Mathematical Modelling Theory and Application 9781470469658, 9781470473891

Table of contents :
Cover
Title page
Contents
Preface
Travelling waves in nonlinear lattices
1. Introduction
2. Metamaterial formed by a discrete array of nonlinear resonators
3. Implicit nonlinear lattices with applications in metamaterials
4. Nonlinear magnetic metamaterials with parity-time-symmetric potentials
5. Discrete nonlinear Schrödinger equations with nonlocal interactions
6. Discrete nonlinear Schrödinger equations with local interactions
7. Forced Fermi-Pasta-Ulam-type lattices
8. Fractional differential equations on lattices
Acknowledgments
References
Dynamics of an SEIR Model for Infectious Diseases in Random Environments
1. Introduction
2. The models
3. Preliminaries
4. Long term dynamics
5. Numerical simulations
6. Closing remarks
Acknowledgment
References
On a Parabolic-ODE chemotaxis system with periodic asymptotic behavior
1. Introduction
2. Global existence of solutions
3. Asymptotic behavior
4. Numerical approach
References
Atrial fibrillation through strange attractor dynamics
1. Introduction
2. Experimental aspects
3. The reconstruction of EKG signals through Scale Relativity Theory
Conclusions
Appendix A
Appendix B
References
Good covers for vortex nerve cell complexes. free group presentation of intersecting nested cycles in planar CW spaces
1. Introduction
2. Preliminaries
3. Rotman Free Group Presentation
4. Main Results
5. Good covers for Cell Complexes
6. Application
7. Conjectures
Acknowledgments
Appendix A. Cell Complexes
References
Back Cover

Citation preview

787

Mathematical Modelling Theory and Application

Hemen Dutta Editor

Mathematical Modelling Theory and Application

Hemen Dutta Editor

787

Mathematical Modelling Theory and Application

Hemen Dutta Editor

EDITORIAL COMMITTEE Michael Loss, Managing Editor John Etnyre

Angela Gibney

Catherine Yan

2020 Mathematics Subject Classification. Primary 34A33, 37C29, 37H10, 92D30, 35B10, 35B40, 92B05, 92C50, 55P57, 55U10.

Library of Congress Cataloging-in-Publication Data Names: Dutta, Hemen, 1981– editor. Title: Mathematical modelling : theory and application / Hemen Dutta, editor. Description: Providence, Rhode Island : American Mathematical Society, [2023] | Series: Contemporary mathematics, 0271-4132 ; volume 787 | Includes bibliographical references. Identifiers: LCCN 2023006327 | ISBN 9781470469658 (paperback) | 9781470473891 (ebook) Subjects: LCSH: Mathematical models. | AMS: Ordinary differential equations – General theory – Lattice differential equations. | Dynamical systems and ergodic theory – Smooth dynamical systems: general theory – Homoclinic and heteroclinic orbits. | Dynamical systems and ergodic theory – Random dynamical systems – Generation, random and stochastic difference and differential equations. | Biology and other natural sciences – Genetics and population dynamics – Epidemiology. | Partial differential equations – Qualitative properties of solutions – Periodic solutions. | Partial differential equations – Qualitative properties of solutions – Asymptotic behavior of solutions. | Biology and other natural sciences – Mathematical biology in general – General biology and biomathematics. | Biology and other natural sciences – Physiological, cellular and medical topics – Medical applications (general). | Algebraic topology – Homotopy theory – Proper homotopy theory. | Algebraic topology – Applied homological algebra and category theory – Simplicial sets and complexes. Classification: LCC QA401 .M39283 2023 | DDC 511/.8–dc23/eng/20230612 LC record available at https://lccn.loc.gov/2023006327

Color graphic policy. Any graphics created in color will be rendered in grayscale for the printed version unless color printing is authorized by the Publisher. In general, color graphics will appear in color in the online version. Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2023 by the American Mathematical Society. All rights reserved.  The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

28 27 26 25 24 23

Contents

Preface

vii

Travelling waves in nonlinear lattices ˇkan Michal Fec

1

Dynamics of an SEIR model for infectious diseases in random environments Yusuke Asai, Jiaqi Cheng, and Xiaoying Han

27

On a Parabolic-ODE chemotaxis system with periodic asymptotic behavior M. Negreanu, J. I. Tello, and A. M. Vargas

55

Atrial fibrillation through strange attractor dynamics Maricel Agop, Alina Gavrilut ¸ , Lucian Eva, and Iulian-Alin Ros¸u

93

Good covers for vortex nerve cell complexes. free group presentation of intersecting nested cycles in planar CW spaces J. F. Peters

v

139

Preface The book covers several topics in areas like travelling waves, epidemiology, the chemotaxis system, atrial fibrillation, and nerve complexes to provide key ideas and theories, as well as their applications in developing various mathematical models. The works of the chapters have also relevance in fields such as physics, biology, health, and sociology. This book is intended for a wide range of readers, including researchers, professionals, educators, and students interested in applied mathematics for modelling various physical phenomena. There are five chapters in this book, and some major aspects of the chapters are presented below. The chapter “Travelling waves in nonlinear lattices” discusses some recent results on travelling wave solutions for differential equations on lattices modelled by metamaterials, discrete nonlinear Schr¨odinger equations, Fermi-Pasta-Ulam models, and fractional differential equations on lattices. Model equations for the propagation of nonlinear electromagnetic waves in metamaterials can be grouped into two classes. The first class assumes that effective metamaterials are homogeneous media with specific physical properties that result in partial differential equations such as coupled short-pulse equations and higher-order nonlinear Schr¨ odinger equations. Metamaterials in the second class are represented by arrays of coupled oscillators, i.e., lattice equations such as a nonlinear Klein-Gordon equation and coupled Klein-Gordon equations. It takes the second approach and considers both local and nonlocal couplings. The chapter “Dynamics of an SEIR model for infectious diseases in random environments” begins with a review of the original deterministic and autonomous SEIR model, followed by a stochastic SEIR model with white noise. It develops a random SEIR model and demonstrates the fundamental difference between the deterministic and random models. The random SEIR model’s well-posedness is established, and it is demonstrated that it has a unique nonnegative and bounded global solution. It also includes preliminaries on canonical noise and random dynamical systems, which are used to investigate the dynamics of solutions to the random SEIR model. Furthermore, numerical simulations for the developed random SEIR model are presented to demonstrate the relevant theoretical results obtained. The chapter “On a parabolic-ODE chemotaxis system with periodic asymptotic behaviour” presents a system of differential equations that models chemotaxis, which is the ability of some living organisms to move towards a higher concentration of a chemical signal. The system is made up of two differential equations, one parabolic that describes the behaviour of a biological species and the other that models the concentration of a chemical substance. The main results are the existence and uniqueness of global and bounded classical solutions, asymptotic behaviour towards a time-periodic function, solution of the corresponding ODE system associated with vii

