Moving kinks and nanopterons in the nonlinear Klein–Gordon lattice

Savin, A. V.; Zolotaryuk, Yaroslav; Eilbeck, J. C.

doi:10.1016/s0167-2789(99)00202-x

Cited by 54 publications

(81 citation statements)

References 23 publications

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“…The existence of traveling kinks showing similar oscillatory tails is reported in several references (see, e.g. [26]). Pulsating solitary waves in the FPU lattice are analyzed by Iooss and one of us (G.J.)…”

Section: Introductionsupporting

confidence: 85%

Center Manifold Theory in the Context of Infinite One-Dimensional Lattices

James

Sire

The Fermi-Pasta-Ulam Problem

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Abstract. Center manifold theory has been used in recent works to analyze small amplitude waves of different types in nonlinear (Hamiltonian) oscillator chains. This led to several existence results concerning traveling waves described by scalar advance-delay differential equations, pulsating traveling waves determined by systems of advance-delay differential equations, and time-periodic oscillations (including breathers) obtained as orbits of iterated maps in spaces of periodic functions. The Hamiltonian structure of the governing equations is not taken into account in the analysis, which heavily relies on the reversibility of the system. The present work aims at giving a pedagogical review on these topics. On the one hand, we give an overview of existing center manifold theorems for reversible infinite-dimensional differential equations and maps. We illustrate the theory on two different problems, namely the existence of breathers in Fermi-Pasta-Ulam lattices and the existence of traveling breathers (superposed on a small oscillatory tail) in semi-discrete KleinGordon equations. IntroductionThe seminal work of Fermi, Pasta and Ulam in 1955 [12] had a broad impact on the development of the theory of nonlinear lattices. The Fermi-PastaUlam (FPU) model consists of a chain of masses nonlinearly coupled to their nearest neighbors. For a general nearest-neighbors interaction potential V , the governing equations readwhere we note x n the displacement of the nth mass with respect to an equilibrium position (in this version of the model the chain is of infinite extent). The anharmonic interaction potential V satisfies V (0) = 0, V (0) = 1. System (6.1) is Hamiltonian, with total energy

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Section: Introductionsupporting

confidence: 85%

Center Manifold Theory in the Context of Infinite One-Dimensional Lattices

James

Sire

The Fermi-Pasta-Ulam Problem

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“…After substituting the ansatz (6) into Eq. (5) Currently it is a well-established fact [10][11][12][13][14][18][19][20] that for the discrete Klein-Gordon equations of the type (3), the continuous family of moving solitons turns into the discrete finite set of monotonic …”

Section: Soliton Mobility In the Hamiltonian Limitmentioning

confidence: 99%

“…If the inductive coupling between the junctions is small, the free fluxon motion becomes impossible. However, the possibility of radiationless propagation of solitary waves in discrete media for selected values of velocity has been reported in the number of papers [10][11][12][13][14][15][16][17][18][19][20]. Also, free propagation of the bound states of several discrete topological solitons has been reported by Peyrard and Kruskal more than 30 years ago [8].…”

Section: Introductionmentioning

confidence: 98%

Embedded soliton dynamics in the asymmetric array of Josephson junctions

Starodub

Zolotaryuk

2017

Low Temperature Physics

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The dc-biased annular array of three-junction asymmetric superconducting quantum interference devices (SQUIDs) is investigated. The existence of embedded solitons (solitons that exist despite the resonance with the linear waves) is demonstrated both in the unbiased Hamiltonian limit and in the dc-biased and damped case on the current-voltage characteristics (CVCs) of the array. The existence diagram on the parameter plane is constructed. The signatures of the embedded solitons manifest themselves as inaccessible voltage intervals on the CVCs. The upper boundary of these intervals is proportional to the embedded soliton velocity.

