Fourth-order Dispersion Research Articles

Blow-up phenomena for the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source

<p style='text-indent:20px;'>This paper deals with the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source. By using some ordinary differential inequalities, the conditions on finite time blow-up of solutions are given with suitable assumptions on initial values. Moreover, the upper and lower bounds of the blow-up time are also investigated.</p>

Brownian Swarm Dynamics and Burgers’ Equation with Higher Order Dispersion

The concept of ranked order probability distribution unveils natural probabilistic interpretations for the kink waves (and hence the solitons) solving higher order dispersive Burgers’ type PDEs. Thanks to this underlying structure, it is possible to propose a systematic derivation of exact solutions for PDEs with a quadratic nonlinearity of the Burgers’ type but with arbitrary dispersive orders. As illustrations, we revisit the dissipative Kotrweg de Vries, Kuramoto-Sivashinski, and Kawahara equations (involving third, fourth, and fifth order dispersion dynamics), which in this context appear to be nothing but the simplest special cases of this infinitely rich class of nonlinear evolutions.

Designing tunable narrowband parametric source in Chalcogenide-based dynamic fiber geometry

We numerically investigate the generation of spectrally isolated narrowband tunable parametric sources by continuous-wave pumping a specially designed gelatin-coated chalcogenide-silica dynamic fiber in the normal dispersion regime. The relative humidity (RH) dependent phase matching dominated by fourth order dispersion has been exploited for the first time, to the best of our knowledge, to produce far-detuned new wavelengths. A tunable source in the range of 0.96–2.5 μm with a tuning rate of 1.3 THz/%RH has been designed by milliwatt pumping a 30 cm long fiber at 1.395 μm. Additionally, the sideband tailorability of about 50 THz close to the visible band was achieved by selectively exciting higher-order fiber modes (HE21) in the anomalous dispersion regime at 0.91 μm. We emphasize that the parametric sources from the proposed host would be extremely useful in short-wave infrared spectroscopy and biomedical applications.

Binary Darboux transformation and multi-dark solitons for a higher-order nonlinear Schrödinger equation in the inhomogeneous optical fiber

Dark solitons in the inhomogeneous optical fiber are studied in this manuscript via a higher-order nonlinear Schrödinger equation, since dark solitons can be applied in waveguide optics as dynamic switches and junctions or optical logic devices. Based on the Lax pair, the binary Darboux transformation is constructed under certain constraints, thus the multi-dark soliton solutions are presented. Soliton propagation and collision are graphically discussed with the group-velocity dispersion, third- and fourth-order dispersions, which can affect the solitons’ velocities but have no effect on the shapes. Elastic collisions between the two dark solitons and among the three dark solitons are displayed, while the elasticity cannot be influenced by the above three coefficients.

Quartic and dipole solitons in a highly dispersive optical waveguide with self-steepening nonlinearity and varying parameters

We show theoretically that highly dispersive optical media characterized by a Kerr nonlinear response may support the existence of quartic and dipole solitons in the presence of the self-steepening effect. The existence and stability properties of these localized pulses are examined in the presence of all the material parameters. Regimes for the modulation instability of a continuous-wave signal propagating inside the nonlinear medium are investigated and an analytic expression for the gain spectrum is obtained and shown to be dependent on the self-steepening parameter in addition to second- and fourth-order group velocity dispersion parameters. Self-similar soliton solutions are constructed for a generalized nonlinear Schr\odinger equation with distributed second-, third-, and fourth-order dispersions, self-steepening nonlinearity, and gain or loss describing ultrashort pulse propagation in the inhomogeneous nonlinear media via the similarity transformation method. The evolutional dynamics of the self-similar structures are investigated in a periodic distributed waveguide system and an exponential dispersion decreasing waveguide.