viii

PREFACE

a time-periodic function approaching a given function acting in the reaction term, and some numerical approximations using the generalized finite difference method, which is based on mesfree, Taylor’s expansion, and the least squares procedure, producing a convergent result. The chapter “Atrial fibrillation through strange attractor dynamics” presents an application of the scale relativity theory in the description of the atrial phenomena. Fractalization is achieved through stochasticization, and it employs two procedures to describe the heart dynamics. In the first operating procedure, the diagnosis and evolution of atrial fibrillation by applying the method of nonlinear dynamics, the values of asymmetry and courtesy are in accordance with the pulse frequency distributions in the histograms of the analysed ECG signal. The second procedure involves the reconstruction of EKG signals using harmonic mappings between ordinary and hyperbolic space. It emphasises that the two operational procedures are complementary, with the goal of obtaining valuable information about fibrillation crises. The chapter “Good covers for vortex nerve complexes. Free group presentation of intersecting nested cycles in planar CW spaces” discusses good covers for cell complexes in the form of path nerve complexes in a planar Whitehead CW space, as well as their Rotman free group presentations. The main results are that every path triangle cluster has a free group presentation, every path triangle cluster has a free group presentation, every path vortex has a free group presentation, every path vortex nerve has a free group presentation, a vortex nerve and the union of the sets in the nerve have the same homotopy type, and every path triangulaton of a cell complex has a good cover. In terms of video frame shape approximation, vortex nerves are applied. I would like to thank the American Mathematical Society (AMS) for agreeing to publish this volume. I would also like to thank the contributors, reviewers, and editors at the AMS for their cooperation in making this volume a success. Hemen Dutta

Contemporary Mathematics Volume 787, 2023 https://doi.org/10.1090/conm/787/15790

Travelling waves in nonlinear lattices Michal Feˇckan Abstract. We present our recent results on travelling wave solutions for differential equations on lattices modeled by metamaterials, discrete nonlinear Schr¨ odinger equations, Fermi-Pasta-Ulam models, and fractional differential equations on lattices. Both local and nonlocal couplings are considered. The obtained results are rather broad and various.

1. Introduction Metamaterials are artificial materials that are engineered to have properties that may not be found in nature [9]. They were predicted theoretically in [23, 44] and demonstrated experimentally in, e.g., [11, 21]. Model equations for the propagation of nonlinear electromagnetic waves in metamaterials can be grouped into two classes. The first approach assumes that effective metamaterials are homogeneous media with specific physical properties resulting in partial differential equations, such as coupled short-pulse equation [42] and higher-order nonlinear Schr¨ odinger equations [43]. In the second class, metamaterials are modelled by arrays of coupled oscillators, i.e., lattice equations, such as a nonlinear Klein-Gordon equation [35] and coupled Klein-Gordon equations [30, 33]. Our study falls into the second approach presented in Sections 2, 3 and 4 based on our papers [1, 2, 10, 14]. Differential equations with nonlocal interactions on lattices have been studied in [3, 37], while DNLS (discrete nonlinear Schr¨odinger) in [7, 8]. Nowadays it is clear that a large number of important models of various fields of physics are based on DNLS type equations with several forms of polynomial nonlinearities starting with the simplest self-focusing cubic (Kerr) nonlinearity, then following with the cubic onsite nonlinearity relevant for Bose-Einstein condensates, then with more general discrete cubic nonlinearity in Salerno model up to cubic-quintic ones. We consider such problems in Section 5 following [15]. Damped and forced DNLS are studied in [20, 29] describing for instance the discrete cavity solitons. Other types of forced differential equations on lattices are investigated for instance in papers [6,46] involving Todda, Ablowitz-Ladik, FrenkelKontorova or sine-Gordon, Klein-Gordon and Fermi-Pasta-Ulam lattice models. Travelling discrete solitary waves of DNLS with vanishing tails are investigated in [36, 38]. Section 6 is devoted to such problems in [16]. 2020 Mathematics Subject Classification. Primary 34A33, 34K27, 37C29. Key words and phrases. Travelling wave, metamaterials, discrete nonlinear Schr¨ odinger equation, Fermi-Pasta-Ulam model, fractional differential equations on lattices. c 2023 American Mathematical Society

1

ˇ MICHAL FECKAN

2

The Fermi-Pasta-Ulam (FPU) model formulated in an attempt to explain heat conduction in non-metallic lattices [18] became a cornerstone in modern statistical mechanics [19]. Since then, the anharmonic oscillators have been used to study various problems, such as the relation between stochastic motions and thermodynamics properties [25, 39] and the heat conduction in a 1D chain [24, 34]. Discrete breathers of FPU models can also be created by applying a sinusoidal drive at one edge of a semi-infinite chain [27]. This direction is followed in Section 7 pursuing [13]. Recently, fractional differential equations (FDE) have proved to be valuable tools in the modeling of many phenomena in various fields of engineering, physics and economics. They have applications in nonlinear oscillations of earthquakes, many physical phenomena such as seepage flow in porous media, and in fluid dynamic traffic models. Actually, FDE are considered as an alternative model to integer differential equations [17, 28, 45]. Section 8 presents some results on FDE on lattices from [12] along with new ones. 2. Metamaterial formed by a discrete array of nonlinear resonators In this section, we consider the propagation of electromagnetic waves in metamaterials given by coupled Klein-Gordon equations [10, 30, 33]

(2.1)

dqn = in , dt

n ∈ Z,

d (λin−1 − in + λin+1 ) = γin − f (t) + ϕ(qn ), dt

with loss coefficient γ, coupling parameter λ, external forcing f and nonlinear function ϕ. We investigate the existence of travelling periodic solutions of system (2.1) with periodic forcing, and the bifurcation of periodic solutions and asymptotic waves of (2.1) with small γ, λ, and f . We consider in particular the case where the external force also varies in space. 2.1. Existence results on periodic solutions. In this subsection, we study the existence of periodic solutions of system (2.1) forced by a travelling wave field where the amplitude of periodic solutions is proportional to the amplitude of forcing. Thus, we consider the equivalent equation (2.2)

qn−1 = γ q˙n + ϕ(qn ) + f cos(ωt + pn), λ¨ qn+1 − q¨n + λ¨

where as above γ ≥ 0 is the dissipative loss of the medium, λ ∈ R is the coupling constant between nearest neighbor resonators, ω > 0 is the external driving frequency, f = 0 is the amplitude of the external force, p = 0 is the wavenumber of the travelling wave field, and ϕ is the nonlinearity of the magnetic material, which is normally assumed to be of Kerr-type [33]. Therefore, we assume that ϕ(z) is an odd analytic function in its variable z with radius of convergence ρ > 0 such that (2.3)

ϕ (0) = 0.

