sine-Gordon Research Articles

Revealing the finite-frequency response of a bosonic quantum impurity

Quantum impurities are ubiquitous in condensed matter physics and constitute the most stripped-down realization of many-body problems. While measuring their finite-frequency response could give access to key characteristics such as excitations spectra or dynamical properties, this goal has remained elusive despite over two decades of studies in nanoelectronic quantum dots. Conflicting experimental constraints of very strong coupling and large measurement bandwidths must be met simultaneously. We get around this problem using cQED tools, and build a precisely characterized quantum simulator of the boundary sine-Gordon model, a non-trivial bosonic impurity problem. We succeeded to fully map out the finite frequency linear response of this system. Its reactive part evidences a strong renormalisation of the nonlinearity at the boundary in agreement with non-perturbative calculations. Its dissipative part reveals a striking many-body broadening caused by multi-photon conversion. The experimental results are matched quantitatively to a perturbative calculation based on a microscopically calibrated model. Furthermore, we push the device into a regime where perturbative calculations break down, which calls for more advanced theoretical tools to model many-body quantum circuits. We also critically examine the technological limitations of cQED platforms to reach universal scaling laws. This work opens exciting perspectives for the future such as quantifying quantum entanglement in the vicinity of a quantum critical point or accessing the dynamical properties of non-trivial many-body problems.

Evolyutsiya nelineynykh volnovykh impul'sov v teorii uravneniya sinus-Gordon

The solution to the Whitham modulation equations for envelopes of one-phase periodic waves evolving according to the sine-Gordon equation is obtained. Using the hodograph method, these equations are reduced to a linear partial differential equation, and the class of solutions to this equation with separation of variables is described. The theory is illustrated by an example in which a complete analytic solution is obtained for the problem of nonlinear wave packet evolution accompanied with self-contraction and a decrease in the number of oscillations in the Whitham nonlinear region.

Numerical solutions of 2D stochastic time‐fractional Sine–Gordon equation in the Caputo sense

AbstractWe study the two‐dimensional stochastic time‐fractional Sine–Gordon equation (2D ST‐FS‐G) and provide a solution for it. We use the new clique polynomial method to obtain a numerical solution to the 2D TFS‐G equation. In this technique, the clique polynomial is considered as a basic function for operational matrices. By converting the 2D ST‐FS‐G equation to algebraic equations, a solution is obtained for the desired equation, which clearly shows that this approach is suitable and accurate for dealing with the equation. We further present the error boundary for the obtained approximation of the desired three‐variable function based on the clique polynomial. Finally, we compare some numerical results obtained with the exact ones by a few practical examples.

A new method of solving the best approximate solution for a nonlinear fractional equation

A new method of solving the best approximate solution for nonlinear fractional equations with smooth and nonsmooth solutions in reproducing kernel space is proposed in the paper. The nonlinear equation outlines some important equations, such as fractional diffusion-wave equation, nonlinear Klein–Gordon equation and time-fractional sine-Gordon equation. By constructing orthonormal bases in reproducing kernel space using Legendre orthonormal polynomials and Jacobi fractional orthonormal polynomials, the best approximate solution is obtained by searching the minimum of residue in the sense of . Numerical experiments verify that the method has higher accuracy.

Evolution of Nonlinear Wave Pulses in the sine-Gordon Equation Theory

We give the solution of the Whitham modulation equations for envelopes of pulses evolving according to the sine-Gordon equation. The Whitham equations are interpreted as the equations of relativistic hydrodynamics and their solving is reduced by the hodograph method to solving a linear partial differential equation. We describe the class of solutions of this equation with separation of variables and illustrate the theory by an example of the nonlinear wave packet evolution accompanied by its shrinking and decrease of the number of oscillations in the Whitham nonlinear region.

