Stability of the Bright-Type Algebraic Solitary-Wave Solutions of Two Extended Versions of the Nonlinear Schrödinger Equation

Fujioka, Jorge; Espinosa, A.

doi:10.1143/jpsj.65.2440

Cited by 13 publications

(13 citation statements)

References 12 publications

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“…22,23,29) Moreover, several families of exact analytical solutions can be obtained using elementary functions. [24][25][26][27][28] The results may be extended to inhomogeneous media. 30) Derivative nonlinear Schrödinger (DNLS) equations constitute another class of physically important evolution systems.…”

Section: Introductionmentioning

confidence: 88%

Propagating Wave Patterns in a Derivative Nonlinear Schrödinger System with Quintic Nonlinearity

Rogers

Malomed

et al. 2012

J. Phys. Soc. Jpn.

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Exact expressions are obtained for a diversity of propagating patterns for a derivative nonlinear Schrödinger equation with the quintic nonlinearity. These patterns include bright pulses, fronts and dark solitons. The evolution of the wave envelope is determined via a pair of integrals of motion, and reduction is achieved to Jacobi elliptic cn and dn function representations. Numerical simulations are performed to establish the existence of parameter ranges for stability. The derivative quintic nonlinear Schrödinger model equations investigated here are relevant in the analysis of strong optical signals propagating in spatial or temporal waveguides.

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Section: Introductionmentioning

confidence: 88%

Propagating Wave Patterns in a Derivative Nonlinear Schrödinger System with Quintic Nonlinearity

Rogers

Malomed

et al. 2012

J. Phys. Soc. Jpn.

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“…Function S(ω) given by Equation (32), with c 0 = 1/40, c 2 = 1/2 and c 3 = 1/50. S(ω) = 0 at the values ω n given in Equation (14).…”

Section: Solutions Of Equationmentioning

confidence: 99%

“…We can see in this figure that S(ω) > 0 if ω < ω 1 or ω 2 < ω < ω 3 , and S(ω) < 0 if ω 1 < ω < ω 2 or ω > ω 3 , where the frequencies ω n are the values shown in (14). Taking into account the signs of S(ω), we can write λ a and λ b in the following way:…”

Section: Solutions Of Equationmentioning

confidence: 99%

“…(a) −iu ttt and/or u 4t : these higher-order derivatives are necessary to describe sub-picosecond pulses [1][2][3][4][5][6][7][8][9][10][11][12]; in particular, conditions for including u 4t and discarding −iu ttt are discussed in [7,9], (b) |u| 4 u: this higher-order nonlinearity is used when we want to describe the propagation of pulses when the light intensity approaches the values which produce the "saturation" of the refractive index [13][14][15][16][17][18][19][20], (c) i(|u| 2 u) t : this term is necessary to describe the self-steepening of the optical pulses [21][22][23], (d) u(|u| 2 ) t : this term is associated with the effect of Raman scattering [24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

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Pulse Propagation Models with Bands of Forbidden Frequencies or Forbidden Wavenumbers: A Consequence of Abandoning the Slowly Varying Envelope Approximation and Taking into Account Higher-Order Dispersion

2017

Self Cite

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Abstract:We study linear and nonlinear pulse propagation models whose linear dispersion relations present bands of forbidden frequencies or forbidden wavenumbers. These bands are due to the interplay between higher-order dispersion and one of the terms (a second-order derivative with respect to the propagation direction) which appears when we abandon the slowly varying envelope approximation. We show that as a consequence of these forbidden bands, narrow pulses radiate in a novel and peculiar way. We also show that the nonlinear equations studied in this paper have exact soliton-like solutions of different forms, some of them being embedded solitons. The solutions obtained (of the linear as well as the nonlinear equations) are interesting since several arguments suggest that the Cauchy problems for these equations are ill-posed, and therefore the specification of the initial conditions is a delicate issue. It is also shown that some of these equations are related to elliptic curves, thus suggesting that these equations might be related to other fields where these curves appear, such as the theory of modular forms and Weierstrass ℘ functions, or the design of cryptographic protocols.

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“…12 Algebraically ͑as opposed to exponentially͒ localized solutions of a NLS equation with both quadratic and cubic terms have recently been explored. 12,13 We predict that quadratic NLS solitary waves can occur in a pendulum lattice with double cylindrical interrupters, or any lattice with nonanalytic nonlinearities in the broad class mentioned above. Furthermore, because the mathematical description of a lattice of coupled oscillators near the lower cutoff mode is closely related to that of acoustic, 14,15 model, 16 and surface [17][18][19][20][21] waveguides near a cutoff mode, these solitary waves can occur in waveguides with similar nonanalytic nonlinearities.…”

Section: And a Force Proportional Tomentioning

confidence: 99%

Nonanalytic nonlinear oscillations: Christiaan Huygens, quadratic Schrödinger equations, and solitary waves

Denardo

1998

The Journal of the Acoustical Society of America

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An example of an oscillatory system with a time-reversible nonanalytic nonlinearity is shown to be a pendulum with a flexible cord sandwiched between two identical circular disks, in contrast to the analytic case of a pendulum interrupted by a single circular disk. The amplitude-dependent frequencies of both cases are perturbatively calculated, and are compared to numerical simulations over the entire range of amplitudes. The nonanalyticity causes the unusual effect of the frequency to vary linearly with amplitude for small amplitudes, which has also been observed in the resonant frequencies of compressional standing waves in sandstone. A general condition for a nonanalytic nonlinearity to yield this behavior is presented. The amplitude-dependent frequency for the double-interrupted pendulum allows an explanation for Huygens'surprising observation that circular interrupters were as effective as cycloidal interrupters in achieving isochronous motion. A lattice of linearly coupled double-interrupted pendulums is described near the lower and upper cutoff modes by quadratic nonlinear Schrödinger ͑NLS͒ equations, in contrast to cubic NLS equations which arise for analytic pendulum lattices as well as typical acoustic and surface waveguides. Solitary breather and kink solutions to the quadratic NLS equations are presented, and are compared to the known soliton solutions of the corresponding cubic NLS equations. Compressional waves in sandstone are shown to be modeled by the inclusion of a nonanalytic quadratic nonlinearity in the stress-strain relationship. Quadratic NLS breathers are predicted to occur in a waveguide of sandstone, and an analysis indicates that such an observation is feasible.

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Stability of the Bright-Type Algebraic Solitary-Wave Solutions of Two Extended Versions of the Nonlinear Schrödinger Equation

Cited by 13 publications

References 12 publications

Propagating Wave Patterns in a Derivative Nonlinear Schrödinger System with Quintic Nonlinearity

Propagating Wave Patterns in a Derivative Nonlinear Schrödinger System with Quintic Nonlinearity

Pulse Propagation Models with Bands of Forbidden Frequencies or Forbidden Wavenumbers: A Consequence of Abandoning the Slowly Varying Envelope Approximation and Taking into Account Higher-Order Dispersion

Nonanalytic nonlinear oscillations: Christiaan Huygens, quadratic Schrödinger equations, and solitary waves

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