A Remark on the Existence of Breather Solutions for the Discrete Nonlinear Schrödinger Equation in Infinite Lattices: The Case of Site-Dependent Anharmonic Parameters

Karachalios, Nikos I.

doi:10.1017/s0013091504001415

Cited by 22 publications

(34 citation statements)

References 27 publications

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“…Using simple arguments based on variational methods [1] or fixed point theorems [10] to establish the existence of solutions (1.7), we were able to show the existence of explicitly given lower bounds on the power of breathers on either finite or infinite DNLS lattices and for different types of nonlinearities (saturable or power). Although some of them depend explicitly on the dimension, a major difference with the excitation threshold of [6,22] existing only when N > N c , is that they are valid for any dimension.…”

Section: Introductionmentioning

confidence: 99%

“…In the focusing case γ > 0, substituting the time-periodic solution 10) to (2.1), the stationary analogue of (2.1) reads as…”

Section: Introductionmentioning

confidence: 99%

“…Proof: We shall follow the variational (minimax) approach of [3,10]. For instance we seek non-trivial breathers as critical points of 15) showing their existence by the mountain pass Theorem (MPT).…”

Section: Introductionmentioning

confidence: 99%

“…For instance we seek non-trivial breathers as critical points of 15) showing their existence by the mountain pass Theorem (MPT). It can be verified as in [2,3,10] that E is a…”

Section: Introductionmentioning

confidence: 99%

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Energy thresholds for the existence of breather solutions and travelling waves on lattices

Cuevas

Karachalios

Palmero

2010

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We discuss the existence of breathers and of energy thresholds for their formation in DNLS lattices with linear and nonlinear impurities. In the case of linear impurities we present some new results concerning important differences between the attractive and repulsive impurity which is interplaying with a power nonlinearity. These differences concern the coexistence or the existence of staggered and unstaggered breather profile patterns.We also distinguish between the excitation threshold (the positive minimum of the power observed when the dimension of the lattice is greater or equal to some critical value) and explicit analytical lower bounds on the power (predicting the smallest value of the power a discrete breather one-parameter family), which are valid for any dimension. Extended numerical studies in one, two and three dimensional lattices justify that the theoretical bounds can be considered as thresholds for the existence of the frequency parametrized families.The discussion reviews and extends the issue of the excitation threshold in lattices with nonlinear impurities while lower bounds, with respect to the kinetic energy, are also discussed for traveling waves in FPU periodic lattices.

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Section: Introductionmentioning

confidence: 99%

“…In the focusing case γ > 0, substituting the time-periodic solution 10) to (2.1), the stationary analogue of (2.1) reads as…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…For instance we seek non-trivial breathers as critical points of 15) showing their existence by the mountain pass Theorem (MPT). It can be verified as in [2,3,10] that E is a…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Energy thresholds for the existence of breather solutions and travelling waves on lattices

Cuevas

Karachalios

Palmero

2010

Applicable Analysis

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“…As a continuation of our previous works on lattice differential equations [18,19], here we consider the following discrete Klein-Gordon equation DKG, considered in higher dimensional lattices (n = (n 1 , n 2 , . .…”

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confidence: 99%

Global Existence in Infinite Lattices of Nonlinear Oscillators: The Discrete Klein-Gordon Equation

Karachalios

2006

Glasgow Math. J.

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Abstract. Pointing out the difference between the Discrete Nonlinear Schrödinger equation with the classical power law nonlinearity -for which solutions exist globally, independently of the sign and the degree of the nonlinearity, the size of the initial data and the dimension of the lattice -we prove either global existence or nonexistence, in time, for the Discrete Klein-Gordon equation with the same type of nonlinearity (but of "blow-up" sign), under suitable conditions on the initial data, and sometimes on the dimension of the lattice. The results consider both the conservative and the linearly damped lattice. Similarities and differences with the continuous counterparts are remarked. We also make a short comment on the existence of excitation thresholds, for forced solutions of damped and parametrically driven Klein-Gordon lattices.2000 Mathematics Subject Classification. 37L60, 37L15, 35Q55, 34C11.

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Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

2012

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We discuss the existence of breathers and lower bounds on their power, in nonlinear Schrödinger lattices with nonlinear hopping. Our methods extend from a simple variational approach to fixed point arguments, deriving lower bounds for the power which can serve as a threshold for the existence of breather solutions. Qualitatively, the theoretical results justify non-existence of breathers below the prescribed lower bounds of the power which depend on the dimension, the parameters of the lattice as well as of the frequency of breathers. In the case of supercritical power nonlinearities we investigate the interplay of these estimates with the optimal constant of the discrete interpolation inequality. Improvements of the general estimates, taking into account the localization of the true breather solutions are derived. Numerical studies in the one dimensional lattice corroborate the theoretical bounds and illustrate that in certain parameter regimes of physical significance, the estimates can serve as accurate predictors of the breather power and its dependence on the various system parameters.

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A Remark on the Existence of Breather Solutions for the Discrete Nonlinear Schrödinger Equation in Infinite Lattices: The Case of Site-Dependent Anharmonic Parameters

Cited by 22 publications

References 27 publications

Energy thresholds for the existence of breather solutions and travelling waves on lattices

Energy thresholds for the existence of breather solutions and travelling waves on lattices

Global Existence in Infinite Lattices of Nonlinear Oscillators: The Discrete Klein-Gordon Equation

Breathers for the Discrete Nonlinear Schrödinger Equation with Nonlinear Hopping

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