Fifth-order Dispersion Research Articles

Stability and non-linear dynamic analysis of Kerr optical frequencycombs in dual-coupled microcavities with high-order dispersion

We investigate the stability and nonlinear dynamics of the Kerr optical frequency comb inside a dual coupled microcavity with high-order dispersion effect based on the theoretical model of coupled nonlinear Schrödinger equation. The effects of different high-order dispersion parameters on the evolution and spectral characteristics of the optical field in the cavity are also explored. Theoretical results indicate that the addition of the third-order dispersion enlarges the stability domain of the parametric space and transforms the periodically varying soliton breathers and chaos into stable bright soliton. In order to obtain an accurate Kerr optical frequency comb spectral envelope, higher-order dispersion should be considered. Moreover, high-order dispersion terms have a significant effect on the spectral characteristics of the optical frequency comb, such as the spectral envelope frequency shift and the dispersive wave spectral position. Specifically, the third-order dispersion and positive fourth-order dispersion can broaden the spectrum and enhance the dispersive waves; while the negative fourth-order dispersion can suppress the dispersive wave generation and obtain a symmetric soliton frequency comb; the fifth-order dispersion can regulate the drift direction and speed of the optical solitons. The theoretical results are of great value for dispersion regulation and design and also for stability studies in double-coupled microcavity experiments.

Traveling wave solutions for the extended modified KORTEWEG-DE VRIES equation

In this paper, we study an extended modified Korteweg-de Vries equation, which contains the relevant higher-order nonlinear terms and fifth-order dispersion. This equation is the extension of the modified Korteweg-de Vries equation and described by the Ablowitz-Kaup-Newell-Segur hierarchy. The standard Korteweg-de Vries equation is the pioneer integrable model in solitary waves theory, which gives rise to multiple soliton solutions. The Korteweg-de Vries equation arises naturally from shallow water, plasma physics, and other fields of science. To obtain exact solutions the sine-cosine method is applied. It is shown that the sine-cosine method provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics. Traveling wave solutions are determined for extended modified Korteweg-de Vries equation. The study shows that the sine–cosine method is quite efficient and practically well suited for use in calculating traveling wave solutions for extended modified Korteweg-de Vries equation.

Dynamic behaviors of general N-solitons for the nonlocal generalized nonlinear Schrödinger equation

The general N-solitons of nonlocal generalized nonlinear Schrodinger equations with third-order, fourth-order and fifth-order dispersion terms and nonlinear terms (NGNLS) are studied. Firstly, the Riemann–Hilbert problem and the general N-soliton solutions of NGNLS equations were given. Then, we study the symmetry relations of the eigenvalues and eigenvectors related to the scattering data which involve the reverse-space, reverse-time and reverse-space-time reductions. Thirdly, some novel solitons and the dynamic behaviors which corresponded to novel eigenvalue configurations and the coefficients of higher-order terms are given. In all the three NGNLS equations, their solutions often collapse periodically, but can remain bounded or nonsingular for wide ranges of soliton parameters as well. In addition, it is found that the higher-order terms of the NGNLS equations not only affect the amplitude variation of the soliton, but also influence the singularity and the motion of the soliton.

Interactions among solitons for a fifth-order variable coefficient nonlinear Schrödinger equation

A fifth-order variable coefficient nonlinear Schrodinger equation based on original inhomogeneous model is proposed and studied. Bright multi-soliton analytic solutions of the equation are calculated through the Hirota method and auxiliary function. The effect of different constraints between fifth-order dispersion with third-order dispersion on soliton transmission is researched. Besides, their propagation and interaction dynamics are analyzed. Moreover, based on the obtained three-soliton solutions, we change the value of $$\beta (x)$$ to discuss the soliton propagation. Three different propagation and interaction structures are derived via adjusting the nonlinear term. The results can enrich the inhomogeneous model and apply in nonlinear optical fiber.

