Search Results (21)

The central objective of this study is to develop some different wave solutions and perform a qualitative analysis on the nonlinear dynamics of the time-fractional chiral nonlinear Schrodinger’s equation (NLSE) in the conformable sense. Combined with the semi-inverse method (SIM) and traveling wave transformation, we establish the variational principle (VP). Based on this, the corresponding Hamiltonian is constructed. Adopting the Galilean transformation, the planar dynamical system is derived. Then, the phase portraits are plotted and the bifurcation analysis is presented to expound the existence conditions of the various wave solutions with the different shapes. Furthermore, the chaotic phenomenon is probed and sensitivity analysis is given in detail. Finally, two powerful tools, namely the variational method (VM) which stems from the VP and Ritz method, as well as the Hamiltonian-based method (HBM) that is based on the energy conservation theory, are adopted to find the abundant wave solutions, which are the bell-shape soliton (bright soliton), W-shape soliton (double-bright solitons or double bell-shaped soliton) and periodic wave solutions. The shapes of the attained new diverse wave solutions are simulated graphically, and the impact of the fractional order $\delta$ on the behaviors of the extracted wave solutions are also elaborated. To the authors’ knowledge, the findings of this research have not been reported elsewhere and can enable us to gain a profound understanding of the dynamics characteristics of the investigative equation. Full article

This paper investigates the bifurcation dynamics and optical soliton solutions of the integrable quintic Kundu–Eckhaus (QKE) equation with an M-fractional derivative. By adding quintic nonlinearity and higher-order dispersion, this model expands on the nonlinear Schrödinger equation, which makes it especially applicable in explaining the propagation of high-power optical waves in fiber optics. To comprehend the behavior of the connected dynamical system, we categorize its equilibrium points, determine and analyze its Hamiltonian structure, and look at phase diagrams. Moreover, integrating along periodic trajectories yields soliton solutions. We achieve this by using the simplest equation approach and the modified extended Tanh method, which allow for a thorough investigation of soliton structures in the fractional QKE model. The model provides useful implications for reducing internet traffic congestion by including fractional temporal dynamics, which enables directed flow control to avoid bottlenecks. Periodic breather waves, bright and dark kinky periodic waves, periodic lump solitons, brilliant-dark double periodic waves, and multi-kink-shaped waves are among the several soliton solutions that are revealed by the analysis. The establishment of crucial parameter restrictions for soliton existence further demonstrates the usefulness of these solutions in optimizing optical communication systems. The theoretical results are confirmed by numerical simulations, highlighting their importance for practical uses. Full article

We study the critical behaviors of systems undergoing fractal time processes above the upper critical dimension. We derive a set of novel critical exponents, irrespective of the order of the fractional time derivative or the particular form of interaction in the Hamiltonian. For fractal time processes, we not only discover new universality classes with a dimensional constant but also decompose the dangerous irrelevant variables to obtain corrections for critical dynamic behavior and static critical properties. This contrasts with the traditional theory of critical phenomena, which posits that static critical exponents are unrelated to the dynamical processes. Simulations of the Landau–Ginzburg model for fractal time processes and the Ising model with temporal long-range interactions both show good agreement with our set of critical exponents, verifying its universality. The discovery of this new universality class provides a method for examining whether a system is undergoing a fractal time process near the critical point. Full article

In this work, we derived a new type model for spatial Hill’s system considering the created perturbation by the parameter effect of the continuation fractional potential. The new model is considered a reduced system from the restricted three-body problem under the same effect for describing Hill’s problem. We identified the associated Lagrangian and Hamiltonian functions of the new system, and used them to verify the existence of the new equations of motion. We also proved that the new model has different six valid solutions under different six symmetries transformations as well as the original solution, where the new model is an invariant under these transformations. The several symmetries of Hill’s model can extremely simplify the calculation and analysis of preparatory studies for the dynamical behavior of the system. Finally, we confirm that these symmetries also authorize us to explore the similarities and differences among many classes of paths that otherwise differ from the obtained trajectories by restricted three-body problem. Full article