Seeking for waves travelling in the same direction as the external drive, we take qn (t) = U (z), z = ωt + pn, in (2.2) with U (z + π) = −U (z) to obtain (2.4)

ω 2 (λU  (z + p) − U  (z) + λU  (z − p)) = γωU  (z) + ϕ(U (z)) + f cos z.

TRAVELLING WAVES IN NONLINEAR LATTICES

3

Note that p ∈ R\{0}. Considering the case p = −π, one will obtain alternating charges between the nearest-neighbor resonators as f cos(ωt + pn) = (−1)n f cos ωt. We take a Banach space     1 (2k+1)ız Y := U ∈ C (R, R) | U (z) = ck e , |2k + 1||ck | < ∞ , k∈Z

with a norm U 1 :=



|2k + 1||ck |,

k∈Z

U 2 :=

k∈Z



(2k + 1)2 |ck |.

k∈Z

Denote B(ρ) := {U ∈ Y | U 1 < ρ}. Now we have the following existence results on (2.4) when all parameters except f are fixed [10]. Theorem 2.1. Assume  2 (2.5) Θ := inf ω 4 (2k + 1)2 (2λ cos(2k + 1)p − 1) + γ 2 ω 2 > 0 k∈Z

along with |f | < |fl |

(2.6) for fl satisfying A(r) := Θr −

∞  |ϕ(k) (0)| k=3

k!

r k = |fl |,

A (r) = 0

for some r ∈ (0, ρ). Then equation (2.4) has a unique solution U (f ) ∈ B(ρf ) in a closed ball where ρf < ρ is the smallest positive root of A(r) = |f |. Moreover, U (f ) can be approximated by an iteration process. Finally, U (f1 ) − U (f2 )1 ≤

Θ−

|f1 − f2 | |ϕ(k) (0)| k−1 k=3 (k−1)! ρmax{|f1 |,|f2 |}

∞

holds for any f1 , f2 ∈ R satisfying (2.6). In the following special cases, (2.5) holds and we can replace Θ with the corresponding Θi in the above considerations: 1. If γ > 0, then Θ ≥ Θ1 := γω > 0. 2. If |λ| < 1/2, then Θ ≥ inf ω 2 |2k + 1|(1 − 2λ cos(2k + 1)p) ≥ ω 2 (1 − 2|λ|) =: Θ2 > 0. k∈Z

1 , we can apply the identity 3. If |λ| > 1/2 and μ := arccos 2λ

cos x − cos y = −2 sin

x−y x+y sin 2 2

to obtain 2λ cos(2k + 1)p − 1 = −4λ sin

(2k + 1)p − μ (2k + 1)p + μ sin . 2 2

Next, if ν := μp ∈ 2Z and p has the form p = 2pp12+1 π with 2p1 + 1 and p2 relatively prime integers (their only common divisor is 1), then we can write (2.7)

|2λ cos(2k + 1)p − 1| = 4|λ|| sin K+ (k)|| sin K− (k)|,

ˇ MICHAL FECKAN

4

where K± (k) :=

(2k+1±ν)(2p1 +1)π . 2p2

So

(odd number)π ∈ / πZ even number for each k ∈ Z. Moreover, note that sin K± (k + p2 ) = − sin K± (k) ∀k ∈ Z. Therefore, it is sufficient to take in (2.5) the inf for k from a set of only |p2 | subsequent integers, i.e., K± (k) =

Θ ≥ inf ω 2 |2k + 1||2λ cos(2k + 1)p − 1| k∈Z

(2.8)

≥ 4ω |λ| 2

min

k=1,...,|p2 |

| sin K+ (k)|| sin K− (k)| =: Θ3 > 0.

1 ,p= 4. If |λ| > 1/2 and ν := μp ∈ R\Q for μ := arccos 2λ prime p1 , p2 ∈ Z\{0}, then we get equality (2.7) where

K± (k) :=

(2.9)

p1 p2 π

with relatively

(2k + 1 ± ν)p1 π ∈ / πZ 2p2

for each k ∈ Z. Furthermore, sin K± (k + p2 ) = (−1)p1 sin K± (k) ∀k ∈ Z. Hence, (2.8) holds with K± (k) given by (2.9). 5. If (2.3) does not hold, i.e., ϕ (0) = 0 then we take 

2 ϕ (0)  := inf Θ + γ 2 ω2 > 0 ω 2 (2k + 1)(2λ cos(2k + 1)p − 1) − k∈Z 2k + 1 instead of Θ. 2.2. Bifurcation of periodic travelling waves. In this subsection, we consider (2.1) with small γ, λ, and f . So we consider the system dqn = in , dt

(2.10)

n ∈ Z,

d (ελin−1 − in + ελin+1 ) = εγin − εh(ωt + pn) + ϕ(qn ), dt for C 2 -smooth and 2π-periodic h, ϕ ∈ C 2 (R, R), and ω > 0, p = 0, and ε = 0 is a small parameter. Equation (2.10) implies (ελ¨ qn−1 − q¨n + ελ¨ qn+1 ) = εγ q˙n − εh(ωt + pn) + ϕ(qn ).

(2.11)

Putting qn (t) = U (ωt + pn) for U ∈ C 2 (R, R) in (2.11), we get (2.12) ω 2 U  (z)+ϕ(U (z))−ελω 2 (U  (z − p) + U  (z + p))+εγωU  (z)−εh(z) = 0. Now, we assume (H1) U  (z) + ϕ(U (z)) = 0 has a T -periodic solution U0 . Remark 2.2. Since U0 (−z + c0 ) also solves U  (z) + ϕ(U (z)) = 0, and there is z0 ∈ R such that U0 (z0 ) = 0, we may suppose that U0 (0) = 0 and then U0 (z) = U0 (−z). Then, Uω (z) := U0 (z/ω) 2

Uω (z)

+ ϕ(Uω (z)) = 0. Note Uω is Tω := T ω-periodic and even. We satisfies ω assume the resonance condition (H2) Tω = 2π uv for u, v ∈ N.

