed frequency Ω >0, and the lower bound on their power, is derived by a simple proof based also on a variational approach. It is shown that the lower bound exhibits an interesting relation between the parameters N, σ, Ω, ² as well as on the excitation threshold for the periodic solutions of the focusing DNLS (1.1). Although the study is limited to the finite dimensional lattice, this case is of importance especially for numerical simulations: since the infinite lattice cannot be modelled numerically, numerical investigations should consider finite lattices with Dirichlet or periodic boundary conditions. The choice of boundary conditions only matters, if the pulse is moving and collides with the boundary. We expect that similar bounds can be derived for the case of periodic boundary conditions, by considering appropriate variational problems, but the details have to be checked. Let us mention at this point, that an analytical and numerical study, on various lower bounds of the power of time periodic solutions, of the DNLS equation with saturable and power nonlinearities in infinite and finite lattices, will be considered in [3]. Periodic solutions of the defocusing DNLS 3 2 A lower bound for time periodic solutions of the defocusing DNLS in a finite lattice: Relation to the excitation threshold of the focusing DNLS Substitution of the solution (1.10) into (1.8)-(1.9) shows that φnsatisfies the system of algebraic equations −²(∆dφ)n−Ωφn=−Λ|φn|2σφn,Ω>0,Λ>0,|n| ≤ K, (2.1) φn= 0,|n|> K. (2.2) The finite dimensional problem (2.1)–(2.2) will be formulated in the finite dimensional subspaces of the sequence spaces `p, 1 ≤p < ∞, `p(ZN K) = {φ∈`p:φn= 0 for |n|> K}.(2.3) Clearly `p(ZN K)≡C(2K+1)Nis endowed with the norm ||φ||p= X |n|≤K |φn|p  1 p . Note that for any 1 ≤p≤q < ∞, there exist constants C1, C2depending on K, such that C1||ψ||p≤ ||ψ||q≤C2||ψ||p.(2.4) The principal eigenvalue of the operator −∆ddenoted by λ1>0, can be characterized as λ1= inf φ∈`2(ZN K) φ6= 0 (−∆dφ, φ)2 P|n|≤K|φn|2,(2.5) Hence (2.5) implies the inequality ²λ1X |n|≤K |φn|2≤²(−∆dφ, φ)2≤4²N X |n|≤K |φn|2.(2.6) Thus from (2.6), we obtain for λ1the bound λ1≤4N. (2.7) In the case of an 1D-lattice n= 1,...,K, the eigenvalues of the discrete Dirichlet problem −∆dφ=λφ, with φreal, are given explicitly by λn= 4 sin2µnπ 4(K+ 1)¶, n = 1,...,K, while for a N-dimensional problem, the eigenvalues are: λ(n1,n2,...,nN)= 4 ·sin2µn1π 4(K+ 1)¶+ sin2µn2π 4(K+ 1)¶+...+ sin2µnNπ 4(K+ 1)¶¸, nj= 1,...,K;j= 1,...,N. In consequence, the principal eigenvalue of the discrete Dirichlet problem −∆dφ=λφ, with φreal, is given by λ1≡λ(1,1,...,1) = 4Nsin2µπ 4(K+ 1)¶. We also mention that the inequality (1.6) holds for any element of the finite dimensional space φ∈`2(ZN K). The result of this note is stated in the following Periodic solutions of the defocusing DNLS 4 Theorem 2.1 We consider the functional E[φ] = ²(−∆φ, φ)2−ΩX n∈ZN |φn|2,Ω>0,(2.8) and the variational problem on `2(ZN K) inf    E[φ] : X |n|≤K |φn|2σ+2 =M > 0   ,(2.9) for some Ω>0. Assume that Ω>4²N. (2.10) Then there exists a minimizer ˆ φ∈`2(ZN K)for the variational problem (2.9) and Λ(M)>0, both satisfying the Euler-Lagrange equation (2.1), and P|n|≤K|ˆ φn|2σ+2 =M. Moreover if σ≥2 N, the power of the minimizer P[ˆ φ]satisfies the lower bound Rthresh ··Ω−4N² 4N²(σ+ 1)2Λ¸1 σ ≤ P[ˆ φ],(2.11) where Rthresh ≡ Rthresh(σ, N, ²)is the excitation threshold of solutions (1.5) of the focusing DNLS (1.1). Proof: We consider the set B=   φ∈`2(ZN K) : X |n|≤K |φn|2σ+2 =M   .(2.12) Clearly E:B→Ris a C1-functional. Also, it is bounded from below: the equivalence of norms (2.4) implies the existence of a N-dependent constant C2such that ||φ||2 2≤C2 2||φ||2 2σ+2,for all φ∈`2(ZN K).(2.13) By using (2.13), we derive the inequality E[φ]≥ −ΩX |n|≤K |φn|2(2.14) ≥ −ΩC2 2 X |n|≤K |φn|2σ+2  1 σ+1 (2.15) ≥ −ΩC2 2M1 σ+1 .(2.16) We are restricted to the finite dimensional space `2(ZN K), and it follows that any minimizing sequence associated with the variational problem (2.9) is precompact: any minimizing sequence has a subsequence, converging to a minimizer. Thus Eattains its infimum at a point ˆ φin B. Now, for the C1-functional LM[φ] = X |n|≤K |φn|2σ+2 −M, (2.17) we get that for any φ∈B hL0 M[φ], φi= 2(σ+ 1) X |n|≤K |φ|2σ+2 >0.(2.18) Thus the Regular Value Theorem ([1, Section 2.9], [7, Appendix A,pg. 556 ]) implies that the set M=L−1 M(0) is aC1-submanifold of `2(ZN K). By applying the Lagrange multiplier rule, we obtain the existence of a parameter λ=λ(M)∈R, such that DE0[ˆ φ]−λL0 M[ˆ φ], ψE= 2²(−∆dˆ φ, ψ)2−2ΩRe X |n|≤K ˆ φnψn −2(σ+ 1)λX |n|≤K |ˆ φn|2σˆ φnψn= 0,for all ψ∈`2(ZN K).(2.19) Periodic solutions of the defocusing DNLS 5 Setting ψ=ˆ φin (2.19), we find that 2E[ˆ φ] = 2²(−∆dˆ φ, ˆ φ)2−2Ω X |n|≤K |ˆ φn|2= 2(σ+ 1)λX |n|≤K |ˆ φ|2σ+2.(2.20) By using inequality (2.6), we obtain that E[ˆ φ]≤4²N X |n|≤K |ˆ φn|2−ΩX |n|≤K |ˆ φn|2.(2.21) Thus E[ˆ φ]<0 if (2.10) is satisfied. Due to the estimate (2.7), the condition (2.10) implies that Ω> ²λ1.(2.22) Then assuming (2.10), we find that λ(M)<0. We set λ=−Λ,Λ>0. Lastly, we assume that the power of the nontrivial minimizer ˆ φis P|n|≤K|ˆ φn|2=R2. Equation (2.20) can be rewritten as 2²(−∆dˆ φ, ˆ φ)2+ 2(σ+ 1)Λ X |n|≤K |ˆ φ|2σ+2 = 2Ω X |n|≤K |ˆ φn|2.(2.23) We shall use (1.6), with the optimal constant (1.7), to estimate the second term on the rhs of (2.23): we have 2²(−∆dˆ φ, ˆ φ)2+ 2(σ+ 1)ΛC∗ÃX n∈ZN |ˆ φn|2!σ (−∆dˆ φ, ˆ φ)2≥2Ω X |n|≤K |ˆ φn|2.(2.24) Since from (2.6) X |n|≤K |ˆ φn|2≥1 4N(−∆dˆ φ, ˆ φ)2, inequality (2.24) becomes 2²(−∆dˆ φ, ˆ φ)2+ 2(σ+ 1)ΛC∗ X |n|≤K |ˆ φn|2  σ (−∆dˆ φ, ˆ φ)2 ≥2Ω 4N(−∆dˆ φ, ˆ φ)2.(2.25) Thus, from (2.25), we get ²+ (σ+ 1)ΛC∗R2σ≥Ω 4N. implying the lower bound ·Ω−4N² 4N(σ+ 1)C∗Λ¸1 σ < R2.