Classical forbidden processes paved the way for the description of mechanical systems with the help of complex Hamiltonians. Fractional integrals of complex order appear as a natural generalization of those of real order. We propose the complex fractional Euler-Lagrange equation, obtained by finding the stationary values associated with the fractional integral of complex order. The complex Hamiltonian obtained from the Lagrangian is suitable for describing nonconservative systems. We conclude by presenting the conserved quantities attached to Noether symmetries corresponding to complex systems. We illustrate the theory with the aid of the damped oscillatory system. Full article

Using general fractional calculus (GFC) of the Luchko form and non-holonomic variational equations of Sedov type, generalizations of the standard action principle and first Noether theorem are proposed and proved for non-local (general fractional) non-Lagrangian field theory. The use of the GFC allows us to take into account a wide class of nonlocalities in space and time compared to the usual fractional calculus. The use of non-holonomic variation equations allows us to consider field equations and equations of motion for a wide class of irreversible processes, dissipative and open systems, non-Lagrangian and non-Hamiltonian field theories and systems. In addition, the proposed GF action principle and the GF Noether theorem are generalized to equations containing general fractional integrals (GFI) in addition to general fractional derivatives (GFD). Examples of field equations with GFDs and GFIs are suggested. The energy–momentum tensor, orbital angular-momentum tensor and spin angular-momentum tensor are given for general fractional non-Lagrangian field theories. Examples of application of generalized first Noether’s theorem are suggested for scalar end vector fields of non-Lagrangian field theory. Full article

The Riesz space-fractional derivative is discretized by the Fourier pseudo-spectral (FPS) method. The Riesz space-fractional nonlinear Klein–Gordon–Zakharov (KGZ) and Klein–Gordon–Schrödinger (KGS) equations are transformed into two infinite-dimensional Hamiltonian systems, which are discretized by the FPS method. Two finite-dimensional Hamiltonian systems are thus obtained and solved by the second-order average vector field (AVF) method. The energy conservation property of these new discrete schemes of the fractional KGZ and KGS equations is proven. These schemes are applied to simulate the evolution of two fractional differential equations. Numerical results show that these schemes can simulate the evolution of these fractional differential equations well and maintain the energy-preserving property. Full article

This work aims to analyze a new system of two fractional Hamiltonian equations. We propose an effective method for transforming the established model into a system of two distinct equations. Two functionals that are connected to the converted system of fractional Hamiltonian systems are introduced together with a new space, and it is demonstrated that these functionals are bounded below on this space. The hypotheses presented here differ from those provided in the literature. Full article

In this manuscript, we are interested in studying the homoclinic solutions of fractional Hamiltonian system of the form $-\mathfrak{D}_{\infty }^{\alpha }{}_{\varsigma }{}^{}\left(\mathfrak{D}_{\varsigma }^{\alpha }{}_{-\infty }{}^{}Z\left(\varsigma \right)\right)-\mathcal{A}\left(\varsigma \right)Z\left(\varsigma \right)+\nabla \omega (\varsigma ,Z\left(\varsigma \right))=0,$ where $\alpha \in (\frac{1}{2},1]$, $Z\in H^{\alpha }(\mathbb{R},{\mathbb{R}}^N)$ and $\omega \in C^1(\mathbb{R}\times {\mathbb{R}}^N,\mathbb{R})$ are not periodic in $\varsigma$. The characteristics of the critical point theory are used to illustrate the primary findings. Our results substantially improve and generalize the most recent results of the proposed system. We conclude our study by providing an example to highlight the significance of the theoretical results. Full article

Singular systems, which can be applied to gauge field theory, condensed matter theory, quantum field theory of anyons, and so on, are important dynamic systems to study. The fractional order model can describe the mechanical and physical behavior of a complex system more accurately than the integer order model. Fractional singular systems within mixed integer and combined fractional derivatives are established in this paper. The fractional Lagrange equations, fractional primary constraints, fractional constrained Hamilton equations, and consistency conditions are analyzed. Then Noether and Lie symmetry methods are studied for finding the integrals of the fractional constrained Hamiltonian systems. Finally, an example is given to illustrate the methods and results. Full article

In this paper, we establish the existence and uniqueness of a strong solution to a fractional mean field games system with non-separable Hamiltonians, where the fractional exponent $\sigma \in (\frac{1}{2},1)$. Our result is new for fractional mean field games with non-separable Hamiltonians, which generalizes the work of D.M. Ambrose for the integral case. The important step is to choose the new appropriate fractional order function spaces and use the Banach fixed-point theorem under stronger assumptions for the Hamiltonians. Full article

