Coupled Ginzburg-Landau Equations Research Articles

Partially “invisible” pulsation of asymmetric soliton molecules

Pulsation is a universal phenomenon in all disciplines of nonlinear science, which generally exhibit various nonlinear soliton dynamic modes in nonlinear optics. With the emergence of real-time spectral measurement technology, a new phenomenon of optical soliton pulsation, “invisible” soliton pulsation, has gradually attracted researchers’ attention. In this paper, by solving the coupled Ginzburg–Landau equation, we have discovered the partially “invisible” pulsation phenomenon of asymmetric soliton molecules (SMs) composed of two unequal-intensity pulses in a normal-dispersion Mamyshev oscillator for the first time. It is indicated that the fluctuation periods of each internal pulse in asymmetric SMs in terms of peak power, energy, and relative phase difference are all four roundtrips. However, the oscillation period of asymmetric SMs energy is two roundtrips, only half of the above period. Further research shows that the oscillation of the relative phase difference between two internal pulses in SMs is related to the variation of their intensity difference, which can also affect the position change of the spectral modulation peak. This work will enrich the research on “invisible” soliton pulsation and is of great significance for promoting the development of nonlinear dissipative systems.

Buildup and synchronization regimes of a vector pure-quartic soliton molecule in a fiber laser cavity.

Pure-quartic solitons (PQSs) have recently received increasing attention due to their energy-width scaling over the traditional soliton, which has expanded our understanding of soliton dynamics with high-order dispersion in nonlinear systems. Here, we numerically reveal the asynchronization and synchronization processes of the sub-pulse within the vector PQS molecule in a mode-locked fiber laser by solving the coupled Ginzburg-Landau equations. During the establishment of a vector PQS molecule, the repulsion, attraction, and finally stabilization processes have been observed. Specifically, sub-pulse disappearance, regeneration, and finally synchronization with the other pulses are also investigated. Our analysis of the pulse energy, time interval, and relative phase evolution dynamics with the round trip indicates that the asynchronization and synchronization within the vector PQS molecule associate tightly with the gain competition and the cross-phase modulation. Our findings provide insights into the internal mutual dynamics within the vector soliton molecule and offer guidance for the applications of PQS.

Coexistence and competition: Double attractors in hybrid mode-locked fiber lasers

The hybrid mode-locking technique has been widely used in enhancing pulse quality, but the specific mechanism of hybrid mode-locking remains unclear. In this work, we experimentally investigate three pulse formation mechanisms in saturable absorbers (SA) and nonlinear polarization evolution (NPE) passively hybrid mode-locked fiber laser, which are SA domination, NPE domination, and SA-NPE co-domination switching and clarified the existent dynamic competition and cooperation between the two mode-locking mechanisms of the hybrid mode-locked fiber laser. The method of simulating the filtered gain spectrum with customized filtering is proposed, and the switching of pulse formation mechanisms is numerically investigated using the coupled Ginzburg–Landau equations. The presence of a double attractor subsystem in a hybrid mode-locked fiber laser is proposed and demonstrated. Our results deepen the understanding of hybrid mode-locked fiber lasers and provide experimental ideas for the study of multi-attractor coexistence.

Dual-loss-modulated QML erbium-doped fiber laser with microfiber and NPR technique

We demonstrated a dual-loss-modulated Q-switched mode-locked (QML) erbium-doped fiber laser based on the nonlinear polarization rotation (NPR) technique with a microfiber. The microfiber served as both a highly nonlinear medium and a saturable absorber, which together with the NPR technique enables stable QML operation under significant pump power fluctuations. We evaluated the nonlinear properties of microfibers with different waist diameters. A maximum QML envelope energy of 467.15 nJ is achieved, corresponding to an average single-pulse energy of 4.06 nJ beneath the Q-switched envelope, which is 2.35 times higher than the maximum pulse energy attained with dissipative soliton mode-locking under similar pump power. In addition, the QML laser was verified by numerical simulation based on the coupled Ginzburg-Landau equation. The microfiber's customizable size enables significant variations in non-linear and saturable absorption characteristics, creating the potential for QML pulse regulation. This research provides novel insights for designing QML lasers.

Vector pure-quartic soliton molecule fiber laser

Pure-quartic solitons (PQSs) balanced by the fourth-order dispersion (FOD) and nonlinearity in weakly birefringent optical fibers exhibit unique characteristics that differ from those of traditional solitons. Owing to the long oscillating tail of the PQS, soliton trapping can be easily promoted either between the sub pulses inside the molecule or along the two polarization axes of the birefringent fiber. Thus, observing the interaction and motion dynamics of vector PQS molecules promotes great opportunities for unveiling new mechanisms of soliton molecular complexity. In this work, we numerically explore the transient dynamics of vector PQS molecules by solving the coupled Ginzburg-Landau equations which are contributed by the FOD. Diverse real-time dynamics of vector PQS molecules are demonstrated for the first time, including stationary and pulsating, which originate from the multi-scale energy exchange inside the molecule and between the orthogonal axes depending on the nonlinear effect. In addition, we reveal that the buildup process of vector PQS molecules includes splitting, pulsating, and synchronization. The related dynamics of the pulse separation and phase difference that constitute the relevant internal degrees of freedom of the molecule are also mapped. These efforts may shed new insights into understanding the internal dynamics of soliton molecular complexes and decomposing the dynamics of complex nonlinear systems.