TRAVELLING WAVES IN NONLINEAR LATTICES

5

Now, following the standard subharmonic Melnikov method of [22] to (2.12), we arrive at the following theorem (see [10] for more details). Theorem 2.3. Suppose (H1) and (H2) hold. If there is a simple zero α0 of a Melnikov function  T (−γωUω (z) + h(z + α)) Uω (z)dz, M u/v (α) := 0

i.e., M (α0 ) = 0 and = 0, then there is a δ > 0 such that for any 0 = ε ∈ (−δ, δ) there is a unique 2πu-periodic solution U (z) of (2.12) with

z − α0 U (z) = U0 + O(ε). ω u/v

d u/v (α0 ) dα M

To illustrate the theory, we consider ϕ(U ) = U + U 3 and so the equation in (H1) is the Duffing equation U  (z) + U (z) + U 3 (z) = 0 possessing a family of periodic solutions U0,a (z) = a cn

 1 + a2 z

4K(k) a , k = √2+2a . Note that U0,a (0) = a and for a > 0 with periods T = T (a) = √ 2 1+a2  U0,a (0) = 0. Here cn is the Jacobi elliptic function, K(k) is the complete elliptic function of the first kind, and k is the elliptic modulus. Now, we take h(z) = cos z. Then setting   √   √ 2K(k) (2 + a2 )K(k) − 2E(k) πK( 1 − k2 )u cosh < 1, Λ(a, u) := 3π 2 u 2K(k) a k=√ , 2 + 2a2 we obtain that 1 γ< Λ(a, u) gives the magnitude for the damping in order to apply Theorem 2.3.

2.3. Bifurcation of asymptotic travelling waves. In this subsection, we first consider, instead of (H1), the following one (C1) ϕ(0) = 0, ϕ (0) < 0 and U  (z)+ϕ(U (z)) = 0 has an asymptotic solution Γ ∈ C 2 (R, R) such that lim|z|→∞ Γ(z) = 0 and lim|z|→∞ Γ (z) = 0. We may again suppose, like in Remark 2.2, that Γ is even. Then Γω (z) := Γ(z/ω) satisfies ω 2 Γω (z) + ϕ(Γω (z)) = 0, lim|z|→∞ Γω (z) = 0, and Γω is even. By following the approach of Subsection 2.2 as in [22], we obtain the following result. Theorem 2.4. Suppose (C1) hold. If there is a simple zero β0 of the Melnikov function  ∞ (−γΓ (z) + h(ωz + β)) Γ (z)dz, M (β) := −∞

ˇ MICHAL FECKAN

6

then there is a θ > 0 such that, for any 0 = ε ∈ (−θ, θ), there is a unique bounded solution U (z) of (2.12) on R with

z − β0 U (z) = Γ + O(ε). ω To illustrate the theory, we consider ϕ(U ) = −U + U 3 and so the equation in (C1) is the Duffing equation U  (z) − U (z) + U 3 (z) = 0 possessing a homoclinic solution Γ(z) =

√ 2 sech z.

Again, h(z) = cos z. If

√ ωπ  3 2 ωπ sech , 4 2 then Theorem 2.4 can be applied. Next, we consider the following condition. (C2) ϕ(±1) = 0, ϕ (±1) < 0, and U  (z)+ϕ(U (z)) = 0 has an asymptotic solution Γ ∈ C 2 (R, R) such that limz→±∞ Γ(z) = ±1 and lim|z|→∞ Γ (z) = 0. Now U  (z) > 0 on R. Next Γ− (z) := Γ(−z) is a solution satisfying limz→±∞ Γ− (z) = ∓1 and lim|z|→∞ Γ− (z) = 0. So Γ and Γ− create a heteroclinic cycle. Next, we can suppose that Γ(0) = 0. If ϕ is odd, then Γ is odd. So Γ is even and Γ is odd. Now, we can repeat the above approach to derive the Melnikov function M (β) :=  ∞   p p  λ Γ z − + Γ z + − γΓ (z) + h(ωz + β) Γ (z)dz. ω ω −∞ γ
0 and varying in the spatial direction with wavenumber p = 0. The governing equation (3.1) is an implicit system of infinite dimensional differential equations on a lattice Z. These kinds of equations appear in many studies [31, 40]. Additionally, we also consider the system d d2 (un − λun−1 − λun+1 ) + γ un + un dt2 dt  γ d 3 1 d2  3 u − h(ωt + pn) = 0 un − λu3n−1 − λu3n+1 − − 2 3 dt 3 dt n that admits periodic and heteroclinic solutions.

(3.2)

3.1. The homoclinic case. Putting un (t) = u(ωt + pn), z = ωt + pn in (3.1), we obtain ω 2 (u(z) − λu(z − p) − λu(z + p)) + γωu (z) + u(z) (3.3) + ω 2 (u2 (z) − λu2 (z − p) − λu2 (z + p)) + γω(u2 (z)) − h(z) = 0. For simplicity, we take ω = 1 in (3.3) and we consider weak couplings and forcing, so in this subsection we look for P −periodic solutions of the advanced-delay equation   (u + u2 ) + u = ελ u(z − p) + u2 (z − p) (3.4)   +ελ u(z + p) + u2 (z + p) − εγ(u + u2 ) + εh(z), where h ∈ C 2 (R, R) is P −periodic and ε = 0 is small. Setting v = (u2 + u) and expanding derivatives in (3.4), we arrive at the system: v , u = 2u +1 (3.5) v  = −u + ε (λv  (z − p) + λv  (z + p) − γv + h(z)) , whose unperturbed equation (i.e., with ε = 0) v , u = 2u + 1  v = −u, has the homoclinic solution 1 z(z 2 − 9) (3 − z 2 ), v¯(z) = , 0 ≤ |z| < 3. u ¯(z) = 12 36 The loop generated by (¯ u, v¯) is filled out with a family of T (c)-periodic solutions {(uc , vc )}c∈(0,1/4) accumulating on (¯ u, v¯) as c → 1/4− (see [14] for more details). Then . lim− T (c) = 6, lim+ T (c) = 2π = 6.28319, c→0

c→ 14

and T (c) is strictly decreasing. Applying the standard method, the Poincar´eMelnikov function of (3.5) is given by  T (c)  T (c) Mc (α) := −γ vc (z)2 dz + h(z + α)vc (z)dz. 0

Hence, the following result is valid.

0

ˇ MICHAL FECKAN

8

Theorem 3.1. Let h ∈ C 2 (R, R) be T (c)−periodic for some c ∈ (0, 1/4), i.e., P = T (c). If there is an α0 ∈ R so that Mc (α0 ) = 0 and Mc (α0 ) = 0, then (3.4) has a T (c)−periodic solution uε (z) for any small ε = 0 such that lim sup |uε (z) − uc (z − α0 )| = 0.