(2.26) Replacing the value C∗, given by (1.2), in inequality (2.26), we find Rthresh ··Ω−4N² 4N²(σ+ 1)2Λ¸1 σ ≤R2, which is the lower bound (2.11). ¦ Periodic solutions of the defocusing DNLS 6 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0 0.5 1 1.5 2 2.5 Ω PΩ σ=2. N=1. ε=0.25 0 0.2 0.4 0.6 0.8 1 1 1.1 1.2 1.3 1.4 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0 0.5 1 1.5 2 2.5 Ω PΩ σ=1. N=2. ε=0.15 0 0.2 0.4 0.6 0.8 1 0.8 1 1.2 1.4 1.6 1.8 Figure 1: Numerical power for solutions (1.10), of the defocusing DNLS (1.8)-(1.9) (a) σ= 2, N= 1 (σ=2 N), (b) σ= 1, N= 2 (σ=2 N). The inset in each case, shows a magnification of the region where the power of periodic solutions (1.5) of the focusing DNLS (1.1), reaches its minimum value. In case (a), Rthresh = 1.009 and in case (b), Rthresh = 0.855. 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Ω PΩ σ=10. N=1. ε=0.25 0.6 0.8 1 1.2 1.4 1.6 1.09 1.1 1.11 1.12 1.13 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 0 0.5 1 1.5 2 2.5 Ω PΩ σ=2. N=2. ε=0.15 0.1 0.2 0.3 0.4 0.5 0.6 1.05 1.1 1.15 1.2 1.25 1.3 Figure 2: Numerical power for solutions (1.10), of the defocusing DNLS (1.8)-(1.9) (a) σ= 10, N= 1 (σ > 2 N), (b) σ= 2, N= 2 (σ > 2 N). The inset in each case, shows a magnification of the region where the power of periodic solutions (1.5) of the focusing DNLS (1.1), reaches its minimum value. In case (a), Rthresh = 1.098 and in case (b), Rthresh = 1.047. Periodic solutions of the defocusing DNLS 7 Numerical study of the lower bound (2.11). We perform a numerical study to test the lower bound (2.11). In figures 1-2, the blue line corresponds to the numerically computed power of periodic solutions (1.10) for the defocusing DNLS (1.8)-(1.9). The inset in each figure shows the numericaly computed power of periodic solutions (1.5) for the focusing DNLS (1.1). The red line corresponds to the estimate (2.11). Figure 1 refers to the cases (a) σ= 2, N = 1, ² = 0.25 and (b) σ= 1, N = 2, ² = 0.15, respectively. Both cases consider the critical value of Theorem 1.1, σ=2 N. The inset in each picture is a numerical verification of Theorem 1.1, demonstrating the region where the numerical power of periodic solutions (1.5) of the focusing DNLS (1.1) for the same values of σ, N, ², reaches the minimum value Rthresh. The numerical value for case (a) is Rthresh = 1.009 and for case (b) is Rthresh = 0.855. These numerical values have been inserted in the estimate (2.11). The numerical study shows that the numerical power of periodic solutions (1.10) of the defocusing DNLS (1.8)-(1.9), fullfils the estimate (2.11). Figure 2 considers the cases (a) σ= 10, N = 1, ² = 0.25 and (b) σ= 2, N = 2, ² = 0.15, examples of the case σ > 2 N. The inset in each picture, is again a numerical verification of Theorem 1.1. The numerical value of Rthresh for case (a) is Rthresh = 1.098 and for case (b) is Rthresh = 1.047. We observe that the lower bound (2.11) is fullfiled. In conclusion, both figures show that the lower bound (2.11), is a quite satisfactory estimate of the power of time periodic solutions (1.10), of the defocusing DNLS (1.8)-(1.9). References [1] S. N. Chow, J. K. 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