The effects of fourth-order dispersion on modulation instabilities in two-core optical fibers with asymmetric CW state

The effects of fourth-order dispersion (FOD) on modulation instabilities in the twin-core optical fibers with the asymmetric CW state are studied systematically. When the product of the group velocity dispersion and FOD is negative, in the anomalous dispersion regime, two new MI bands in the high frequency are generated with one band either having the equivalent gain with that of original band or dominating the MI depending on the FOD value; in the normal dispersion, a new dominant MI band in the high frequency is always generated, and the original MI bands vanish in the presence of FOD at total power above a critical value. When the product of the group velocity dispersion and FOD is positive, the main effect of FOD is to shift the MI band to the low frequency and reduce the bandwidth simultaneously with minor effects on the instability gain. We also find that the effects of linear coupling coefficient dispersion can be significantly suppressed in the presence of FOD especially for the case of negative product of the group velocity dispersion and FOD. Finally, the effects of FOD on MI are verified numerically by launching continuous waves with the wave propagation method. Our results should provide very benificial information for successfully observing MI in TCFs or MI related applications with TCFs.

A novel design of double chirped pulse amplification laser systems for fourth-order dispersion control.

A novel design of double chirped pulse amplification laser systems implementing a combination of negatively and positively chirped pulse amplification is proposed for the first time. Without utilizing any extra dispersion compensation element, this design can sufficiently cancel out the second-, third- and especially fourth-order dispersion simultaneously, just by optimizing the parameters of the stretcher and compressor in first chirped pulse amplification stage which applies negatively chirped pulse amplification. The numerical results indicate that near Fourier-transform-limited pulse duration about 20fs can be achieved in high-peak-power femtosecond laser systems up to multi-Petawatt level. This design not only provides a feasible solution for the dispersion control in high-contrast and high-peak-power femtosecond laser systems, but also can avoid the degradation of temporal contrast induced by seed energy loss in the presence of additional dispersion compensation components.

Modified linear stability analysis for quantitative dynamics of a perturbed plane wave.

We develop linear stability analysis (LSA) to quantitatively predict the dynamics of a perturbed plane wave in nonlinear systems. We take a nonintegrable fiber model with purely fourth-order dispersion as an example to demonstrate this method's effectiveness. For a Gaussian-type initial perturbation with cosine-type modulation on a plane wave, its propagation velocities, periodicity, and localization are predicted successfully, and the range of application is discussed. Importantly, the modulation-instability-induced growth of localized perturbation is proved different from the one of purely periodic perturbation and requires the modification of gain value for more accurate prediction. The method offers a needful supplement and improvement for LSA and paves a way to study the dynamics of a perturbed plane wave in more practical nonlinear systems.

Impact of higher-order effects on dissipative soliton in metamaterials

We study the influence of higher-order effects such as third order dispersion (TOD), fourth order dispersion (FOD), quintic nonlinearity (QN), self steepening (SS) and second order nonlinear dispersion (SOND) on the dynamics of dissipative soliton (DS) in metamaterials. Considering each higher-order effect as a perturbation to the system and following Lagrangian variational method, we demonstrate stable dynamics of DS as a result of the interplay between different higher-order effects. We also perform numerical analysis to confirm the analytical results.

Influence of fourth-order dispersion on the Anderson localization

Anderson localization, which consists in the absence of wave dispersion due to disorder, brought to the field of optics and matter waves has greatly improved our understanding of fundamental processes, such as transport and multiple dispersion of light. In this paper, we investigate the existence of Anderson localization in the interplay of a standard group velocity dispersion and a fourth-order dispersion term in the nonlinear Schrodinger (NLS) equation in the presence of a quasiperiodic linear coefficient. We employ a variational approach with a Gaussian ansatz to describe the center of the localized state, while the tails are studied by direct numerical simulations of the NLS equation. These two approaches are important to distinguish the region of the solution presenting exponential decay, which is the main signature of the Anderson localization. The existence of Anderson localization has been confirmed by numerical simulations even when the system presents a small defocusing nonlinearity. In this sense, we reported how the fourth-order dispersion effect changes the existence region of the localized state by changing the critical values for the transition between the localized and delocalized states.

The pure-quartic soliton laser

Ultrashort pulse generation hinges on the careful management of dispersion. Traditionally, this has exclusively involved second-order dispersion, with higher-order dispersion treated as a nuisance to be minimized. Here, we show that this higher-order dispersion can be strategically leveraged to access an uncharted regime of ultrafast laser operation. In particular, our mode-locked laser—with an intracavity spectral pulse shaper—emits pure-quartic soliton pulses that arise from the interaction of fourth-order dispersion and the Kerr nonlinearity. Phase-resolved measurements demonstrate that their pulse energy scales with the third power of the inverse pulse duration. This implies a strong increase in the energy of short pure-quartic solitons compared with conventional solitons, for which the energy scales as the inverse of the pulse duration. These results not only demonstrate a novel approach to ultrafast lasers, but more fundamentally, they clarify the use of higher-order dispersion for optical pulse control, enabling innovations in nonlinear optics and its applications. By suppressing the second- and third-order intracavity dispersion using an intracavity spectral pulse shaper, a mode-locked laser that emits pure-quartic soliton pulses that arise from the interaction of the fourth-order dispersion and the Kerr nonlinearity is demonstrated.