ε→0 z∈R

Applying Theorem 3.1, we have the following theorem. Theorem 3.2. If 45 . = 1.45132 π3 π and 0 < 3 − δ is sufficiently small and fixed, then (3.4) with h(z) = cos δz has a 2π δ −periodic solution for any λ > 0, p = 0 fixed and ε = 0 small. 0 0, p = 0 are parameters and h ∈ C(R, R) is such that   hk ekız , |hk | < ∞. h(z) = k∈Z

k∈Z

Now we take Banach spaces  X := ⎧ ⎨ Z :=

⎩

U ∈ C(R, R) | U (z) =



kız

ck e

k∈Z

U ∈ C 2 (R, R) | U (z) =



with the norms U  :=



|ck |,

U 2 := |c0 | +

k∈Z

 |ck | < ∞ ,

k∈Z

ck ekız ,

k∈Z

,



 k∈Z\{0}



⎫ ⎬ k2 |ck | < ∞ ⎭

k2 |ck |,

k∈Z\{0}

respectively. First, we present the following existence result for (3.3) when all parameters except h are fixed. Theorem 3.5. Assume ⎧  ⎫

2 ⎨ 2 ω2 ⎬ 1 γ (3.9) Θ := inf + ω 2 (2λ cos kp − 1) + 2 1, >0 k2 k ⎭ k∈Z\{0} ⎩ for a constant depending on γ, λ, ω, and p, along with 0 < h
12 , then Θ ≥ Θ1 := min{1, ω 2 (2λ − 1)} > 0. 2. Let γ = 0. If p ∈ πQ, then we can write p = pp12 π for some p1 ∈ Z, p2 ∈ N, where p1 , p2 are relatively prime (their only common divisor is 1). Moreover,   kπ (3.11) M1 := {cos kp | k ∈ Z\{0}} ⊂ cos | k ∈ {0, 1, . . . , 2p2 − 1} =: M2 . p2 Indeed, for each k ∈ Z there exist i, j ∈ Z such that 0 ≤ i ≤ 2p2 − 1 and kp1 = 2jp2 + i. Hence, for cos kp ∈ M1 , we have kp1 π iπ cos kp = cos = cos ∈ M2 . p2 p2 

To find a better relationship between M1 and M2 , we need to solve cos kp1 p2 1 π = cos kp22π for  k1 ∈ Z\{0} and k2 ∈ {0, 1, . . . , 2p2 − 1}. This is equivalent to kp1 p2 1 π = k2 π 1 , i.e., k1 p1 = k2 + 2lp2 . If p1 is odd, then p1 + 2πl for l ∈ Z and k1 = ±k p2

ˆ ˆl ∈ Z such that and 2p2 are relatively prime. Thus, we know that there exist k, ˆ 2 p1 = k2 +2ˆlk2 p2 and M1 = M2 . If p1 is even, then ˆ 1 = 1+2ˆlp2 . Consequently, kk kp p1 and k1 2¯ p1 = k2 + 2lp2 , so k2 = 2k¯2 and we have k1 p¯1 = k¯2 + lp2 . Clearly, p1 = 2¯ ¯ ¯ ¯ = 1 + ¯lp2 . p¯1 and p2 are relatively prime. Thus, there  exist k, l ∈ Z such that k p¯1  2kπ ¯ ¯ ¯ ¯ ¯ Then, k k2 p¯1 = k2 + lk2 p2 . Hence, M1 = cos p2 | k ∈ {0, 1, . . . , p2 − 1}  M2 . Nevertheless, assuming   1 (3.12) ω2 ∈ / k2 (1 − 2λ cos kp) k∈Z\{0} and (3.13)

0∈ / 2λM2 − 1

for M2 defined in (3.11), we get the existence of δ > 0 such that   

2   1 γ 2 ω2 2 1 2 ≥δ>0 + ω (2λ cos kp − 1) + ≥ ω + 2λ cos kp − 1   k2 k2 ω2 k2 for each k ∈ Z\{0}. Thus, if p ∈ πQ and (3.12), (3.13) are valid, then     1  Θ ≥ min 1, inf ω 2  2 2 + 2λ cos kp − 1 =: Θ2 > 0. ω k k∈Z\{0} Note that k2 (1−2λ1 cos kp) → 0 if |k| → ∞. So if p ∈ πQ and (3.13) holds, there is at most a finite number of resonant modes k0 ∈ Z\{0} determined by equation 1 (3.14) ω2 = 2 . k0 (1 − 2λ cos k0 p)

TRAVELLING WAVES IN NONLINEAR LATTICES

11

More precisely, for given λ, ω there is not more than 2(p2 + 1) resonant modes k0 . Indeed, let {k0j }Jj=1 be an increasing sequence of positive resonant modes corresponding to the fixed λ, ω. Then, {1 − 2λ cos k0j p}Jj=1 has to be decreasing, since 1 2 (1 − 2λ cos k0j p) = 2 = const. ∀j = 1, . . . , J, k0j ω i.e., {cos k0j p}Jj=1 is increasing. From properties of cos x and (3.11), it follows that the longest sequence of the form cos k0j p has the values {cos kπ p2 | k = p2 , . . . , 2p2 }, which has p2 + 1 elements. Finally, to each k0j corresponds the resonant mode −k0j . 3. If γ > 0, condition (3.9) is satisfied even without the non-resonance condition (3.12), since in each resonant mode k0 , we have 

2 1 γ 2 ω2 γω 2 (2λ cos k p − 1) > 0, + ω + 2 = 0 2 k0 k0 |k0 | and there is only a finite number of resonant modes. For each non-resonant mode k,   

2 1  1 γ 2 ω2 2 2   > 0. + ω (2λ cos kp − 1) + ≥ + ω (2λ cos kp − 1)   k2 k2 k2 Summarizing, if γ > 0, p ∈ πQ, and (3.13) holds, condition (3.9) is satisfied and      γω 2 1 Θ ≥ min 1, min , inf ω  2 2 + 2λ cos kp − 1 =: Θ3 > 0, k∈M |k| k∈Z\({0}∪M ) ω k where M is the set of resonant modes k0 . 3.4. Simple resonances. In this subsection, we investigate the bifurcation of a small solution of (3.3) under the assumption of a simple resonance. So, we assume: (R) p ∈ πQ, condition (3.13) holds, for some k0 ∈ Z\{0} equation (3.14) holds and has the only solutions ±k0 . We consider the equation (3.15)