Singular systems play an important role in many fields, and some new fractional operators, which are general, have been proposed recently. Therefore, singular systems on the basis of the mixed derivatives including the integer order derivative and the generalized fractional operators are studied. Firstly, Lagrange equations within mixed derivatives are established, and the primary constraints are presented for the singular systems. Then the constrained Hamilton equations are constructed by introducing the Lagrange multipliers. Thirdly, Noether symmetry, Lie symmetry and the corresponding conserved quantities for the constrained Hamiltonian systems are investigated. And finally, an example is given to illustrate the methods and results. Full article

In this paper, we consider the topological abelian BF theory with radial boundary on a generic 3D manifold, as we were motivated by the recently discovered accelerated edge modes on certain Hall systems. Our aim was to research if, where, and how the boundary keeps the memory of the details of the background metrics. We discovered that some features were topologically protected and did not depend on the bulk metric. The outcome was that these edge excitations were $accelerated$, as a direct consequence of the non-flat nature of the bulk spacetime. We found three possibilities for the motion of the edge quasiparticles: same directions, opposite directions, and a single-moving mode. However, requiring that the Hamiltonian of the 2D theory is bounded by below, the case of the edge modes moving in the same direction was ruled out. Systems involving parallel Hall currents (for instance, a fractional quantum Hall effect with $\nu =2/5$) cannot be described by a BF theory with the boundary, independently from the geometry of the bulk spacetime, because of positive energy considerations. Thus, we were left with physical situations characterized by edge excitations moving with opposite velocities (for example, the fractional quantum Hall effect with $\nu =1-1/n$, with the n positive integer, and the helical Luttinger liquids phenomena) or a single-moving mode (quantum anomalous Hall). A strong restriction was obtained by requiring time reversal symmetry, which uniquely identifies modes with equal and opposite velocities, and we know that this is the case of topological insulators. The novelty, with respect to the flat bulk background, is that the modes have local velocities, which correspond to topological insulators with $accelerated$ edge modes. Full article

Inspired by special and general relativistic systems that can have Hamiltonians involving square roots or more general fractional powers, in this article, we address the question of how a suitable set of coherent states for such systems can be obtained. This becomes a relevant topic if the semiclassical sector of a given quantum theory is to be analysed. As a simple setup, we consider the toy model of a deparametrised system with one constraint that involves a fractional power of the harmonic oscillator Hamiltonian operator, and we discuss two approaches to finding suitable coherent states for this system. In the first approach, we consider Dirac quantisation and group averaging, as have been used by Ashtekar et al., but only for integer powers of operators. Our generalisation to fractional powers yields in the case of the toy model a suitable set of coherent states. The second approach is inspired by coherent states based on a fractional Poisson distribution introduced by Laskin, which however turn out not to satisfy all properties to yield good semiclassical results for the operators considered here and in particular do not satisfy a resolution of identity as claimed. Therefore, we present a generalisation of the standard harmonic oscillator coherent states to states involving fractional labels, which approximate the fractional operators in our toy model semiclassically more accurately and satisfy a resolution of identity. In addition, motivated by the way the proof of the resolution of identity is performed, we consider these kind of coherent states also for the polymerised harmonic oscillator and discuss their semiclassical properties. Full article

A general approach to the construction of non-Markovian quantum theory is proposed. Non-Markovian equations for quantum observables and states are suggested by using general fractional calculus. In the proposed approach, the non-locality in time is represented by operator kernels of the Sonin type. A wide class of the exactly solvable models of non-Markovian quantum dynamics is suggested. These models describe open (non-Hamiltonian) quantum systems with general form of nonlocality in time. To describe these systems, the Lindblad equations for quantum observable and states are generalized by taking into account a general form of nonlocality. The non-Markovian quantum dynamics is described by using integro-differential equations with general fractional derivatives and integrals with respect to time. The exact solutions of these equations are derived by using the operational calculus that is proposed by Yu. Luchko for general fractional differential equations. Properties of bi-positivity, complete positivity, dissipativity, and generalized dissipativity in general non-Markovian quantum dynamics are discussed. Examples of a quantum oscillator and two-level quantum system with a general form of nonlocality in time are suggested. Full article