Tunneling induced two-dimensional phase grating in a quantum well nanostructure via third and fifth orders of susceptibility

We study the nonlinear optical properties in an asymmetric double AlGaAs/GaAs quantum well nanostructure by using an external control field and resonant tunneling effects. It is found that the resonant tunneling can modulate the third-order and fifth-order of susceptibilities via detuning frequency of coupling light. In presence of the resonant tunneling and when the coupling light is in resonance with the corresponding transition, the real parts of third-order and fifth-order susceptibilities are enhanced which are accompanied by nonlinear absorption. However, in off-resonance of coupling light, real parts of third-order and fifth-order susceptibilities enhance while the nonlinear absorption vanishes. We investigate also the two-dimensional electromagnetically induced grating (2D-EIG) of the weak probe light by modulating the third-order and fifth-order susceptibilities. In resonance of coupling light, both amplitude and phase grating are formed in the medium due to enhancement of third-order and fifth-order probe absorption and dispersion. When the coupling light is out of resonance, most of probe energy is transferred from zero-order to higher-order directions due to resonant tunneling effect. The efficiency of phase grating for third-order of susceptibility is higher than phase grating for fifth-order susceptibility. Our proposed model may be useful for optical switching and optical sensors based on semiconductor nanostructures.

Soliton and breather solutions for a fifth-order modified KdV equation with a nonzero background

We investigate an integrable fifth-order modified KdV equation, which contains the fifth-order dispersion and relevant higher-order nonlinear terms, on the line with a nonzero constant background via the Riemann–Hilbert method. As an application, the explicit soliton and breather solutions are obtained. Some figures are also presented to depict the soliton and breather behaviors of the fifth-order modified KdV equation.

Bilinear forms, modulational instability and dark solitons for a fifth-order variable-coefficient nonlinear Schrödinger equation in an inhomogeneous optical fiber

In this paper, a fifth-order variable-coefficient nonlinear Schrödinger equation in an inhomogeneous optical fiber is investigated. Bilinear forms, which are different from those previously reported, are obtained under certain vraible-coefficient constraints. Modulational instability is shown to be related to the group velocity dispersion, Kerr nonlinearity and fifth-order dispersion. Dark soliton solutions are presented and discussed: Soliton velocity is related to the Kerr nonlinearity and fifth-order dispersion, while soliton amplitude is independent of them. Interactions between the dark two solitons are elastic, possibly the overtaking or head-on interactions. Soliton stability is also discussed via the numerical simulation, and the latter is verified through the independence verification.

Breathers, solitons and rogue waves of the quintic nonlinear Schrödinger equation on various backgrounds

We investigate the generation of breathers, solitons, and rogue waves of the quintic nonlinear Schrodinger equation (QNLSE) on uniform and elliptical backgrounds. The QNLSE is the general nonlinear Schrodinger equation that includes all terms up to the fifth-order dispersion. We use Darboux transformation to construct initial conditions for the dynamical generation of localized high-intensity optical waves. The condition for the breather-to-soliton conversion is provided with the analysis of soliton intensity profiles. We discover a new class of higher-order solutions in which Jacobi elliptic functions are set as background seed solutions of the QNLSE. We also introduce a method for generating a new class of rogue waves—the periodic rogue waves—based on the matching of the periodicity of higher-order breathers with the periodicity of the background dnoidal wave.

Dynamics of superregular breathers in the quintic nonlinear Schrödinger equation

In this paper, we consider an extended nonlinear Schrodinger equation that includes fifth-order dispersion with matching higher-order nonlinear terms. Via the modified Darboux transformation and Joukowsky transform, we present the superregular breather (SRB), multipeak soliton and hybrid solutions. The latter two modes appear as a result of the higher-order effects and are converted from a SRB one, which cannot exist for the standard NLS equation. These solutions reduce to a small localized perturbation of the background at time zero, which is different from the previous analytical solutions. The corresponding state transition conditions are given analytically. The relationship between modulation instability and state transition is unveiled. Our results will enrich the dynamics of nonlinear waves in a higher-order wave system.