ω 2 (u(z) − λu(z − p) − λu(z + p)) + εγωu (z) + u(z) + ω 2 (u2 (z) − λu2 (z − p) − λu2 (z + p)) + εγω(u2 (z)) − εh(z) = 0,

for u and ε small. We have the following results [1]. Theorem 3.6. Let hk0 = μk0 + ıνk0 and assume (R) holds. Then we have • If (μk0 , νk0 ) = (0, 0), then equation (3.15) has a solution u ∈ Z close to 0 for any ε = 0 sufficiently small. • If γ = 0 and with εh instead of h in (3.15) and (μk0 , νk0 ) = (0, 0), then equation (3.15) has a solution u ∈ Z close to 0 for any ε = 0 sufficiently small. • If (μk0 , νk0 ) = (0, 0), then equation (3.15) has a solution u ∈ Z close to 0 for any ε = 0 sufficiently small provided 2 2 (4μ20 + γ 2 ω 2 k02 ) − 4(μ22k0 + ν2k ) = 0. δ2k 0 0

The order of this solution is O(ε). If, in addition, 2 2 (4μ20 + γ 2 ω 2 k02 ) − 4(μ22k0 + ν2k ) < 0, δ2k 0 0

ˇ MICHAL FECKAN

12

then (3.15) has another solution. This one is of order O( |ε|). Our analytical results are found to be in good agreement with the direct numerical computations presented in [1].

4. Nonlinear magnetic metamaterials with parity-time-symmetric potentials Consider a one-dimensional array of dimers, each comprising two nonlinear split-ring resonators (SRRs): one with loss and the other with an equal amount of gain as sketched in [32]. In the equivalent circuit model, extended for the PT dimer chain, the dynamics of the charge qn in the capacitor of the n-th SRR is governed by [2] q¨2n+1 + q2n+1 + λM q¨2n + λM q¨2n+2 + λE q2n + λE q2n+2 (4.1)

2 3 = ε sin(pn + Ωt) − aq2n+1 − βq2n+1 − γ q˙2n+1 ,

q¨2n+2 + q2n+2 + λM q¨2n+1 + λM q¨2n+3 + λE q2n+1 + λE q2n+3 2 3 = ε sin(pn + Ωt) − aq2n+2 − βq2n+2 + γ q˙2n+2 ,

where λM , λM and λE , λE are the magnetic and electric interaction coefficients, respectively, between nearest neighbours, a and β are nonlinear coefficients, γ is the gain or loss coefficient (γ > 0), ε is the amplitude of the external driving voltage, √ while Ω is the driving frequency normalized to ω0 = 1/ LC0 , and t temporal variable normalized to ω0−1 , with C0 being the linear capacitance. Replacing q2n+1 = Un and q2n+2 = Vn in (4.1), we obtain ¨n + Un + aUn2 + βUn3 U (4.2)

= −λM V¨n−1 − λM V¨n − λE Vn−1 − λE Vn − γ U˙ n + ε sin(pn + Ωt), V¨n + Vn + aVn2 + βVn3 ¨n − λM U ¨n+1 − λE Un − λE Un+1 + γ V˙ n + ε sin(pn + Ωt). = −λM U

Looking for travelling waves and hence setting z = pn + Ωt, U (z) = Un (t) and V (z) = Vn (t), (4.2) becomes Ω2 Uzz (z) + U (z) + aU 2 (z) + βU 3 (z) = −λM Ω2 Vzz (z − p) − λM Ω2 Vzz (z) − λE V (z − p) (4.3)

−λE V (z) − γΩUz (z) + ε sin z, 2

Ω Vzz (z) + V (z) + aV 2 (z) + βV 3 (z) = −λM Ω2 Uzz (z) − λM Ω2 Uzz (z + p) − λE U (z) −λE U (z + p) + γΩVz (z) + ε sin z.

To perform a Melnikov analysis, see more details in [2], we consider (4.3) for weak coupling, forcing and damping, which is expressed, after scaling λM → ελM ,

TRAVELLING WAVES IN NONLINEAR LATTICES

13

λM → ελM , λE → ελE , λE → ελE , and z → z/Ω, as Uzz (z) + U (z) + aU 2 (z) + βU 3 (z) = ε − λM Vzz (z − p) − λM Vzz (z)

 −λE V (z − p) − λE V (z) − γUz (z) + sin Ωz , (4.4)

Vzz (z) + V (z) + aV 2 (z) + βV 3 (z) = ε − λM Uzz (z) − λM Uzz (z + p) − λE U (z)  −λE U (z + p) + γVz (z) + sin Ωz ,

where the parameter |ε|  1 indicates the pertubative character in the above equations. For ε = 0 the unperturbed system is (4.5)

Uzz (z) + U (z) + aU 2 (z) + βU 3 (z) = 0, Vzz (z) + V (z) + aV 2 (z) + βV 3 (z) = 0.

We consider the following cases: a) a < 0 and β = 0, and b) a = 0 and β = −1, which represent systems with non-topological and topological localised waves.  Case (a). Both equations in (4.5) have a hyperbolic equilibrium (pi , q i ) =  4.1. 0, − a1 , i = 1, 2 where i = 1 and i = 2 refer to the first and second equation of (4.5), q1 = U , q2 = V , connected by a homoclinic solution 2

1 3 sech z2 . qi (z) = − + a 2a So in the full space R4 , the system (4.5) has a hyperbolic equilibrium pi = q˙i ,

(p1 , q 1 , p2 , q 2 ) connected by the homoclinic trajectory (p1h (z), q1h (z), p2h (z), q2h (z)) =   2 2 3 sech2 z2 tanh z2 3 sech2 z2 tanh z2 1 3 sech z2 1 3 sech z2 − ,− + ,− ,− + . 2a a 2a 2a a 2a Theorem 4.1. Let a < 0 and β = 0. The following condition γ | sinh Ωπ| < −5aΩ2 π is sufficient for the persistence of a homoclinic type solution of (4.4) for ε = 0 small. 4.2. Case (b). The equations of (4.5) possess heteroclinic solutions z pi = q˙i , qi (z) = tanh √ . 2 So in the full space R4 , the system (4.5) has hyperbolic equilibria ± ± ± (p± 1 , q 1 , p2 , q 2 ) = (0, ±1, 0, ±1)

connected by the heteroclinic trajectory   2 z sech2 √z2 z sech √2 z √ , tanh √ , √ , tanh √ . (p1h (z), q1h (z), p2h (z), q2h (z)) = 2 2 2 2