Effect of Fourth-Order Dispersion on Solitonic Interactions

Solitons became important in optical communication systems thanks to their robust nature. However, the interaction of solitons is considered as a bad effect. To avoid interactions, the obvious solution is to respect the temporal separation between two adjacent solitons determined as a bit rate. Nevertheless, many better solutions exist to decrease the bit rate error. In this context, the aim of our work is to study the possibility to delete the interaction of adjacent solitons, by using a special dispersion management system, precisely by introducing both of the third- and fourth-order dispersions in the presence of a group velocity dispersion. To study the influence of the fourth- and third-order dispersions, we use the famous non-linear Schr¨odinger equation solved with the Fast Fourier Transform method. The originality of this work is to bring together the dispersion of the fourth, third, and second orders to separate two solitons close enough to create the Kerr-induced interaction and consequently to improve the propagation by decreasing the bit rate error. This study illustrates the influence of the fourth-order dispersion on one single soliton and two co-propagative solitons with different values of the temporal separation. Then the third order dispersion is introduced in the presence of the fourth-order dispersion in the propagation of one and two solitons in order to study its influence on the interaction. Finally, we show the existence of a precise dispersion management system that allows one to avoid the interaction of solitons.

Propagation of dipole solitons in inhomogeneous highly dispersive optical-fiber media.

We consider ultrashort light pulse propagation through an inhomogeneous monomodal optical fiber exhibiting higher-order dispersive effects. Wave propagation is governed by a generalized nonlinear Schrödinger equation with varying second-, third-, and fourth-order dispersions, cubic nonlinearity, and linear gain or loss. We construct a type of exact self-similar soliton solution that takes the structure of a dipole via a similarity transformation connected to the related constant-coefficients one. The conditions on the optical-fiber parameters for the existence of these self-similar structures are also given. The results show that the contribution of all orders of dispersion is an important feature to form this kind of self-similar dipole pulse shape. The dynamic behaviors of the self-similar dipole solitons in a periodic distributed amplification system are analyzed. The significance of the obtained self-similar pulses is also discussed. By performing numerical simulations, the self-similar soliton solutions are found to be stable under slight disturbance of the constraint conditions and the initial perturbation of white noise.

Effect of power-law nonlinearity on -symmetric optical system with fourth-order diffraction

Gaussian-type soliton solutions of the nonlinear Schrödinger (NLS) equation with fourth order dispersion, and power law nonlinearity in the novel parity-time ()-symmetric quartic Gaussian potential are derived analytically and numerically. The exact analytical expressions of the solutions are obtained in the first two-dimensional (1D and 2D) power law NLS equations. By means of the linear stability analysis, the effect of power law nonlinearity on the stability of Gauss type solitons in different nonlinear media is carried out. Numerical investigations do confirm the stability of our soliton solutions in both focusing and defocusing cases, specially around the propagation parameters.

Impact of fourth order dispersion on modulational instabilities in asymmetrical three-core optical fiber

In this paper we investigate the impact of linear coupling coefficient and coupling coefficient dispersion on modulational instability (MI) in the presence of fourth order dispersion (FOD) in an asymmetric three core optical fiber under normal and anomalous dispersion regime. In addition, we also considered the impact of symmetric and asymmetric power ratio on MI characteristics. It is found that there is significant influence by linear coupling coefficient and coupling coefficient dispersion (CCD) on MI under different dispersion regimes. In anomalous dispersion regime, at β4 < 0 the generation of smaller MI secondary side bands were observed at higher frequencies while at normal dispersion the effect of FOD leads to breaking of MI gain at different values of coupling coefficient. Though the appearance of the MI gain at different frequencies is highly controlled by FOD, but the amplitude of MI gain found to be reduced with increasing FOD under each dispersion regimes. In case of CCD, the impact of FOD found to be quite insensitive and found no MI gain in the absence of CCD under symmetric power ratio under both dispersion regimes.