Matricials methods applied to the solution of the Ginzburg-Landau equations

In this contribution we propose two numerical methods for the solution of the system of two non-linear coupled differential Ginzburg-Landau equations. These proposals are based firstly on taking a matrix view considering the quasi-linear coupled system, as a second option, considering the computational molecule with its respective restriction of the values and eigen-vectors of the matrix. We clearly and concisely obtained the eigenvalues that lead us to an optimal spatio-temporal convergence solution of said system of equations. We compare the numerical convergence times obtained by these three methods with the known time found by applying the link variable method.

Vector pulsating solitons and soliton molecules under higher-order effects in passively mode-locked fiber lasers

We numerically study the transport dynamics of vector solitons in ytterbium-doped fiber lasers based on a coupled Ginzburg-Landau equation with higher-order effects. By adjusting the small signal gain coefficient and intracavity birefringence, the number of soliton molecules can be manipulated, the transmission of synchronous and asynchronous vector solitons and the generation of chaotic pulse sequences can be realized. Synchronous vector solitons can be generated when the intracavity birefringence is relatively low, the stable transmission of multi-soliton molecules and the interaction between soliton molecules can be achieved. By changing the value of the phase delay, periodic pulsating solitons can be produced. Furthermore, the phenomenon of the pulse train from the period-doubling to chaos can also be observed. With the enhancement of the intracavity birefringence, asynchronous vector solitons can also achieve stable transmission in the cavity, which the amplitudes and widths of two components are opposite with the different phases. The proposed model has universal application value and the numerical results have guiding significance for studying polarization tunable lasers and exploring the nonlinear mechanism in fiber lasers.

Dynamics of frequency detuning in a hybrid Er-doped mode-locked fiber laser

Frequency detuning of mode-locked fiber lasers displays many remarkable nonlinear dynamical behaviors. Here we report for the first time the evolution of pulses from mode-locking through period pulsation to Q -switched mode-locking for three fundamental cases. Our experiments are performed in a hybrid actively and passively amplitude-modulated all-fiber polarization-maintaining mode-locked fiber laser, where the amplitude modulation frequency artificially deviates from the fundamental frequency of the cavity. We design and numerically simulate the laser with coupled Ginzburg–Landau equations. The experimentally observed dynamics of the mode detuning process is discussed with the assistance of the fitted model and numerical simulations, showing the generalizability of the optical mode detuning variation process. Our work provides fundamental insights for understanding perturbations in nonlinear optical resonant cavities and expands the ideas for studying chaotic path theory in hybrid mode-locked fiber lasers.

Variable-time-step BDF2 nonconforming VEM for coupled Ginzburg-Landau equations

In this paper, we solve the coupled Ginzburg-Landau equations by using a linearized variable-time-step second order backward differentiation formula in time combining with a nonconforming virtual element method in space. Based on the techniques of the discrete orthogonal convolution kernels and the discrete complementary convolution kernels, the unconditional optimal L2-norm error estimate of the fully discrete scheme was presented by using the error splitting technique under the mild restriction on the ratio of adjacent time-steps ratios. Finally, numerical experiments illustrate the correctness of our theoretical analysis.

A multiphase phase-field study of three-dimensional martensitic twinned microstructures at large strains

The nanoscale multiphase phase-field model for stress and temperature-induced multivariant martensitic transformation under large strains developed by the authors in Basak and Levitas (J Mech Phys Solids 113:162–196, 2018) is revisited, the issues related to the gradient energy and coupled kinetic equations for the order parameters are resolved, and a thermodynamically consistent non-contradictory model for the same purpose is developed in this paper. The model considers \(N+1\) order parameters to describe austenite and N martensitic variants. One of the order parameters describes austenite\(\leftrightarrow \)martensite transformations, and the remaining N order parameters, whose summation is constrained to the unity, describe the transformations between the variants. A non-contradictory gradient energy is used within the free energy of the system to account for the energies of the interfaces. In addition, a kinetic relationship for the rate of the order parameters versus thermodynamic driving forces is suggested, which leads to a system of consistent coupled Ginzburg–Landau equations for the order parameters. An approximate general crystallographic solution for twins within twins is presented, and the explicit solution for the cubic to tetragonal transformations is derived. A large strain-based finite element method is developed for solving the coupled Ginzburg–Landau and elasticity equations, and it is used to simulate a 3D complex twins within twins microstructure. A comparative study between the crystallographic solution and the simulation results is presented.

