Nonintegrable three mode interaction in the Zakharov equations

Oliveira, G. I. de; Rizzato, Felipe Barbedo; Oliveira, Luiz Paulo Luna de

doi:10.1016/s0167-2789(96)00294-1

Cited by 17 publications

(15 citation statements)

References 15 publications

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“…Coherence is lost if the interaction time exceeds the inverse frequency widths of the modes 16 and it also fails when wave amplitude are large and the interaction time drops to a value comparable with wave periods. 17 In Sec. IV some numerical experiments integrating the DNLS equation were carried out in order to check the validity of the truncation model.…”

Section: Introductionmentioning

confidence: 99%

The truncation model of the derivative nonlinear Schrödinger equation

2009

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The derivative nonlinear Schrödinger (DNLS) equation is explored using a truncation model with three resonant traveling waves. In the conservative case, the system derives from a time-independent Hamiltonian function with only one degree of freedom and the solutions can be written using elliptic functions. In spite of its low dimensional order, the truncation model preserves some features from the DNLS equation. In particular, the modulational instability criterion fits with the existence of two hyperbolic fixed points joined by a heteroclinic orbit that forces the exchange of energy between the three waves. On the other hand, numerical integrations of the DNLS equation show that the truncation model fails when wave energy is increased or left-hand polarized modulational unstable modes are present. When dissipative and growth terms are added the system exhibits a very complex dynamics including appearance of several attractors, period doubling bifurcations leading to chaos as well as other nonlinear phenomenon. In this case, the validity of the truncation model depends on the strength of the dissipation and the kind of attractor investigated.

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Section: Introductionmentioning

confidence: 99%

The truncation model of the derivative nonlinear Schrödinger equation

2009

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“…Esse aumento no espaçamen-to basal é um forte indício que a nanoargila está de forma intercalada na matriz do hidrogel, sendo que intercalações promovem a diminuição no ângulo 2θ e, consequentemente, o aumento do espaçamento basal. [29][30][31][32] Este aumento do espaçamento interlamelar pode estar atribuído à dimensão das cadeias poliméricas do hidrogel, que para os autores o hidrogel está alojando entre as plaquetas, expulsando os íons de sódio, como ilustrado na Figura 7.…”

Section: Difração De Raios-xunclassified

Otimização da síntese de hidrogéis nanocompósitos intercalados para possível aplicação na área médica

Fernandes¹,

Moura²,

Aouada³

2016

Quim. Nova

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. In this study, the hydrogel formulations based on polysaccharide sodium alginate were investigated for further development of intercalated nanocomposite hydrogels, for purposes of future applications in the areas of health as a carrier in drug delivery systems. Thus, the objective of the work dealt with in the optimization of hydrogels synthesis and obtaining nanostructured hydrogels with clay in an intercalated conformation. The optimization of the hydrogel synthesis was successful, and it is possible to define the materials and / or solutions to be used and concentrations of these solutions to the best formulations of hydrogels. The choice of the concentration of the polysaccharide, the type of crosslinking agent and nanofiller to continued development work, it was determined following criteria as the degree of swelling, and in some cases the physical properties, contributed to determine the best hydrogels. The development of intercalated nanocomposites hydrogel beads was confirmed by the X-ray diffraction technique, which thus could be used to observe evidence of effective incorporation or intercalation of the nanoclay in the hydrogel matrix.Keywords: hydrogels; nanocomposites; sodium alginate; nanofillers; intercalation. INTRODUÇÃOTendo em vista a necessidade de equipamentos sofisticados na área biomédica, os que são disponibilizados para utilização apresentam alto custo, sendo este um impasse no que diz respeito à aquisição. Deste modo, os biomateriais surgem como uma opção interessante, já que desde a última década vem se estudando e trabalhando com este tipo de material, principalmente na área médica. Existem diferentes maneiras de se definir o termo biomateriais. Dentre essas, os biomateriais são definidos como materiais utilizados na substituição de algum órgão vital que perdeu sua atividade no organismo. Atualmente o campo de aplicação engloba desde as próteses até a utilização para regeneração óssea. 1 Dentro da classe dos biomateriais, existem os polímeros, que são macromoléculas que se permitem serem moldadas de acordo com a área a ser empregada. 2 A utilização de polímeros está em ascendência, e se mostra altamente proveitoso na biomedicina principalmente quando se trata de polímeros naturais. A motivação está fundamentada nas propriedades como biodegradabilidade e biocompatibilidade. Dessa maneira, o alginato de sódio (sendo um polímero natural), em razão das propriedades já mencionadas e pelo fato de ser atóxico, termoestável e de baixo custo, vem recebendo grande atenção para tal finalidade. 3 O alginato de sódio é um polissacarídeo obtido por meio da extração de algas marrons (Phaeophyceae). As cadeias macromoleculares são lineares com segmento rígidos e flexíveis. 4 O segmento rígido é atribuído aos blocos G formado pelo monossacarídeo ácido α-L-gulurônico e o segmento flexível está associado aos blocos M que são constituídos de unidades monossacarídicas de ácido β-D-manurônico. 4 O alginato de sódio pode apresentar cadeias com apenas blocos M ou apenas blocos G, ou pode ser constituída por ...

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“…Quite different models apply if the coherence is lost, leading to a random-phase approximation, 23 or if nonlinearities are large. 24 Writing a j →a j exp(i j ) above, with a j and j now real and slowly time dependent, one obtains a fourth-order system of equations describing the slow evolution of a 1 Since the difference a 3 2 Ϫa 2 2 would now vanish exponentially with time, one could just take a 3 ϭa 2 in studying the asymptotic behavior of Eqs. ͑1a͒-͑1d͒.…”

Section: Theoretical Modelmentioning

confidence: 99%

Sudden transition to chaos in plasma wave interactions

1998

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The coherent three-wave interaction, with linear growth in the higher frequency wave and damping in the two other waves, is reconsidered; for equal dampings, the resulting three-dimensional ͑3-D͒ flow of a relative phase and just two amplitudes behaved chaotically, no matter how small the growth of the unstable wave. The general case of different dampings is studied here to test whether, and how, that hard scenario for chaos is preserved in passing from 3-D to four-dimensional flows. It is found that the wave with higher damping is partially slaved to the other damped wave; this retains a feature of the original problem ͑an invariant surface that meets an unstable fixed point, at zero growth rate͒ that gave rise to the chaotic attractor and determined its structure, and suggests that the sudden transition to chaos should appear in more complex wave interactions.

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Nonintegrable three mode interaction in the Zakharov equations

Cited by 17 publications

References 15 publications

The truncation model of the derivative nonlinear Schrödinger equation

The truncation model of the derivative nonlinear Schrödinger equation

Otimização da síntese de hidrogéis nanocompósitos intercalados para possível aplicação na área médica

Sudden transition to chaos in plasma wave interactions

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