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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

• f 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 ≥ Ncrit

Practical applications:

◮ Energy thresholds can assist in choosing a proper energy range for the detection

• f 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 ˙ ψn + ǫ(∆dψ)n + Λ|ψn|2σψn = 0, Λ > 0, σ > 0, (1)

◮ The question was resolved for breathers in the ansatz of time-periodic

solutions ψn(t) = eiΩtφn, Ω > 0, (2) spatially localized in the sense ||ψ||ℓ2 → 0, as |n| → ∞, (n = (n1, n2, . . . , nN) ∈ ZN).

• 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(∆dφ)n + Ωφn − Λ|φn|2σφn = 0, n ∈ ZN. (3) We can associate a power to any solution of the form (2), defined as R[φ] = X

n∈ZN

|φn|2. (4) The power (4) together with the Hamiltonian H[φ] = ǫ(−∆dφ, φ)2 −

1 σ+1

P

n∈ZN |φn|2σ+2,

(5) are the fundamental conserved quantities for (1).

Theorem

(Weinstein 1999) Let σ ≥ 2

N . Then there exists a ground state excitation

threshold Rthresh > 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 X

n∈ZN

|φn|2σ+2 ≤ C @ X

n∈ZN

|φn|2 1 A

σ

(−∆dφ, φ)2, σ ≥ 2 N , (6) and its optimal constant. The optimal constant C∗ of (6) has the variational characterization 1 C∗ = inf φ ∈ ℓ2 φ = 0 `P

n∈ZN |φn|2´σ (−∆dφ, φ)2

P

n∈ZN |φn|2σ+2

.

• 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 Rthresh > 0

Consider the variational problem IR = inf {H[φ] : R[φ] = R} . (7)

◮ If R > Rthresh then IR < 0, and a ground state breather exists that is,

minimizer of the variational problem (7).

◮ On the other hand, if R < Rthresh there is no ground state minimizer of

(7).

◮ The excitation threshold satisfies

(σ + 1)ǫ (Rthresh)−σ = C∗. (8)

σ < 2

N : Non-existence of excitation threshold

◮ For σ < 2 N there is no ground-state excitation threshold (Rthresh = 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

Definition of the excitation threshold Upper and lower estimates for the excitation threshold Figures from numerical studies Discussion Auxiliary Lemmas: Ω-Lower bound 1 Auxiliary Lemmas: Ω-Lower bound 2 Derivation of the bounds 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

Rthresh depending on lattice parameters. Both interests are connected due to (8).

Proposition

Let σ ≥ 2/N. There exist κcrit > 1/2 such that »√2κcrit − 1 κcrit · 4Nǫ(σ + 1) 2σ + 1 – 1

σ

< Rthresh < [4ǫN(σ + 1)]

1 σ .

(9)

• 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 Auxiliary Lemmas: Ω-Lower bound 1 Auxiliary Lemmas: Ω-Lower bound 2 Derivation of the bounds 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 Rmin,1(Ω) := Rthresh · » Ω 4ǫΛN(σ + 1) – 1

σ

< R[φ] for all Ω > 0. (10)

◮ Remark The excitation threshold Rthresh is the power of the nontrivial

breather solution ψ∗

n (t) = eiΩthreshtφ∗ n, Ωthresh > 0,

where φ∗ is the nontrivial minimizer. Since (10) is satisfied by the power

• f any nontrivial solution (2) for any Ω > 0, it also holds that

Rmin,1(Ωthresh) < Rthresh(Ωthresh). This implies an upper estimate for the frequency Ωthresh Ωthresh < 4ǫΛN(σ + 1). (11)

• 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 Auxiliary Lemmas: Ω-Lower bound 1 Auxiliary Lemmas: Ω-Lower bound 2 Derivation of the bounds 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 Rmin,2(κ, Ω) := »√2κ − 1 κ · Ω Λ(2σ + 1) – 1

σ

< R[φ] for all Ω > 0. (12)

◮ Remark An explicit Ω-independent estimation for Rthresh will be derived

immediately by an ordering of Rmin,2(κ, Ω) and Rmin,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

