Search Results (12)

The weakly nonlinear stability analysis of a convective flow in a planar vertical fluid layer is performed in this paper. The base flow in the vertical direction is generated by internal heat sources distributed within the fluid. The system of Navier–Stokes equations under the Boussinesq approximation and small-Prandtl-number approximation is transformed to one equation containing a stream function. Linear stability calculations with and without a small-Prandtl-number approximation lead to the range of the Prantdl numbers for which the approximation is valid. The method of multiple scales in the neighborhood of the critical point is used to construct amplitude evolution equation for the most unstable mode. It is shown that the amplitude equation is the complex Ginzburg–Landau equation. The coefficients of the equation are expressed in terms of integrals containing the linear stability characteristics and the solutions of three boundary value problems for ordinary differential equations. The results of numerical calculations are presented. The type of bifurcation (supercritical bifurcation) predicted by weakly nonlinear calculations is in agreement with experimental data. Full article

In this study, we use an analytical method tailored for the in-depth exploration of coupled nonlinear partial differential equations (NLPDEs), with a primary focus on the dynamics of solitons. Traditional methods are quite effective for solving individual nonlinear partial differential equations (NLPDEs). However, their performance diminishes notably when addressing systems of coupled NLPDEs. This decline in effectiveness is mainly due to the complex interaction terms that arise in these coupled systems. Commonly, researchers have attempted to simplify coupled NLPDEs into single equations by imposing proportional relationships between various solutions. Unfortunately, this simplification often leads to a significant deviation from the true physical phenomena that these equations aim to describe. Our approach is distinctively advantageous in its straightforwardness and precision, offering a clearer and more insightful analytical perspective for examining coupled NLPDEs. It is capable of concurrently facilitating the propagation of different soliton types in two distinct systems through a single process. It also supports the spontaneous emergence of similar solitons in both systems with minimal restrictions. It has been extensively used to investigate a wide array of new coupled progressive solitons in birefringent fibers, specifically for complex Ginzburg–Landau Equations (CGLEs) involving Hamiltonian perturbations and Kerr law nonlinearity. The resulting solitons, with comprehensive 2D and 3D visualizations, showcase a variety of coupled soliton configurations, including several that are unprecedented in the field. This innovative approach not only addresses a significant gap in existing methodologies but also broadens the horizons for future research in optical communications and related disciplines. Full article

In planar superconductor thin films, the places of nucleation and arrangements of moving vortices are determined by structural defects. However, various applications of superconductors require reconfigurable steering of fluxons, which is hard to realize with geometrically predefined vortex pinning landscapes. Here, on the basis of the time-dependent Ginzburg–Landau equation, we present an approach for the steering of vortex chains and vortex jets in superconductor nanotubes containing a slit. The idea is based on the tilting of the magnetic field $\mathbf{B}$ at an angle $\alpha$ in the plane perpendicular to the axis of a nanotube carrying an azimuthal transport current. Namely, while at $\alpha =0^{\circ }$, vortices move paraxially in opposite directions within each half-tube; an increase in $\alpha$ displaces the areas with the close-to-maximum normal component $|B_{\mathrm{n}}|$ to the close(opposite)-to-slit regions, giving rise to descending (ascending) branches in the induced-voltage frequency spectrum $f_{\mathrm{U}}\left(\alpha \right)$. At lower B values, upon reaching the critical angle ${\alpha }_{\mathrm{c}}$, the close-to-slit vortex chains disappear, yielding $f_{\mathrm{U}}$ of the $n{f}_1$ type ($n\ge 1$: an integer; $f_1$: the vortex nucleation frequency). At higher B values, $f_{\mathrm{U}}$ is largely blurry because of multifurcations of vortex trajectories, leading to the coexistence of a vortex jet with two vortex chains at $\alpha ={90}^{\circ }$. In addition to prospects for the tuning of GHz-frequency spectra and the steering of vortices as information bits, our findings lay the foundation for on-demand tuning of vortex arrangements in 3D superconductor membranes in tilted magnetic fields. Full article

Chains of coupled van der Pol equations are considered. The main assumption that motivates the use of special asymptotic methods is that the number of elements in the chain is sufficiently large. This allows moving from a discrete system of equations to the use of a continuity argument and obtaining an integro-differential boundary value problem as the initial model. In the study of the behaviour of all its solutions in a neighbourhood of the equilibrium state, infinite-dimensional critical cases arise in the problem of the stability of solutions. The main results include the construction of special families of quasi-normal forms, namely non-linear boundary value problems of either Schrödinger or Ginzburg–Landau type. Their solutions make it possible to determine the main terms of the asymptotic expansion of both regular and irregular solutions to the original system. The main goal is the study of chains with diffusion- and advective-type couplings, as well as fully connected chains. Full article

The well-known Van der Pol equation with delayed feedback is considered. It is assumed that the delay factor is large enough. In the study of the dynamics, the critical cases in the problem of the stability of the zero equilibrium state are identified. It is shown that they have infinite dimension. For such critical cases, special local analysis methods have been developed. The main result is the construction of nonlinear evolutionary boundary value problems, which play the role of normal forms. Such boundary value problems can be equations of the Ginzburg–Landau type, as well as equations with delay and special nonlinearity. The nonlocal dynamics of the constructed equations determines the local behavior of the solutions to the original equation. It is shown that similar normalized boundary value problems also arise for the Van der Pol equation with a large coefficient of the delay equation. The important problem of a small perturbation containing a large delay is considered separately. In addition, the Van der Pol equation, in which the cubic nonlinearity contains a large delay, is considered. One of the general conclusions is that the dynamics of the Van der Pol equation in the presence of a large delay is complex and diverse. It fundamentally differs from the dynamics of the classical Van der Pol equation. Full article