ˇ MICHAL FECKAN

14

Theorem 4.2. Let a = 0 and β = −1. The following condition  √   2γ  Ωπ Ωπ   0 < ±(λE + λE ) −  < √ csch √  3  2 2 for either sign is sufficient for the persistence of a heteroclinic type solution in (4.4) for ε = 0 small. We refer to [2] for interesting numerical simulations. Our analytical results are found to be in good agreement with direct numerical computations. For the particular nonlinearity admitting travelling kinks, numerically we observe homoclinic snaking in the bifurcation diagram. The Melnikov analysis yields a good approximation to one of the boundaries of the snaking profile. 5. Discrete nonlinear Schr¨ odinger equations with nonlocal interactions In this section, we consider the discrete nonlinear Schr¨ odinger equations on the lattice Z (DNLS) with nonlocal interactions of the form [15]  aj Δj un + f (|un |2 )un , n ∈ Z, (5.1) ıu˙ n = j∈N

where un ∈ C, Δj un := un+j + un−j − 2un are 1-dimensional discrete Laplacians and  (H1) f ∈ C(R+ , R) for R+ := [0, ∞), f (0) = 0, and aj ∈ R with j∈N |aj | < ∞. Moreover, there are constants s > 0, μ > 1, c1 > 0, c2 > 0, and r¯ > 0 such that |f (w)| ≤ c1 (ws + 1),

c2 (ws+1 − 1) ≤ F (w), μF (w) − r¯ < f (w)w w for any w ≥ 0, where F (w) = 0 f (z)dz. Furthermore, lim supw→0+ f (w)/ws < ∞ for a constant s > 0. Of course we suppose that not all aj are zero. Note any polynomial f (w) = p1 w + · · · + ps ws , s ∈ N with ps > 0 satisfies (H1). Furthermore, (5.1) can be rewritten into a standard form  (5.2) ıu˙ n = a|m−n| (um − un ) + f (|un |2 )un , n ∈ Z. m=n

It is well known that (5.2) conserves two dynamical invariants  |un |2 − the norm,  n∈Z

n∈Z

⎡

⎤  ⎣− 1 a|m−n| |um − un |2 + F (|un |2 )⎦ 2

− the energy.

m=n

We are interested in the existence of traveling wave solutions un (t) = U (n − νt) of (5.1) with a quasi periodic function U (z), z = n − νt and some ν = 0. First, we introduce the function #x $ 4 aj sin2 j . Φ(x) := x 2 j∈N

Clearly Φ ∈ C(R\ {0}, R), Φ is odd, Φ(2πk) = 0 for any k ∈ Z \ {0}, and Φ(x) → 0 as |x| → ∞. If j∈N j|aj | < ∞, then Φ ∈ C(R, R), and if j∈N j 2 |aj | < ∞, then

TRAVELLING WAVES IN NONLINEAR LATTICES

15

Φ ∈ C 1 (R, R). Consequently, the range RΦ := Φ(R \ {0}) is either an interval ¯ R] ¯ or (−R, ¯ R) ¯ here with possibility that R ¯ = ∞ (see Section 2.4 of [15] for [−R, concrete examples). Now we can state the following existence result. Theorem 5.1. Let (H1) hold and T > 0. Then for almost each ν ∈ R \ {0} and any rational r ∈ Q ∩ (0, 1), there is a nonzero periodic traveling wave solution un (t) = U (n − νt) of (5.1) with U ∈ C 1 (R, C) and such that U (z + T ) = e2πrı U (z), ∀z ∈ R . Moreover, for any ν ∈ R \ {0}, there is at most a finite number of constants r¯1 , r¯2 , · · · , r¯m ∈ (0, 1) such that the equation

2π (¯ rj + k) −ν = Φ T has a solution k ∈ Z. Then, for any r ∈ (0, 1) \ {¯ r1 , r¯2 , · · · , r¯m }, there is a nonzero quasi periodic traveling wave solution un (t) = U (n − νt) with the above properties. ¯ and r ∈ (0, 1), there is such a nonzero quasi periodic In particular, for any |ν| > R traveling wave solution. If a nonresonance condition of Theorem 5.1 fails, then we have the following bifurcation results. Theorem 5.2. Suppose f ∈ C 2 (R+ , R) with f (0) = 0. Assume there are r¯1 ∈ (0, 1), ν ∈ RΦ \ {0} and T > 0 such that all solutions k1 , k2 , · · · , km1 ∈ Z of equation

2π (¯ r1 + k) −ν = Φ T are either nonnegative or negative, and m1 > 0. Then for any ε > 0 small, there are m1 branches of nonzero quasi periodic traveling wave solutions un,j,ε (t) = Uj,ε (n − νε t) of (5.1) with Uj,ε ∈ C 1 (R, C), j = 1, 2, · · · , m1 , and nonzero velocity νε satisfying Uj,ε (z + T ) = e2π¯r1 ı Uj,ε (z), ∀z ∈ R along with νε → ν and Uj,ε ⇒ 0 uniformly on R as ε → 0. If aj ≥ 0 for all j ∈ N, then the assumptions of Theorem 5.2 are satisfied for T any ν ∈ RΦ \ {0} such that 2π Φ−1 (−ν) \ Z = ∅, and so there are bifurcations of quasi periodic traveling waves in the generic resonant cases. On the other hand, T Φ−1 (−ν) ⊂ Z, then Theorem 5.1 is applicable for any if ν ∈ RΦ \ {0} with 2π r ∈ (0, 1). Theorem 5.2 is a Lyapunov center theorem for traveling wave solutions. A discussion is given in Section 4 of [15] on the extension of these results of (5.1) on the lattices Z2 and Z3 (see also [7]). 6. Discrete nonlinear Schr¨ odinger equations with local interactions In this section, we consider forced DNLS (FDNLS) on the lattice Z of the form   (6.1) ıu˙ n = un+1 + un−1 − 2un + f |un |2 un + c, n ∈ Z , 0 = c ∈ C . Some known forms of f (for c = 0 - conservative case) are listed below: fDN LS (w) = w f3−5 (w) = w + αw 1 fsat (w) = 1+w

2

−

DNLS equation,

−

Cubic-quintic DNLS,

−

Saturable DNLS.