One-Parameter Finite Difference Methods and Their Accelerated Schemes for Space-Fractional Sine-Gordon Equations with Distributed Delay

One-Parameter Finite Difference Methods and Their Accelerated Schemes for Space-Fractional Sine-Gordon Equations with Distributed Delay

Barycentric Lagrange interpolation collocation method for solving the Sine–Gordon equation

This paper mainly discusses the numerical solution of Sine–Gordon (SG) equation which is widely used in engineering field. Different from the classical numerical calculation method, we choose the barycentric Lagrange interpolation collocation method (BLICM) to solve the SG equation. Firstly, the barycentric Lagrange interpolation is introduced and its differential matrix is given. Secondly, two linearized iterative schemes namely direct linearized iterative scheme and Newton linearized iterative scheme to solve SG equation are constructed, the matrix equation of the two iterative scheme is obtained. Thirdly, the Newton–Raphson iterative scheme for SG equation is also presented and the detailed solution process is given. Finally, several numerical examples with exact solution are given to show the accuracy of BLICM, the results of three iterative schemes with equidistant nodes and Chebyshev nodes are compared. Further, some examples with line and ring solitons are given to simulate by BLICM, the accurate numerical results are obtained to support our experiments.

Functional-renormalization-group approach to circuit quantum electrodynamics

A nonperturbative approach is developed to analyze superconducting circuits coupled to quantized electromagnetic continuum within the framework of the functional renormalization group. The formalism allows us to determine complete physical pictures of equilibrium properties in the circuit quantum electrodynamics (cQED) architectures with high-impedance waveguides, which have recently become accessible in experiments. We point out that nonperturbative effects can trigger a breakdown of the supposedly effective descriptions, such as the spin-boson and boundary sine-Gordon models, and lead to qualitatively unique phase diagrams. The origin of the failure of conventional understandings is traced to strong renormalizations of circuit parameters at low-energy scales. Our results indicate that a nonperturbative analysis is essential for a comprehensive understanding of cQED platforms consisting of superconducting circuits and long high-impedance transmission lines.

New optical solitons of double Sine-Gordon equation using exact solutions methods

In this study, disparate analytical methodologies like the tan method, rational Exp-function method, sech method, extended tan method, and sine–cosine method are implemented to solve the double Sine-Gordon equation (DSG). Subsequently, the dynamical attributes of the obtained results have been underlined and depicted in terms of 3D and 2D graphical illustrations, likewise, different soliton solutions such as kink, bright-dark, and bright are depicted under an appropriate selection of parameters, and the existing criteria of these solitons are also given. A Survey of the literature shows that these methodologies are potent enough to extract exact solutions of differential equations applied in engineering mathematics.

Efficient energy-preserving eighth-order compact finite difference schemes for the sine-Gordon equation

In this paper, three efficient energy conserving compact finite difference schemes are developed to numerically solve the sine-Gordon equation with homogeneous Dirichlet boundary conditions. To be specific, the spatial discretization is carried out by a novel eighth-order accurate compact difference scheme in which the fast discrete sine transform can be utilized for efficient implementation. The second-order conservative Crank–Nicolson scheme is considered in the temporal direction. Then the conservative property and convergence of the first scheme in two-dimensional space are discussed. A linearized iteration based on the fast discrete sine transform technique is devised to solve the nonlinear system effectively. Since the resultant algorithm does not use matrix inversion, it is computationally efficient in long-time calculations. Furthermore, the two other schemes are constructed based on improved scalar auxiliary variable approaches by converting the sine-Gordon equation into an equivalent new system which involves solving linear systems with constant coefficients at each time step. Finally, numerical experiments are presented to validate the correctness of the theoretical findings and demonstrate the excellent performance in long-time conservation of the schemes.

Solutions of the sinh-Gordon and sine-Gordon equations and applications

Solutions of the sinh-Gordon and sine-Gordon equations and applications

Magnetic field induced deformation of the spin density wave microphases in Ca3Co2O6

The frustrated triangular Ising magnet Ca$_3$Co$_2$O$_6$ has long been known for an intriguing combination of extremely slow spin dynamics and peculiar magnetic orders, such as the evenly-spaced non-equilibrium metamagnetic magnetization steps and the long-wavelength spin density wave (SDW) order, the latter of which is essentially an emergent crystal of solitons. Recently, an elaborate field-cooling protocol to bypass the low-field SDW phase was proposed to overcome the extraordinarily long timescale of spin relaxation that impeded previous experimental studies in equilibrium, which may point to a deep connection between the low-temperature slow relaxation and the cooling process passing through the low-field SDW phase. As the first step to elucidate the conjectured connection, we investigate the magnetic field-induced deformation of the SDW state and incommensurate-commensurate transitions, thereby mapping out the equilibrium in-field phase diagram for a realistic three-dimensional lattice spin model by using Monte Carlo simulations. We also discuss Ginzburg-Landau theory that includes several Umklapp terms as well as an effective sine-Gordon model, which can qualitatively explain the observed magnetic field-induced deformation of the SDW microphases.