Solitary wave and periodic solutions of nonlinear Schrödinger equation including higher order dispersions

The solitary wave solution and periodic solutions expressed in terms of elliptic Jacobi functions are obtained for the nonlinear Schrödinger equation governing the propagation of pulses in optical fibers including the effects of second, third and fourth order dispersion. The approach is based on the reduction of the generalized nonlinear Schrödinger equation to an ordinary nonlinear differential equation. The periodic solutions obtained form one-parameter family which depend on an integration constant p. The solitary wave solution with sech2 shape is the limiting case of this family with p=0. The solutions obtained describe also a train of soliton-like pulses with sech2 shape. It is shown that the bounded periodic solutions arise for special domains of integration constant.

Optical soliton perturbation with polynomial and triple-power laws of refractive index by semi-inverse variational principle

Abstract This paper secures perturbed bright 1–soliton solution with polynomial and triple–power law nonlinearities. These are dispersive solitons that come with third and fourth order dispersions. The application of semi–inverse variational principle leads to the analytic closed–form expression for such a soliton with necessary parameter restrictions for them to exist and sustain.

Exact Soliton Solutions to the Cubic-Quartic Non-linear Schrödinger Equation With Conformable Derivative

The research paper aims to investigate the space-time fractional cubic-quartic nonlinear Schrodinger equation in the appearance of the third, and fourth-order dispersion impacts without both group velocity dispersion, and disturbance with parabolic law media by utilizing the extended sinh-Gordon expansion method. This method is one of the strongest methods to find the exact solutions to the nonlinear partial differential equations. In order to confirm the existing solutions, the constraint conditions are used. We successfully construct various exact solitary wave solutions to the governing equation, for example, singular, and dark-bright solutions. Moreover, the 2D, 3D, and contour surfaces of all obtained solutions are also plotted. The finding solutions have justified the efficiency of the proposed method.

Lax pair and vector semi-rational nonautonomous rogue waves for a coupled time-dependent coefficient fourth-order nonlinear Schrödinger system in an inhomogeneous optical fiber**Project supported by the BUPT Excellent Ph.D. Students Foundation (Grant No. CX2019201), the National Natural Science Foundation of China (Grant Nos. 11772017 and 11805020), the Fund of State Key Laboratory of Information Photonics and Optical Communications (Beijing University of

Optical fibers are seen in the optical sensing and optical fiber communication. Simultaneous propagation of optical pulses in an inhomogeneous optical fiber is described by a coupled time-dependent coefficient fourth-order nonlinear Schrödinger system, which is discussed in this paper. For such a system, we work out the Lax pair, Darboux transformation, and corresponding vector semi-rational nonautonomous rogue wave solutions. When the group velocity dispersion (GVD) and fourth-order dispersion (FOD) coefficients are the constants, we exhibit the first- and second-order vector semi-rational rogue waves which are composed of the four-petalled rogue waves and eye-shaped breathers. Both the width of the rogue wave along the time axis and temporal separation between the adjacent peaks of the breather decrease with the GVD coefficient or FOD coefficient. With the GVD and FOD coefficients as the linear, cosine, and exponential functions, we respectively present the first- and second-order periodic vector semi-rational rogue waves, first- and second-order asymmetry vector semi-rational rogue waves, and interactions between the eye-shaped breathers and the composite rogue waves.

High-order dispersion mapping of an optical fiber.

We report on measurements of high-order dispersion maps of an optical fiber, showing how the ratio between the third and fourth-order dispersion (β3/β4) and the zero-dispersion wavelength (λ0) vary along the length of the fiber. Our method is based on Four-Wave Mixing between short pulses derived from an incoherent pump and a weak laser. We find that the variations in the ratio β3/β4 are correlated to those in λ0. We present also numerical calculations to illustrate the limits on the spatial resolution of the method. Due to the good accuracy in measuring λ0 and β3/β4 (10 -3% and 5% relative error, respectively), and its simplicity, the method can be used to identify fiber segments of good uniformity, suitable to build nonlinear optical devices such as parametric amplifiers and frequency comb generators.