Conservation laws, periodic and rational solutions for an extended modified Korteweg–de Vries equation

We study an integrable extended modified Korteweg–de Vries equation, which contains the fifth-order dispersion and relevant higher-order nonlinear terms. The infinitely many conservation laws are constructed based on the Lax pair. A general Nth-order periodic solution is obtained by means of the N-fold Darboux transformation (DT), and a simple representation of the Nth-order rational solution is derived from the generalized DT by using the limit approach. As an application, the explicit periodic and rational solutions from first to second order are given, and some typical nonlinear wave patterns such as the doubly periodic lattice-like and doubly localized high-peak waves are shown. It is interestingly found that, the doubly localized high-peak wave can be converted into a W-shaped soliton in the second-order rational solution due to the existence of higher-order terms.

Solitons, breather and bound waves for a generalized higher-order nonlinear Schrödinger equation in an optical fiber or a planar waveguide

In this paper, a generalized higher-order nonlinear Schrodinger equation with variable coefficients is investigated, which could describe the attosecond pulses in an optical fiber or a continuous wave beam in a planar waveguide. Based on the transformation, symbolic computation and Hirota method, the new bilinear forms, one-, two-, three-soliton solutions are obtained under certain constraints. We see that: i) the fifth-order dispersion may affect the soliton velocity as well as the velocities of the breathers and bound solitons; ii) the fifth-order dispersion may influence the shapes of the breathers and bound solitons; iii) the gain function has an impact on the intensities of the solitons, breathers and bound solitons; iv) the gain function may change the breathers’ and bound solitons’ periods and v) the solitons can change into the breathers under the influence of a certain gain function.

Bright optical solitons with Kerr law nonlinearity and fifth order dispersion

In this work, we investigate optical solitons with fifth-order dispersion and Kerr nonlinearity. By means of the coupled amplitude-phase formulation, we derive the basic nonlinear equation describing the dynamics of the pulse amplitude as it propagates in nonlinear dispersive media. The ansatz approach is utilized to analytically solve the derived equation with fifth-degree nonlinear term. Two families of bright soliton solutions with different amplitudes are obtained in the most general case, when all terms related to even and odd hierarchical orders are present in the model. These localized structures exist due to a balance among several higher order dispersion and nonlinear terms of different nature. The constraint conditions for the existence of soliton pulses are presented.

Refractive–diffractive generation of ultrashort Airy–Bessel wave packets with orbital angular momentum: a comprehensive analysis

Bessel beams belong to the class of “non-diffracting” beams, which means that the field distribution near the optical axis does not broaden as the field propagates. Higher order Bessel beams have helical wavefronts and carry an orbital angular momentum. Such non-diffracting “vortex beams” play an important role in physics and engineering. Orbital angular momentum using ultrashort wave packets has been demonstrated in recent years. However, because of spatiotemporal dispersive coupling, spatial beam shaping affects the temporal properties of the pulse. To overcome this, dispersion compensation techniques may be applied, for example, by the combination of refraction and diffraction. Here, we present a comprehensive analysis of the generation of ultrashort Airy–Bessel wave packets taking into account third-order effects, in particular. Thus, one can explore the limits of this technique. It is shown that, for pulses with a duration of less than 10 fs, third-order dispersion has a significant impact on the pulse shape: a Gaussian profile (in time) is transformed into a temporal Airy profile. Our analysis also reveals that terms of higher order, i.e., fourth- and fifth-order dispersion, etc., are less significant.

Radiating dispersive shock waves in non-local optical media.

We consider the step Riemann problem for the system of equations describing the propagation of a coherent light beam in nematic liquid crystals, which is a general system describing nonlinear wave propagation in a number of different physical applications. While the equation governing the light beam is of defocusing nonlinear Schrödinger (NLS) equation type, the dispersive shock wave (DSW) generated from this initial condition has major differences from the standard DSW solution of the defocusing NLS equation. In particular, it is found that the DSW has positive polarity and generates resonant radiation which propagates ahead of it. Remarkably, the velocity of the lead soliton of the DSW is determined by the classical shock velocity. The solution for the radiative wavetrain is obtained using the Wentzel-Kramers-Brillouin approximation. It is shown that for sufficiently small initial jumps the nematic DSW is asymptotically governed by a Korteweg-de Vries equation with the fifth-order dispersion, which explicitly shows the resonance generating the radiation ahead of the DSW. The constructed asymptotic theory is shown to be in good agreement with the results of direct numerical simulations.