Z_{3} Parafermion in the Double Charge Kondo Model.

Quantum impurity models with frustrated Kondo interactions can support quantum critical points with fractionalized excitations. Recent experiments [W. Pouse et al., Nat. Phys. (2023)NPAHAX1745-247310.1038/s41567-022-01905-4] on a circuit containing two coupled metal-semiconductor islands exhibit transport signatures of such a critical point. Here, we show using bosonization that the double charge-Kondo model describing the device can be mapped in the Toulouse limit to a sine-Gordon model. Its Bethe-ansatz solution shows that a Z_{3} parafermion emerges at the critical point, characterized by a fractional 1/2ln(3) residual entropy, and scattering fractional charges e/3. We also present full numerical renormalization group calculations for the model and show that the predicted behavior of conductance is consistent with experimental results.

Anomalous scaling at nonthermal fixed points of the sine-Gordon model

We extend the theory of non-thermal fixed points to the case of anomalously slow universal scaling dynamics according to the sine-Gordon model. This entails the derivation of a kinetic equation for the momentum occupancy of the scalar field from a non-perturbative two-particle irreducible effective action, which re-sums a series of closed loop chains akin to a large-$N$ expansion at next-to-leading order. The resulting kinetic equation is analyzed for possible scaling solutions in space and time that are characterized by a set of universal scaling exponents and encode self-similar transport to low momenta. Assuming the momentum occupancy distribution to exhibit a scaling form we can determine the exponents by identifying the dominating contributions to the scattering integral and power counting. If the field exhibits strong variations across many wells of the cosine potential, the scattering integral is dominated by the scattering of many quasiparticles such that the momentum of each single participating mode is only weakly constrained. Remarkably, in this case, in contrast to wave turbulent cascades, which correspond to local transport in momentum space, our results suggest that kinetic scattering here is dominated by rather non-local processes corresponding to a spatial containment in position space. The corresponding universal correlation functions in momentum and position space corroborate this conclusion. Numerical simulations performed in accompanying work yield scaling properties close to the ones predicted here.

ОДНОМЕРНАЯ ДИНАМИКА ДОМЕННОЙ ГРАНИЦЫ В СЕМИСЛОЙНОЙ ФЕРРОМАГНИТНОЙ СТРУКТУРЕ

The dynamics of the domain boundary is considered using the example of a seven-layer ferromagnetic structure with three thin and four wide magnetic layers. The structure of the domain boundary is represented as a kink solution of the sine-Gordon equation. The equation of motion for magnetization was solved numerically using an explicit scheme. The discretization of the equation was carried out according to a standard five-point scheme of the "cross" type. The paper shows the features of the dynamics of the domain boundary in a multilayer magnetic system in the presence of thin magnetic layers with an increased value of the magnetic anisotropy constant. Thin layers with an increased value of the magnetic anisotropy constant compared to the homogeneous state represent potential barriers to the moving domain boundary. Thin layers with an increased magnitude of magnetic anisotropy compared to a homogeneous state represent potential barriers to a moving domain boundary. A diagram of possible scenarios of the dynamics of the domain boundary is constructed depending on the initial velocity of its movement and the distance between three thin magnetic layers. The maximum value of the kink velocity for reflection from all potential barriers, depending on their size, is obtained. With an increase in the height and width of the barrier, the value of such a threshold maximum reflection velocity of the domain boundary increases nonlinearly. With a sufficiently high barrier height, there is already an almost linear dependence on the width of this threshold velocity. With a slight increase in the speed of movement of the domain boundary, the kink can pass through the first barrier, but it is reflected from the second barrier. There is also a case of kink oscillation between the second and third potential barriers. Such fluctuations are clearly inharmonious. The dependence of the threshold velocity on the distance between the barriers is obtained. As the distance between the barriers increases, the threshold speed value tends to a value equal to the threshold speed for one barrier. In the work, the minimum value of the speed of the domain boundary of the passage of all layers, depending on the parameters of potential barriers, is obtained. It is also found that there is a critical distance separating the dynamics of the domain boundary into two regions with qualitatively different behavior of the system.