Attractors of Modified Coupled Ginzburg–Landau Model

Bose–Einstein condensation is a gaseous, superfluid state of matter exhibited by bosons as they cool to near absolute zero, which was discovered as early as 1924 but was not experimentally realized until 1995. In 2006, Machida and Koyama developed the corresponding Ginzburg–Landau model for superfluid and Bose–Einstein condensation‐spanning phenomena. We mainly consider the global attractor for the initial boundary value problem of the modified coupled Ginzburg–Landau equations, which come from the BCS‐BEC crossover model. Combining Gronwall inequality, properties of the binomial function, with some suitable a priori estimates, we establish the existence of global attractors. The attractor results obtained in this paper can provide a strong theoretical basis for the experimental realization of the BCS‐BEC spanning phenomenon, and the adopted research method can also serve as a reference for analysing other types of partial differential equation attractors.

Asymptotic analysis to domain walls between traveling waves modeled by real coupled Ginzburg-Landau equations

We apply an asymptotic analysis to one-dimensional domain wall between traveling waves. The governing partial differential equations are firstly transformed into a real coupled nonlinear ordinary equations system, and then the homotopy renormalization (HTR) method is used to solve it. These solutions are analytical and under some special conditions of the parameters, are exact. In particular, the phenomena revealed by the solutions, namely the sinks and sources, are discussed and it can be seen easily that the source is narrower than the sink, which agrees well with the experimental results. Moreover, compared with the numerical results graphically, we can guarantee that the obtained simple analytical solutions are accurate in the whole domain.

Asymptotic behavior of random coupled Ginzburg\u2013Landau equation driven by colored noise on unbounded domains

In this paper, a random coupled Ginzburg–Landau equation driven by colored noise on unbounded domains is considered, in which the nonlinear term satisfies a local Lipschitz condition. It is shown that the random attractor of such a coupled Ginzburg–Landau equation is a singleton set, and the components of solutions are very close when the coupling parameter becomes large enough.

Dark-bright soliton trapping in a fiber laser.

We report, to the best of our knowledge, the first experimental observation of coexistence of scalar dark and bright solitons in a fiber laser operating in the near zero-dispersion regime. We show that because of the incoherent cross coupling, under suitable conditions a bright soliton formed in the net anomalous cavity dispersion regime could bind with a dark soliton formed in the net normal cavity dispersion regime in a fiber laser. The properties of the dark and bright solitons, as well as their bound states, are experimentally investigated. The numerical simulations based on the coupled Ginzburg-Landau equations have reproduced the experimental results well.

XPM‐Induced Vector Asymmetrical Soliton with Spectral Period Doubling in Mode‐Locked Fiber Laser

AbstractCross phase modulation (XPM) can induce soliton trapping in nonlinear medium, which has been employed to achieve vector soliton, optical switching, and optical analog of gravity‐like potentials. Here, the first observation of a novel soliton operation is reported whose wavelength exhibits redshift and blueshift periodically in a mode‐locked fiber laser under appropriate birefringence and dispersion map. XPM dominates the dynamics by inducing and sustaining the vector asymmetrical soliton (VAS) with spectral period doubling, and exhibits a unique trajectory of pulse trapping. The XPM‐induced VAS is an idiosyncratic steady state featured with asymmetry in the model based on the coupled Ginzburg–Landau equation, which depicts the distinct pulse features in comparison with vector soliton, soliton molecule, and traditional period‐doubling evolutions. The two orthogonal polarized directions of mode‐locked fiber laser can be regarded as the XPM‐coupled dissipative resonators guiding the further study of the optical nonlinear dynamics and chaos for soliton, which is helpful to laser design and brings useful insights into nonlinear science and applications.

Transmission stability of chirped dark vector quasi-solitons in birefringent fiber system with nonlinear gain

In this article, we consider the coupled Ginzburg–Landau equation with variable coefficients including the nonlinear gain and obtain the exact solutions of chirped dark vector quasi-solitons via the ansatz method. Next, the propagation of chirped dark vector quasi-solitons is discussed to verify whether they can be transmitted stably in the birefringent optical fiber system. The numerical simulation shows that this can be achieved. We deeply add the small perturbation to the transmission of dark vector quasi-solitons to make the results above more general. The results further prove the correctness of our solutions.