Generalized Frobenius Integrable Decompositions for (2 + 1)-Dimensional Partial Differential Equations

Frobenius integrable decompositions are introduced for partial differential equations. Then, such integrable decompositions are generalized, which are applied to (2+1)-dimensional partial differential equations. Some generalized soliton equations are obtained which possess generalized Frobenius integrable decompositions, such as (2+1)-dimensional KdV equation, (2+1)-dimensional Burgers equation, (2+1)-dimensional dispersion equation, etc. Meanwhile, their special cases are just well-known KdV equation, Burgers equation, fifth-order dispersion equation, etc.

An optimized dispersion–relation-preserving combined compact difference scheme to solve advection equations

In this study, we first present an improved version of the classical sixth-order combined compact difference (CCD6) scheme to enhance the convective stability of advection equations through an increased dispersion accuracy. This improved fifth-order dispersion–relation-preserving combined compact difference scheme (DRPCCD5) has been rigorously analyzed through the dispersion, phase speed anisotropy and stability analyses. We then couple the DRPCCD5 scheme with the previous fifth-order compact-reconstruction weighted essentially non-oscillatory (CRWENO5) scheme using a hybrid strategy based on the monotonicity–maintenance criteria. To verify the resulting “optimized” hybrid scheme (ODRPCCD5), several benchmark problems with available exact solution are investigated. The comparison to the previous fifth-order WENO (WENO5) scheme shows that the ODRPCCD5 avoids numerical oscillation around discontinuities, handles large gradients well, and is much faster at the same accuracy because a coarser mesh can be used.

Breather-to-soliton conversions described by the quintic equation of the nonlinear Schrödinger hierarchy.

We analyze the quintic integrable equation of the nonlinear Schrödinger hierarchy that includes fifth-order dispersion with matching higher-order nonlinear terms. We show that a breather solution of this equation can be converted into a nonpulsating soliton solution on a background. We calculate the locus of the eigenvalues on the complex plane which convert breathers into solitons. This transformation does not have an analog in the standard nonlinear Schrödinger equation. We also study the interaction between the new type of solitons, as well as between breathers and these solitons.

Breathers and rogue waves of the fifth-order nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain

One-dimensional anisotropic Heisenberg ferromagnetic spin chain can be described by the fifth-order nonlinear Schrodinger equation, which is investigated in this paper. Through the Darboux transformation, we obtain the Akhmediev breathers (ABs), Kuznetsov–Ma (KM) solitons and rogue-wave solutions. Effects of the coefficients of the fourth-order dispersion, $$\gamma $$ , and of the fifth-order dispersion, $$\delta $$ , on the properties of ABs, KM solitons and rogue waves are discussed: (1) With $$\gamma $$ increasing, the AB exhibits stronger localization in time; (2) The propagation directions of an AB and a KM soliton change with the presence of $$\delta $$ ; and (3) Enhancement of $$\gamma $$ makes the existence time of the rogue waves shorter, while enhancement of $$\delta $$ increases the existence time of the rogue waves.

Breather solutions of the integrable quintic nonlinear Schrödinger equation and their interactions.

We present breather solutions of the quintic integrable equation of the Schrödinger hierarchy. This equation has terms describing fifth-order dispersion and matching nonlinear terms. Using a Darboux transformation, we derive first-order and second-order breather solutions. These include first- and second-order rogue-wave solutions. To some extent, these solutions are analogous with the corresponding nonlinear Schrödinger equation (NLSE) solutions. However, the presence of a free parameter in the equation results in specific solutions that have no analogues in the NLSE case. We analyze new features of these solutions.

Structure of internal solitary waves in two-layer fluid at near-critical situation

A new model equation describing weakly nonlinear long internal waves at the interface between two thin layers of different density is derived for the specific relationships between the densities, layer thicknesses and surface tension between the layers. The equation derived and dubbed here the Gardner–Kawahara equation represents a natural generalisation of the well-known Korteweg–de Vries (KdV) equation containing the cubic nonlinear term as well as fifth-order dispersion term. Solitary wave solutions are investigated numerically and categorised in terms of two dimensionless parameters, the wave speed and fifth-order dispersion. The equation derived may be applicable to wave description in other media.