16

ˇ MICHAL FECKAN

Note (6.1) is non-Hamiltonian for c = 0 and c represents an external force or field. We consider travelling wave solutions of (6.1) of the form (6.2)

un (t) = U (n − νt) ,

where ν ∈ R, ν = 0 is a parameter. Inserting (6.2) into (6.1), we obtain a differential advance-delay equation:   (6.3) −ıνU  (z) = U (z + 1) + U (z − 1) − 2U (z) + f |U (z)|2 U (z) + c for z = n − νt. We are interested in the existence of 2π-periodic solutions U (z) of (6.3), i.e., U (z + 2π) = U (z), ∀z ∈ R . Similarly, we can study the existence and bifurcations of T -periodic solutions of (6.3) for some T > 0. We can take ce(βn−φ(t))ı in (6.1) instead of c and consider un (t) = U (n − νt)e(βn−φ(t))ı , where β ∈ R is a constant and φ ∈ C 1 (R, R) is such that φ is periodic. We should deal with more general form of (6.1), namely with      g (un−1 , un , un+1 ) = ∂x G |un |2 , |un+1 |2 + ∂y G |un−1 |2 , |un |2 un     instead of f |un |2 un for a function G(x, y) ∈ C 2 R2 , R . So (6.3) now has the form −ıνU  (z) = U (z + 1) + U (z − 1) − 2U (z)      + ∂x G |U (z)|2 , |U (z + 1)|2 + ∂y G |U (z − 1)|2 , |U (z)|2 U (z) + c for c ∈ C. But we study (6.3) for simplicity. For  theconservative case c = 0, (6.3) has plane waves un (t) = Aeı(qn−νt) with ν = f |A|2 − 4 sin2 q2 and A, q ∈ R. 6.1. Existence of periodic travelling waves. Clearly (6.3) has trivial travˆ0 = −A0 c/|c| with f (A20 )A0 = |c|, A0 ∈ R. The elling waves only constant ones U aim of this subsection is to show results on nonconstant periodic travelling waves (see [16] for more details). Two types of nonlinearity f are considered. 1. Polynomial types nonlinearities We make the following assumption: (H1) f ∈ C 2 (R+ , R) for R+ := [0, ∞). Moreover, there are constants s > 0, sˆ > 0, μ > 1, c1 > 0, c2 > 0 and r¯ > 0 such that |f (w)| ≤ c1 (ws + 1),

|f  (w)| ≤ c1 (wsˆ + 1) ,

c2 (ws+1 − 1) ≤ F (w), μF (w) − r¯ < f (w)w w for any w ≥ 0, where F (w) = 0 f (z)dz. Note any polynomial f (w) = p0 + p1 w + · · · + ps ws , s ∈ N with ps > 0 satisfies (H1), so FDNLS and cubic-quintic   FDNLS equations are included in our study. Next, (H1) implies limw→∞ f w2 w = ∞, so f (A20 )A0 = |c| has a solution A0 ∈ R. Now we can state the following result [16]. Theorem 6.1. Let (H1) hold. If f (A%20 )A 0=  |c|& = 0 has only   A0 ∈  one solution d f A2 A A=A0 = f A20 + 2f  A20 A20 = R, which, in addition, is simple, i.e. dα 0, then for each ν = 0 such that (6.4)



 2 2  2  2 1 2 k 2 k 2  ν = 2 f A0 − 4 sin f A0 + 2f A0 A0 − 4 sin ∀k ∈ Z \ {0}, k 2 2 there is a nonconstant 2π-periodic travelling wave solution of (6.1).

TRAVELLING WAVES IN NONLINEAR LATTICES

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2. Bounded nonlinearities In this part we assume: (H2) f ∈ C 2 (R+ , R) and limx→∞ f (x) = f (∞) ∈ R exists. Moreover, there is a constant sˆ > 0 such that |f  (w)| ≤ c1 (wsˆ + 1) for any w ≥ 0. 1 w Note that either f (w) = 1+w or f (w) = 1+μw , μ > 0, are such nonlinearities. So saturable FDNLS equations are included in our study.

Definition 6.2. The dimension of the subspace of all negative eigenvalues of a symmetric matrix L is called as the Morse index of L and it is denoted by M− (L). We are ready to state the following result [16]. Theorem 6.3. Suppose (H2) with f (∞) = 0. Assume for a ν = 0

1 2 k ν = f (∞) − 4 sin ∀k ∈ Z \ {0} . k 2 ˆi = −Ai c/|c|, Let f (A2i )Ai = |c| = 0, i = 1, 2, · · · , k, be the only real solutions. Set U i = 1, 2, · · · , k. If (see [16] for the next notations) #   $ ˆi ) , M− Hess J(U ˆi ) + dim ker Hess J(U ˆi ) M− (A) ∈ / M− Hess J(U for any i = 1, 2, · · · , k, then there is a nonconstant 2π-periodic travelling wave solution of (6.1). This result can be applied to saturable FDNLS equations with external potentials [26] of the form ıu˙ n = un+1 + un−1 − 2un +

un − γun + c, 1 + |un |2

n ∈ Z,

with parameters γ ∈ R \ [0, 1], 0 = c ∈ C. 6.2. Bifurcation of periodic travelling waves. In this section we consider (6.3) in case nonresonance condition (6.4) fails. So we consider (6.3) with f ∈ C 2 (R+ , R) and 0 = c ∈ C. Theorem 6.4. Suppose that A0 is a simple root of f (A20 )A0 = |c| = 0. Let ν = 0 be such that



 2 2  2  2 2 k 2 k 2 2  (6.5) ν k = f A0 − 4 sin f A0 + 2f A0 A0 − 4 sin 2 2 has a solution k = k1 ∈ N and let k = k1 , k2 , · · · , km ∈ N be all solutions of (6.5) in Z0 .     k If all 4 sin2 2j − f A20 − f  A20 A20 , j = 1, 2, · · · , m, are either positive or negative, then for any ε > 0 small, there are m branches of nonconstant 2π-periodic travelling wave solutions Uj,ε , j = 1, 2, · · · , m, of (6.1) with nonzero velocity νε such ˆ0 = −A0 c uniformly on R as ε → 0. that νε → ν and Uj,ε ⇒ U |c| The above results are directly extended in [16] to higher-dimensional FDNLS.

ˇ MICHAL FECKAN

18

7. Forced Fermi-Pasta-Ulam-type lattices We consider a 1D damped FPU lattice forced by a travelling wave field: (7.1)

u ¨n = α (un+1 + un−1 − 2un ) + β(un+1 − un )3 + β(un−1 − un )3 −γ u˙ n + f cos(ωt + pn) ,

where α > 0, β > 0, γ ≥ 0, ω > 0, p = 0, f = 0 are parameters. Putting un (t) = U (ωt + pn) in (7.1), we obtain (7.2)

ω 2 U  (z) = α (U (z + p) + U (z − p) − 2U (z)) +β(U (z + p) − U (z))3 + β(U (z − p) − U (z))3 − γωU  (z) + f cos z,

and seek solutions of the advance-delay equation satisfying the property U (z + π) = −U (z).

(7.3)

We take the same Banach space Y as in Subsection 2.1 with the same norm. The following results hold [13]. Theorem 7.1. Assume (7.4)

α

4 sin2 2k+1 γω 2 p ı = 0 − ω2 + (2k + 1)2 2k + 1

along with

 |f |