This article develops dual variational formulations for a large class of models in variational optimization. The results are established through basic tools of functional analysis, convex analysis and duality theory. The main duality principle is developed as an application to a Ginzburg–Landau-type system in superconductivity in the absence of a magnetic field. In the first section, we develop new general dual convex variational formulations, more specifically, dual formulations with a large region of convexity around the critical points, which are suitable for the non-convex optimization for a large class of models in physics and engineering. Finally, in the last section, we present some numerical results concerning the generalized method of lines applied to a Ginzburg–Landau-type equation. Full article

This article develops an approximate proximal approach for the generalized method of lines. We recall that for the generalized method of lines, the domain of the partial differential equation in question is discretized in lines (or in curves) and the concerning solution is developed on these lines, as functions of the boundary conditions and the domain boundary shape. Considering such a context, in the text we develop an approximate numerical procedure of proximal nature applicable to a large class of models in physics and engineering. Finally, in the last sections, we present numerical examples and results related to a Ginzburg–Landau-type equation. Full article

In this paper, chains of coupled logistic equations with delay are considered, and the local dynamics of these chains is investigated. A basic assumption is that the number of elements in the chain is large enough. This implies that the study of the original systems can be reduced to the study of a distributed integro–differential boundary value problem that is continuous with respect to the spatial variable. Three types of couplings of greatest interest are considered: diffusion, unidirectional, and fully connected. It is shown that the critical cases in the stability of the equilibrium state have an infinite dimension: infinitely many roots of the characteristic equation tend to the imaginary axis as the small parameter tends to zero, which characterizes the inverse of the number of elements of the chain. In the study of local dynamics in cases close to critical, analogues of normal forms are constructed, namely quasinormal forms, which are boundary value problems of Ginzburg–Landau type or, as in the case of fully connected systems, special nonlinear integro–differential equations. It is shown that the nonlocal solutions of the obtained quasinormal forms determine the principal terms of the asymptotics of solutions to the original problem from a small neighborhood of the equilibrium state. Full article

In this study, our attention is focused on deriving integrals of motion (conservation laws; invariants) for the problem of an optical pulse propagation in an optical fiber containing an optical amplifier or attenuator because, to date, such invariants are absent in the literature. The knowledge of a problem’s invariants allows us develop finite-difference schemes possessing the conservativeness property, which is crucial for solving nonlinear problems. Laser pulse propagation is governed by the nonlinear Ginzburg–Landau equation. Firstly, the problem’s conservation laws are developed for the various parameters’ relations: for a linear case, for a nonlinear case without considering the linear absorption, and for a nonlinear case accounting for the linear absorption and homogeneous shift of the pulse’s phase. Hereafter, the Crank–Nicolson-type scheme is constructed for the problem difference approximation. To demonstrate the conservativeness of the constructed implicit finite-difference scheme in the sense of preserving difference analogs of the problem’s invariants, the corresponding theorems are formulated and proved. The problem of the finite-difference scheme’s nonlinearity is solved by means of an iterative process. Finally, several numerical examples are presented to support the theoretical results. Full article

The article produces a brief review of some recent results which predict stable propagation of solitons and solitary vortices in models based on the nonlinear Schrödinger equation (NLSE) including fractional one-dimensional or two-dimensional diffraction and cubic or cubic-quintic nonlinear terms, as well as linear potentials. The fractional diffraction is represented by fractional-order spatial derivatives of the Riesz type, defined in terms of the direct and inverse Fourier transform. In this form, it can be realized by spatial-domain light propagation in optical setups with a specially devised combination of mirrors, lenses, and phase masks. The results presented in the article were chiefly obtained in a numerical form. Some analytical findings are included too, in particular, for fast moving solitons and the results produced by the variational approximation. Moreover, dissipative solitons are briefly considered, which are governed by the fractional complex Ginzburg–Landau equation. Full article

In this paper, bright-dark, multi solitons, and other solutions of a (3 + 1)-dimensional cubic-quintic complex Ginzburg–Landau (CQCGL) dynamical equation are constructed via employing three proposed mathematical techniques. The propagation of ultrashort optical solitons in optical fiber is modeled by this equation. The complex Ginzburg–Landau equation with broken phase symmetry has strict positive space–time entropy for an open set of parameter values. The exact wave results in the forms of dark-bright solitons, breather-type solitons, multi solitons interaction, kink and anti-kink waves, solitary waves, periodic and trigonometric function solutions are achieved. These exact solutions have key applications in engineering and applied physics. The wave solutions that are constructed from existing techniques and novel structures of solitons can be obtained by giving the special values to parameters involved in these methods. The stability of this model is examined by employing the modulation instability analysis which confirms that the model is stable. The movements of some results are depicted graphically, which are constructive to researchers for understanding the complex phenomena of this model. Full article

In this work, we investigate the conformable space–time fractional complex Ginzburg–Landau (GL) equation dominated by three types of nonlinear effects. These types of nonlinearity include Kerr law, power law, and dual-power law. The symmetry case in the GL equation due to the three types of nonlinearity is presented. The governing model is dealt with by a straightforward mathematical technique, where the fractional differential equation is reduced to a first-order nonlinear ordinary differential equation with solution expressed in the form of the Weierstrass elliptic function. The relation between the Weierstrass elliptic function and hyperbolic functions enables us to derive two types of optical soliton solutions, namely, bright and singular solitons. Restrictions for the validity of the optical soliton solutions are given. To shed light on the behaviour of solitons, the graphical illustrations of obtained solutions are represented for different values of various parameters. The symmetrical structure of some extracted solitons is deduced when the fractional derivative parameters for space and time are symmetric. Full article