Lower and upper estimates on the excitation threshold for DNLS - PowerPoint PPT Presentation

Definition of the excitation threshold Upper and lower estimates for the excitation threshold Figures from numerical studies Discussion Lower and upper estimates on the excitation threshold for DNLS lattices J. Cuevas, N.I. Karachalios and F. Palmero Departamento de F´ ısica Aplicada I, Escuela Universitaria Polit´ enica, C/ Virgen de Africa, 7, University of Sevilla, 41011 Sevilla, Spain Department of Mathematics, University of the Aegean, Karlovassi, 83200 Samos, Greece Departamento de F´ ısica Aplicada I, ETSI Inform´ atica, Avd. Reina Mercedes s/n, University of Sevilla, 41012 Sevilla, Spain July 15, 2009 J. Cuevas, N.I. Karachalios and F. Palmero Lower and upper estimates on the excitation threshold for DNLS lattices

Definition of the excitation threshold Upper and lower estimates for the excitation threshold Figures from numerical studies Discussion Introduction Summary of the presentation ◮ We propose analytical lower and upper estimates on the excitation threshold for breathers (in the form of spatially localized and time periodic solutions) in DNLS lattices with power nonlinearity. The estimation depending explicitly on the lattice parameters, is derived by a combination of a comparison argument on appropriate lower bounds depending on the frequency of each solution with a simple and justified heuristic argument. ◮ The numerical studies verify that the analytical estimates can be usefull, as a simple analytical detection of the activation energy for breathers in DNLS lattices. J. Cuevas, N.I. Karachalios and F. Palmero Lower and upper estimates on the excitation threshold for DNLS lattices

Definition of the excitation threshold Upper and lower estimates for the excitation threshold Figures from numerical studies Discussion Introduction The excitation threshold in DNLS lattices The excitation threshold for discrete breathers: The positive lower energy bound possessed by discrete breather families (S. Flach, K. Kladko, R. S. MacKay. Phys. Rev. Lett. 78 (1997), 1207-1210) ◮ The excitation threshold for a discrete breather family was observed when lattice dimension N ≥ N crit Practical applications: ◮ Energy thresholds can assist in choosing a proper energy range for the detection of discrete breathers in real experiments or computer experiments. ◮ The energy threshold can be considered as the activation energy for localized excitations in nonlinear lattices. J. Cuevas, N.I. Karachalios and F. Palmero Lower and upper estimates on the excitation threshold for DNLS lattices

Definition of the excitation threshold Upper and lower estimates for the excitation threshold Figures from numerical studies Discussion Introduction Existence of energy threshold in DNLS lattices: The result of M. I. Weinstein M. I. Weinstein (Nonlinearity 12 (1999), 673-691) resolved the question of the existence of the excitation threshold for the DNLS equation i ˙ ψ n + ǫ (∆ d ψ ) n + Λ | ψ n | 2 σ ψ n = 0 , Λ > 0 , σ > 0 , (1) ◮ The question was resolved for breathers in the ansatz of time-periodic solutions ψ n ( t ) = e i Ω t φ n , Ω > 0 , (2) spatially localized in the sense || ψ || ℓ 2 → 0 , as | n | → ∞ , ( n = ( n 1 , n 2 , . . . , n N ) ∈ Z N ). J. Cuevas, N.I. Karachalios and F. Palmero Lower and upper estimates on the excitation threshold for DNLS lattices

Definition of the excitation threshold Upper and lower estimates for the excitation threshold Figures from numerical studies Discussion Introduction Existence of energy threshold in DNLS lattices: The result of M. I. Weinstein Solutions (2) of (1) satisfy the infinite system of algebraic equations − ǫ (∆ d φ ) n + Ω φ n − Λ | φ n | 2 σ φ n = 0 , n ∈ Z N . (3) We can associate a power to any solution of the form (2), defined as X | φ n | 2 . R [ φ ] = (4) n ∈ Z N The power (4) together with the Hamiltonian 1 n ∈ Z N | φ n | 2 σ +2 , H [ φ ] = ǫ ( − ∆ d φ, φ ) 2 − P (5) σ +1 are the fundamental conserved quantities for (1). Theorem (Weinstein 1999) Let σ ≥ 2 N . Then there exists a ground state excitation threshold R thresh > 0 . J. Cuevas, N.I. Karachalios and F. Palmero Lower and upper estimates on the excitation threshold for DNLS lattices

Definition of the excitation threshold Upper and lower estimates for the excitation threshold Figures from numerical studies Discussion Introduction Existence of energy threshold in DNLS lattices: The role of the discrete interpolation inequality ◮ The excitation threshold is defined through the discrete version of a Sobolev-Gagliardo-Nirenberg inequality σ 0 1 ( − ∆ d φ, φ ) 2 , σ ≥ 2 | φ n | 2 σ +2 ≤ C X @ X | φ n | 2 N , (6) A n ∈ Z N n ∈ Z N and its optimal constant. The optimal constant C ∗ of (6) has the variational characterization n ∈ Z N | φ n | 2 ´ σ ( − ∆ d φ, φ ) 2 `P 1 C ∗ = inf . P n ∈ Z N | φ n | 2 σ +2 φ ∈ ℓ 2 φ � = 0 J. Cuevas, N.I. Karachalios and F. Palmero Lower and upper estimates on the excitation threshold for DNLS lattices