Wave plate-dependent lasing regimes transitions in an all-normal-dispersion fiber laser mode-locked by nonlinear polarization rotation

We address the transitions between three different working regimes of an all-normal-dispersion pulsed fiber laser, mode-locked by nonlinear polarization rotation (NPR). In particular, we investigate numerically the dynamics and study in details the transitions induced by solely rotating the cavity wave plates. We analyze the phemomenon by putting forward a model for all-normal-dispersion NPR-moded-locked fiber laser, based on the coupled Ginzburg-Landau equations and Jones matrix. Numerical results show that depending on the values of the cavity parameters, the transitions between the dissipative soliton regime and amplifier similariton one, or the transition between the dissipative soliton regime and the splitting pulse one may be realized by simply rotating the cavity wave plates. We found that the transitions are intrinsically related to the NPR mode-locking mechanism by explicitly evaluating the polarization states across the pulse at different locations of the cavity. Our numerical results are in very good agreement with experimental results reported in the relevant literature, and provide evidence that besides gain saturation and spectral filtering, the orientations of wave plates may be exploited as a suitable degree of freedom to obtain the desired pulse regimes in all-normal-dispersion fiber lasers.

Pulse Dynamics of an All-Normal-Dispersion Ring Fiber Laser Under Four Different Pulse Regimes

Based on the coupled Ginzburg-Landau equations and Jones matrices of the waveplates considered, a numerical model of an all-normal-dispersion fiber laser mode-locked by nonlinear polarization rotation has been proposed. The operating characteristics of the fiber laser discussed were studied numerically. It has been found that the proposed all-normal-dispersion mode-locked fiber laser (AND-MLFL) could deliver dissipative solitons (DSs) with a M-shaped and U-shaped spectrum, the splitting pulse with a divided spectrum and the amplifier similaritons. The evolution of the intra-cavity pulse and spectrum has been calculated under different regimes and the effects of group velocity dispersion (GVD) and nonlinearity are analyzed. When the fiber laser delivers DSs or causes pulse splitting, nonlinear effects dominate the pulse evolution. With the increase of the accumulated nonlinear phase shift, the operation states change from DS with a M-shaped spectrum to a U-shaped spectrum, and then to the splitting pulse. In the case of amplifier similaritons, both the GVD and nonlinearity play important roles in pulse evolution. The effect of nonlinear polarization rotation and filtering on the pulse reshaping has been analyzed. When the fiber laser delivers DSs with a M-shaped spectrum, the filter has a very weak effect on the pulse and on spectral reshaping. However, when the fiber laser operates in the amplifier similariton state, the filter plays a key role in pulse and spectral reshaping, whereas the nonlinear polarization rotation become less dominant. The dependence of the operational states on the filter bandwidth, fiber length, small signal gain coefficient and orientation of waveplates has also been calculated. A Yb-doped doubled-cladding fiber laser, mode-locked by nonlinear polarization rotation, has also been demonstrated and all of the four pulse regimes are obtained experimentally.

Spatio-temporal numerical solutions of the coupled real and complex Ginzburg-Landau amplitude equations for one-dimensional systems near the Turing-Hopf bifurcation

In this paper, the solutions of the coupled Ginzburg-Landau equations are numerically studied with the aim to describe the dynamics of systems close to the Turing-Hopf bifurcation. We found that the spatial modulations of the Turing and Hopf amplitudes increase with the domain size due to inhomogeneous perturbations. By measuring the growth of spatial Fourier modes in systems initialized with random initial conditions, the effective size of the domain where the generalized Eckhaus and Benjamin-Feir-Newell instabilities occur was determined. Besides, we have numerically corroborated that such instabilities can be quantified by recent theoretical results on secondary instabilities of the Mixed mode solution. Furthermore, this study exemplifies diverse spatiotemporal patterns related to intermittency and chaos previously studied for the uncoupled complex Ginzburg-Landau equation as well a new features related to the interaction of Turing and Hopf modes, as the appearance of backbones patterns, foliar figures and wormlike structures in the space-time maps.

A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection

In this paper we investigate the transition from periodic solutions to spatiotemporal chaos in a system of four globally coupled Ginzburg Landau equations describing the dynamics of instabilities in the electroconvection of nematic liquid crystals, in the weakly nonlinear regime. If spatial variations are ignored, these equations reduce to the normal form for a Hopf bifurcation with \begin{document}$O(2) × O(2)$\end{document} symmetry. Both the amplitude system and the normal form are studied theoretically and numerically for values of the parameters including experimentally measured values of the nematic liquid crystal Merck I52. Coexistence of low dimensional and extensive spatiotemporal chaotic patterns, as well as a temporal period doubling route to spatiotemporal chaos, corresponding to a period doubling cascade towards a chaotic attractor in the normal form, and a kind of spatiotemporal intermittency that is characteristic for anisotropic systems are identified and characterized. A low-dimensional model for the intermittent dynamics is obtained by perturbing the eight-dimensional normal form by imperfection terms that break a continuous translation symmetry.