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“…Hence, exponentially localized fundamental (single-humped) moving discrete solitons cannot be constructed. While these results settle a longstanding controversy-see, e.g., [17][18][19]-they are also somewhat unsatisfactory since they do not give conditions under which moving discrete solitons might exist for generic, nonintegrable lattices.In the arguably simpler problem of kinks in FrenkelKontorova lattices (and, more generally, in discrete KleinGordon models), genuinely traveling fundamental (''charge one'') topological solitons are known not to occur unless there is a ''competing'' nonlinearity that causes vanishings of the PN barrier; see, e.g., [20,21]. Drawing an analogy from this, for (the more complicated, complex field) DNLS models with pure cubic nonlinearity, the PN barrier never disappears, so one might not expect genuinely localized moving solitons (in some sense anticipating the above-mentioned negative result).…”

mentioning

confidence: 99%

“…In the arguably simpler problem of kinks in FrenkelKontorova lattices (and, more generally, in discrete KleinGordon models), genuinely traveling fundamental (''charge one'') topological solitons are known not to occur unless there is a ''competing'' nonlinearity that causes vanishings of the PN barrier; see, e.g., [20,21]. Drawing an analogy from this, for (the more complicated, complex field) DNLS models with pure cubic nonlinearity, the PN barrier never disappears, so one might not expect genuinely localized moving solitons (in some sense anticipating the above-mentioned negative result).…”

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confidence: 99%

Radiationless Traveling Waves in Saturable Nonlinear Schrödinger Lattices

et al. 2006

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The long-standing problem of moving discrete solitary waves in nonlinear Schrödinger lattices is revisited. The context is photorefractive crystal lattices with saturable nonlinearity whose grand-canonical energy barrier vanishes for isolated coupling strength values. Genuinely localized traveling waves are computed as a function of the system parameters for the first time. The relevant solutions exist only for finite velocities. DOI: 10.1103/PhysRevLett.97.124101 PACS numbers: 42.65.Tg, 05.45.Yv, 63.20.Pw Recently, the topic of discrete solitons (intrinsic localized modes) in photorefractive materials has received significant attention; see [1] for a review. This interest was initialized by the experimental realization of twodimensional periodic lattices in photorefractive crystals [2] in which solitons were observed [3,4]. Further work has revealed a wealth of additional coherent structures such as dipoles, quadrupoles, soliton trains, vector, necklace, and ring solitons; see, e.g., [5][6][7][8]. Since photorefractive materials feature the so-called saturable nonlinearity, these results for periodic lattices have spawned a parallel interest in genuinely discrete saturable nonlinear lattices [9][10][11]. One particularly intriguing result of these studies is that the stability properties of the localized modes are substantially different from their regular discrete nonlinear Schrödinger (DNLS) analogs. In DNLS, it is well known [12 -14] that site-centered localized modes are always stable, while intersite-centered modes are unstable (and are stabilized only in the continuum limit where these two branches degenerate into the well-known continuum sech-soliton of the integrable cubic NLS). For the photorefractive nonlinearity (i.e., the so-called Vinetskii-Kukhtarev model originally proposed in Ref. [15] and revisited in Refs. [9][10][11]), depending on the coupling strength, the intersite-centered modes may have lower energy than their on site counterparts. Hence, one should expect that the ensuing sign reversal of the so-called Peierls-Nabarro (PN) energy barrier E E IS ÿ E OS (where the subscripts denote intersite and on site, respectively) should cause an exchange between the linear stability properties of the two modes.A related, even more fundamental, question in nonlinear lattice models of DNLS type is whether exponentially localized self-supporting excitations that move with a constant wave speed can exist, the so-called moving discrete solitons. For continuum models, this question is in some sense trivial since the equations posed in a moving frame remain of the same fundamental type. Yet for lattices, passing to a moving frame leads to so-called advance-delay equations that are notoriously hard to analyze. One recent (negative) result in this direction [16] for the so-called Salerno model shows that, starting from the integrable Ablowitz-Ladik equation, mobile discrete solitary waves acquire nonvanishing tails as soon as parameters deviate from the integrable limit. Hence, exponentially localized fu...

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Moving kinks and nanopterons in the nonlinear Klein–Gordon lattice

Cited by 54 publications

References 23 publications

Center Manifold Theory in the Context of Infinite One-Dimensional Lattices

Center Manifold Theory in the Context of Infinite One-Dimensional Lattices

Embedded soliton dynamics in the asymmetric array of Josephson junctions

Radiationless Traveling Waves in Saturable Nonlinear Schrödinger Lattices

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