On Linearized Korteweg-de Vries Equations

Villanueva, Alfredo

doi:10.5539/jmr.v4n1p2

Cited by 2 publications

(1 citation statement)

References 18 publications

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“…As a small digression it should be mentioned that equation (3.73) sometimes is referred to as the linearized Korteweg de Vries equation [55]. Since its solution may be expressed in terms of the Airy function Ai, it is also sometimes called the Airy equation [56].…”

Section: Large Squeezing Approximationmentioning

confidence: 99%

Enhancing the Formation of Wigner Negativity in a Kerr Oscillator via Quadrature Squeezing

Rosiek¹

2022

Preprint

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Motivated by quantum experiments with nanomechanical systems, the evolution of a Kerr oscillator with focus on creation of states with a negative Wigner function is investigated. Using the phase space formalism, results are presented that demonstrate an asymptotic behavior in the large squeezing regime for the negativity of a squeezed vacuum state under unitary evolution. The analysis and model are extended to squeezed vacuum states of open systems, adding the decoherence effects of damping and dephasing. To increase experimental relevance, the regime of strong damping is considered. These effects are investigated, yielding similar asymptotic results for the behavior of these effects in the large squeezing regime. Combining these results, it is shown that a weak nonlinearity as compared to damping may be improved by increasing the squeezing of the initial state. It is also shown that this may be done without exacerbating the effects of dephasing.

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Section: Large Squeezing Approximationmentioning

confidence: 99%

Enhancing the Formation of Wigner Negativity in a Kerr Oscillator via Quadrature Squeezing

Rosiek¹

2022

Preprint

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Existence, compactness, estimates of eigenvalues and s-numbers of a resolvent for a linear singular operator of the Korteweg-de Vries type

Muratbekov

Suleimbekova

2022

Filomat

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In this paper, we consider a linear operator of the Korteweg-de Vries type Lu = ?u ?y + R2(y)?3u ?x3 + R1(y)?u ?x + R0(y)u initially defined on C? 0,?(??), where ?? = {(x, y) : ?? ? x ? ?,?? < y < ?}. C? 0,?(??) is a set of infinitely differentiable compactly supported function with respect to a variable y and satisfying the conditions: u(i) x (??, y) = u(i) x (?, y), i = 0, 1, 2. With respect to the coefficients of the operator L , we assume that these are continuous functions in R(??,+?) and strongly growing functions at infinity. In this paper, we proved that there exists a bounded inverse operator and found a condition that ensures the compactness of the resolvent under some restrictions on the coefficients in addition to the above conditions. Also, two-sided estimates of singular numbers (s-numbers) are obtained and an example is given of how these estimates allow finding estimates of the eigenvalues of the considered operator.

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On Linearized Korteweg-de Vries Equations

Cited by 2 publications

References 18 publications

Enhancing the Formation of Wigner Negativity in a Kerr Oscillator via Quadrature Squeezing

Enhancing the Formation of Wigner Negativity in a Kerr Oscillator via Quadrature Squeezing

Existence, compactness, estimates of eigenvalues and s-numbers of a resolvent for a linear singular operator of the Korteweg-de Vries type

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