Definition of the excitation threshold Upper and lower estimates for the excitation threshold Figures from numerical studies Discussion Auxiliary Lemmas: Ω-Lower bound 1 Auxiliary Lemmas: Ω-Lower bound 2 Derivation of the bounds Observations

Estimation of the excitation threshold

◮ Lower bound for Rthresh: limκ→∞ Rmin,2(κ, Ωthresh) = 0

= ⇒ there exists a κcrit > 1/2 such that Rmin,2(κcrit, Ω) < Rmin,1(Ω) = ⇒ LHS of (9)

◮ Upper bound for Rthresh:

sup φ ∈ ℓ2 φ = 0 P

n∈ZN |φn|2σ+2

`P

n∈ZN |φn|2´σ+1 = 1, for all σ ≥ 0.

(13) and P

n∈ZN |φn|2σ+2

`P

n∈ZN |φn|2´σ+1 ≤ 4NC ∗, for all σ ≥ 2/N, φ ∈ ℓ2.

(14) (13)+(14)+(8) = ⇒ 1 ≤ 4NC∗ = 4ǫN(σ + 1)R−σ

thresh

• 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 Auxiliary Lemmas: Ω-Lower bound 1 Auxiliary Lemmas: Ω-Lower bound 2 Derivation of the bounds Observations

Determination of κcrit-Numerical studies

◮ Set

Rlb = »√2κcrit − 1 κcrit · 4Nǫ(σ + 1) 2σ + 1 – 1

σ

.

◮ (8) =

⇒ Rthresh = »(σ + 1)ǫ C∗ – 1

σ

.

◮ Independently of the choice of κ, ǫ, N

lim

σ→∞ Rlb = lim σ→∞ Rthresh = 1.

(15)

Determination of κcrit

(15) justifies that even the first choice κcrit = 1 is valid for “sufficiently large” σ.

• 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 κcrit = 1, N = 1, 2 κcrit = 1, N = 3 κcrit = 2, N = 1, 2 κcrit = 2, N = 3 κcrit = 3, N = 1, 2 κcrit = 3, N = 3

Determination of κcrit-Numerical studies

Numerical study for κcrit = 1

»4Nǫ(σ + 1) 2σ + 1 – 1

σ

< Rthresh < [4ǫN(σ + 1)]

1 σ .

(16)

2 3 4 5 6 7 8 9 10 1 1.5 2 2.5 3 3.5

σ Rthresh

N=1. κ=1

1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16

σ Rthresh

N=2. κ=1

Figure 1 (a) σ ≥ 2. (b) σ ≥ 1.

• 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 κcrit = 1, N = 1, 2 κcrit = 1, N = 3 κcrit = 2, N = 1, 2 κcrit = 2, N = 3 κcrit = 3, N = 1, 2 κcrit = 3, N = 3

Determination of κcrit-Numerical studies

Numerical study for κcrit = 1

1 2 3 4 5 6 7 8 9 10 10 20 30 40 50 60 70 80 90

σ Rthresh

N=3. κ=1

0.7 0.8 0.9 1 1.1 1.2 5 10 15 20 25

Figure 2 σ ≥ 2/3. Formula (16) is valid for the case N = 1, 2 but also of good accuracy for N = 2 and very good for N = 3 for σ ≥ 1 with a discrepancy regarding the prediction of the lower bound Rlb appearing in the interval σ ∈ (2/3, 1). In the light of (15), κcrit = 1 is effective for all σ ≥ 1.

• 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 κcrit = 1, N = 1, 2 κcrit = 1, N = 3 κcrit = 2, N = 1, 2 κcrit = 2, N = 3 κcrit = 3, N = 1, 2 κcrit = 3, N = 3

Determination of κcrit-Numerical studies

Numerical study for κcrit = 2

»2 √ 3Nǫ(σ + 1) 2σ + 1 – 1

σ

< Rthresh < [4ǫN(σ + 1)]

1 σ .

(17)

2 3 4 5 6 7 8 9 10 1 1.5 2 2.5 3 3.5

σ Rthresh

N=1. κ=2

1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16

σ Rthresh

N=2. κ=2

Figure 3 (a) σ ≥ 2. (b) σ ≥ 1.