ВЗАИМОДЕЙСТВИЕ И ДИНАМИКА ТОПОЛОГИЧЕСКИХ ДЕФЕКТОВ В ДОМЕННОЙ СТРУКТУРЕ ЗАКРУЧЕННОГО НЕМАТИКА

The dynamics and interaction of topological defects in the domain structure of π/2 twisted nematic liquid crystal are studied. A feature of Williams domains in twisted nematics is that hydrodynamic flows in them, along with the tangential velocity component, also have an axial component, the direction of which is opposite in neighboring domains. The axial velocity component arises as a result of the strong coupling between the initial twisted director orientation n and the hydrodynamic flow velocity. In domain structures of π/2 twisted nematics, as well as in planar oriented ones, edge dislocations with topological charges S =±1 arise. With increasing applied voltage density of dislocations grows. Dislocations move both along the Williams domains ( climb ) and perpendicular to them ( glide ). At a certain rate of the increasing applied voltage U > U c, distortions of the domain structure begin to propagate on both sides of the defect core. This leads to the formation of a special topological defects - dislocations with a “diffused” core, or linear defects, which are oriented perpendicular to the Williams domains and move mainly along them. The arising linear defect has the same topological charge as the initial dislocation. It is shown that their interaction is qualitatively well described by the perturbed sine-Gordon equation.

Equilibrium states for the massive Sine-Gordon theory in the Lorentzian signature

In this paper we investigate the massive Sine-Gordon model in the ultraviolet finite regime in thermal states over the two-dimensional Minkowski spacetime. We combine recently developed methods of perturbative algebraic quantum field theory with techniques developed in the realm of constructive quantum field theory over Euclidean spacetimes to construct the correlation functions of the equilibrium state of the Sine-Gordon theory in the adiabatic limit. First of all, the observables of the Sine-Gordon theory are seen as functionals over the free configurations and are obtained as a suitable combination of the S−matrices of the interaction Lagrangian restricted to compact spacetime regions over the free massive theory. These S−matrices are given as power series in the coupling constant with values in the algebra of fields over the free massive theory. Adapting techniques like conditioning and inverse conditioning to spacetimes with Lorentzian signature, we prove that these power series converge when evaluated on a generic field configuration. The latter observation implies convergence in the strong operator topology in the GNS representations of the considered states. In the second part of the paper, adapting the cluster expansion technique to the Lorentzian case, we prove that the correlation functions of the interacting equilibrium state at finite temperature (KMS state) can be constructed also in the adiabatic limit, where the interaction Lagrangian is supported everywhere in space.

On the Cauchy Problem for the mKdV–Sine-Gordon Equation with an Additional Term

On the Cauchy Problem for the mKdV–Sine-Gordon Equation with an Additional Term

Funnel control for a class of nonlinear infinite-dimensional systems

An adaptive funnel control method is considered for the regulation of the output for a class of nonlinear infinite-dimensional systems on real Hilbert spaces. After a decomposition of the state space and some change of variables related to the Byrnes–Isidori form, it is shown that the funnel controller presented in Berger et al. (2020) achieves the control objective under some assumptions on the nonlinear system dynamics, like well-posedness and Bounded-Input-State Bounded-Output (BISBO) stability. The theory is applied to the regulation of the temperature in a chemical plug-flow tubular reactor whose reaction kinetics are modeled by the Arrhenius nonlinearity. Furthermore a damped sine–Gordon model is shown to fit the required assumptions as well. The theoretical results are illustrated by means of numerical simulations.

Classical solutions of the degenerate fifth Painlevé equation

In this paper classical solutions of the degenerate fifth Painlevé equation are classified, which include hierarchies of algebraic solutions and solutions expressible in terms of Bessel functions. Solutions of the degenerate fifth Painlevé equation are known to be expressible in terms of solutions of the third Painlevé equation. The classification and description of the classical solutions of the degenerate fifth Painlevé equation is done using the Hamiltonian associated with third Painlevé equation. Two applications of these classical solutions are discussed, deriving exact solutions of the complex sine-Gordon equation and of the coefficients in the three-term recurrence relation associated with generalised Charlier polynomials.