Definition of the excitation threshold Upper and lower estimates for the excitation threshold Figures from numerical studies Discussion Introduction Existence of energy threshold in DNLS lattices: The role of the discrete interpolation inequality σ ≥ 2 N : Characterization of the excitation threshold R thresh > 0 Consider the variational problem I R = inf {H [ φ ] : R [ φ ] = R} . (7) ◮ If R > R thresh then I R < 0, and a ground state breather exists that is, minimizer of the variational problem (7). ◮ On the other hand, if R < R thresh there is no ground state minimizer of (7). ◮ The excitation threshold satisfies ( σ + 1) ǫ ( R thresh ) − σ = C ∗ . (8) σ < 2 N : Non-existence of excitation threshold ◮ For σ < 2 N there is no ground-state excitation threshold ( R thresh = 0). Ground states of arbitrary power R exist. J. Cuevas, N.I. Karachalios and F. Palmero Lower and upper estimates on the excitation threshold for DNLS lattices

Auxiliary Lemmas: Ω -Lower bound 1 Definition of the excitation threshold Upper and lower estimates for the excitation threshold Auxiliary Lemmas: Ω -Lower bound 2 Figures from numerical studies Derivation of the bounds Discussion Observations Estimation of the excitation threshold Statement of the result Motivation ◮ Mathematical interest on determining the optimal constants of Sobolev type inequalities. Here, the constant C ∗ of the discrete inequality (6). ◮ Physical interest on giving explicit estimation of the excitation threshold R thresh depending on lattice parameters. Both interests are connected due to (8). Proposition Let σ ≥ 2 / N. There exist κ crit > 1 / 2 such that » √ 2 κ crit − 1 – 1 · 4 N ǫ ( σ + 1) σ 1 σ . < R thresh < [4 ǫ N ( σ + 1)] (9) κ crit 2 σ + 1 J. Cuevas, N.I. Karachalios and F. Palmero Lower and upper estimates on the excitation threshold for DNLS lattices

Auxiliary Lemmas: Ω -Lower bound 1 Definition of the excitation threshold Upper and lower estimates for the excitation threshold Auxiliary Lemmas: Ω -Lower bound 2 Figures from numerical studies Derivation of the bounds Discussion Observations Estimation of the excitation threshold Frequency dependent lower bounds on the power Lemma The power of a nontrivial breather solution (2) of (1), satisfies the lower bound – 1 » Ω σ R min , 1 (Ω) := R thresh · < R [ φ ] for all Ω > 0 . (10) 4 ǫ Λ N ( σ + 1) ◮ Remark The excitation threshold R thresh is the power of the nontrivial breather solution ψ ∗ n ( t ) = e i Ω thresh t φ ∗ n , Ω thresh > 0 , where φ ∗ is the nontrivial minimizer. Since (10) is satisfied by the power of any nontrivial solution (2) for any Ω > 0, it also holds that R min , 1 (Ω thresh ) < R thresh (Ω thresh ) . This implies an upper estimate for the frequency Ω thresh Ω thresh < 4 ǫ Λ N ( σ + 1) . (11) J. Cuevas, N.I. Karachalios and F. Palmero Lower and upper estimates on the excitation threshold for DNLS lattices

Auxiliary Lemmas: Ω -Lower bound 1 Definition of the excitation threshold Upper and lower estimates for the excitation threshold Auxiliary Lemmas: Ω -Lower bound 2 Figures from numerical studies Derivation of the bounds Discussion Observations Estimation of the excitation threshold Frequency dependent lower bounds on the power Lemma Let κ > 1 2 , arbitrary. Then every non-trivial breather solution (2) of (1) has power satisfying » √ 2 κ − 1 – 1 Ω σ R min , 2 ( κ, Ω) := · < R [ φ ] for all Ω > 0 . (12) κ Λ(2 σ + 1) ◮ Remark An explicit Ω-independent estimation for R thresh will be derived immediately by an ordering of R min , 2 ( κ, Ω) and R min , 1 (Ω) which will eliminate Ω. ◮ Ω-dependent lower bounds for DNLS with power or saturable nonlinearity: (a) J.C. Eilbeck, J. Cuevas, NK, Discrete Cont. Dyn. Syst. 21 (2) (2008), 445-475. (b) J.C. Eilbeck, J. Cuevas, NK, Dyn. Partial Differ. Equ. 5 (2008), no. 1, 69–85. For DNLS systems involving linear and nonlinear impurities: J. Cuevas, F. Palmero, NK (2009). J. Cuevas, N.I. Karachalios and F. Palmero Lower and upper estimates on the excitation threshold for DNLS lattices