• 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 κcrit = 1, N = 1, 2 κcrit = 1, N = 3 κcrit = 2, N = 1, 2 κcrit = 2, N = 3 κcrit = 3, N = 1, 2 κcrit = 3, N = 3

Determination of κcrit-Numerical studies

Numerical study for κcrit = 2

1 2 3 4 5 6 7 8 9 10 10 20 30 40 50 60 70 80 90

σ Rthresh

N=3. κ=2

0.7 0.8 0.9 1 1.1 1.2 5 10 15 20 25

Figure 4 σ ≥ 2/3. The discrepancy now is observed in the interval σ ∈ (2/3, 0.72) which is reduced in comparison with Figure 2.

• 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 κcrit = 1, N = 1, 2 κcrit = 1, N = 3 κcrit = 2, N = 1, 2 κcrit = 2, N = 3 κcrit = 3, N = 1, 2 κcrit = 3, N = 3

Determination of κcrit-Numerical studies

Numerical study for κcrit = 3

»√ 5 3 · 4Nǫ(σ + 1) 2σ + 1 – 1

σ

< Rthresh < [4ǫN(σ + 1)]

1 σ .

(18)

2 3 4 5 6 7 8 9 10 1 1.5 2 2.5 3 3.5

σ Rthresh

N=1. κ=3

1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16

σ Rthresh

N=2. κ=3

Figure 5 (a) σ ≥ 2. (b) σ ≥ 1.

• 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 κcrit = 1, N = 1, 2 κcrit = 1, N = 3 κcrit = 2, N = 1, 2 κcrit = 2, N = 3 κcrit = 3, N = 1, 2 κcrit = 3, N = 3

Determination of κcrit-Numerical studies

Numerical study for κcrit = 2

1 2 3 4 5 6 7 8 9 10 10 20 30 40 50 60 70 80 90

σ Rthresh

N=3. κ=3

Figure 6 σ ≥ 2/3. The discrepancy observed in Figures 2 and 4 dissapears.

• 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

Comments

◮ Letting κ ∈ Z+ and N being fixed, we observe that since

limκ→∞ Rmin,2(κ) = 0, we can always find κcrit(N) ≥ N such that "p 2κcrit(N) − 1 κcrit(N) · 4Nǫ(σ + 1) 2σ + 1 # 1

σ

< Rthresh < [4ǫN(σ + 1)]

1 σ ,

(19) for all 1 ≤ N ≤ κcrit(N).

◮ The numerical study for the cases N = 1, 2, 3 and (19) suggest that it is

justified to consider κcrit(N) = N and that »√ 2N − 1 N · 4Nǫ(σ + 1) 2σ + 1 – 1

σ

< Rthresh < [4ǫN(σ + 1)]

1 σ , for all 1 ≤ N ≤ 3,

(20) which is of valuable accuracy for N = 2, 3.

• 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

Comments

◮ It would be interesting to compare with the approximating formulas for the

excitation threshold of J. Dorignac, J. Zhou and D.K. Campbell, Physica D 237 (2008), 486–504. Rthresh ≃ " 2ǫ Λ √ σN − 1 „σN − 1 σN − 2 « σN

2 −1# 1 σ−1

, σ ∈ N, σ > 2/N. (exponential ansatz φn = „1 − λ2 1 + λ2 «N/2 |λ||n|, |n| =

N

X

i

|ni|, (21) and the single nonlinear impurity model (SNI)).

• 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

Comments

◮ According to J. Dorignac, J. Zhou and D.K. Campbell:

“In 1D, the exponential ansatz is more accurate than the SNI solution close to the anti-continuum limit, while the opposite result holds in higher dimensions. The excitation thresholds predicted by the SNI solution are in excellent agreement with the exact results but cannot be obtained analytically except in

• 1D. An EA approach to the SNI problem provides an approximate analytical

solution that is asymptotically exact as σ tends to infinity. But the EA result degrades as the dimension N, increases. This is in contrast to the exact SNI solution which improves as σ and/or N increase.”

• 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

And....

Many thanks and congratulations to the organizing commitee for the excellent conference!!!

• J. Cuevas, N.I. Karachalios and F. Palmero

Lower and upper estimates on the excitation threshold for DNLS lattices