Constant Backgrounds Research Articles

Magnetic soliton and breather interactions for the higher-order Heisenberg ferromagnetic equation via the iterative N-fold Darboux transformation

This paper focuses on a higher-order Heisenberg ferromagnetic equation, which may describe the motion of the magnetic vector of isotropic ferromagnetism. The iterative N-fold Darboux transformation is first constructed to generate the dark and anti-dark magnetic solitons on the non-zero constant backgrounds, bright and dark breathers on the trigonometric function and non-zero constant backgrounds as well as breathers on the trigonometric function and vanishing backgrounds. We discover that the soliton structures of three different components can generate rotation with different constant seed solutions. Meanwhile, the trajectory curve and the direction of the magnetic vector are also discussed from the perspective of magnetism, we find that for constant seed solutions, the motion of the magnetic vector is limited to the hemisphere, while for trigonometric seed solutions, the motion of the magnetic vector can be distributed throughout the whole sphere. These novel phenomena may be helpful to understand the dynamics of the magnetic vector in the magnetic materials.

Multiple localized waves to the (2+1)-dimensional shallow water waveequation on non-flat constant backgrounds and their applications

In this paper, a new general bilinear Bäcklund transformation and Lax pair for the (2+1)-dimensional shallow water wave equation are given in terms of the binary Bell polynomials. Based on this transformation along with introducing an arbitrary function, the multi-kink soliton, line breather, and multi-line rogue wave solutions on a non-flat constant background plane are derived. Further, we found that the dynamic pattern of line breather on the background of periodic line waves are similar to the two-periodic wave solutions obtained through a multi-dimensional Riemann theta function. Also, the generation mechanism and smooth conditions of the line rogue waves on the periodic line wave background are presented with long-wave limit method. Additionally, a family of new rational solutions, consisting of line rogue waves and line solitons, are derived, which have never been reported before. Furthermore, the present work can be directly applied to other nonlinear equations.

Rogue wave excitations and hybrid wave structures of the Heisenberg ferromagnet equation with time-dependent inhomogeneous bilinear interaction and spin-transfer torque.

In this paper, we focus on the localized rational waves of the variable-coefficient Heisenberg spin chain equation, which models the local magnetization in ferromagnet with time-dependent inhomogeneous bilinear interaction and spin-transfer torque. First, we establish the iterative generalized (m,N-m)-fold Darboux transformation of the Heisenberg spin chain equation. Then, the novel localized rational solutions (LRSs), rogue waves (RWs), periodic waves, and hybrid wave structures on the periodic, zero, and nonzero constant backgrounds with the time-dependent coefficients α(t) and β(t) are obtained explicitly. Additionally, we provide the trajectory curves of magnetization and the variation of the magnetization direction for the obtained nonlinear waves at different times. These phenomena imply that the LRSs and RWs play the crucial roles in changing the circular motion of the magnetization. Finally, we also numerically simulate the wave propagations of some localized semi-rational solutions and RWs.

Rational and semi-rational solutions to the nonlocal Davey–Stewartson III equation

We utilize the bilinear method and Kadomtsev–Petviashvili (KP) hierarchy reduction technique to obtain solutions for the nonlocal DS III equation. These solutions are elegantly expressed in terms of an N×N determinant, incorporating various parameter reduction constraints. It is important to note that these solutions are nontrivial and do not necessarily satisfy the local DS III equation. In the case of an even value of N, we derive solutions on constant backgrounds, while for odd values of N, solutions on periodic backgrounds are generated. To gain a deeper understanding of the dynamics involved, we conduct a thorough analysis of the derived solutions, accompanied by informative plots. Overall, our findings showcase the power of the bilinear method and KP hierarchy reduction technique in obtaining meaningful solutions for the nonlocal DS III equation, shedding light on its complex behavior and enriching our understanding of the system.

Dynamical analysis of multi-soliton and breather solutions on constant and periodic backgrounds for the (2+1)-dimensional Heisenberg ferromagnet equation

Dynamical analysis of multi-soliton and breather solutions on constant and periodic backgrounds for the (2+1)-dimensional Heisenberg ferromagnet equation

The two-dimensional Leznov lattice equation and its various solutions

In this paper, the two-dimensional Leznov lattice equation is studied by Hirota’s bilinear method and KP-Toda hierarchy reduction approach. We present solutions of the lattice equation in the ratio of τ functions given by the N×N Gram-type determinants. Solitons, breathers and rational solutions on constant and periodic backgrounds are derived. In particular, we find three types of breather solutions that are periodic in x-direction, y-direction and a general oblique line, respectively. Algebraic solitons are obtained by introducing two differential operators applied to the soliton solutions. We investigate the asymptotic behavior and interactions of algebraic soliton. The interaction of two algebraic solitons are analyzed and proved to be completely elastic. We demonstrate that the asymptotic algebraic solitons in the straight lines well approach the exact ones. Multiple algebraic solitons on the constant and periodic backgrounds are derived with higher-order determinant size. We introduce Schur polynomials to generate explicit expression for fundamental and higher-order lump solutions. Dynamics of lump solutions are analyzed with plots.

General Soliton and (Semi-)Rational Solutions of a (2+1)-Dimensional Sinh-Gordon Equation

In this paper, we investigate solutions of a (2+1)-dimensional sinh-Gordon equation. General solitons and (semi-)rational solutions are derived by the combination of Hirota’s bilinear method and Kadomtsev-Petviashvili hierarchy reduction approach. General solutions are expressed as Ntimes N Gram-type determinants. When the determinant size N is even, we generate solitons, line breathers, and (semi-)rational solutions located on constant backgrounds. In particular, through the asymptotic analysis we prove that the collision of solitons are completely elastic. When N is odd, we derive exact solutions on periodic backgrounds. The dynamical behaviors of those derived solutions are analyzed with plots. For rational solutions, we display the interaction of lumps. For semi-rational solutions, we find the interaction solutions between lumps and solitons.

Rogue waves and breathers of the derivative Yajima-Oikawa long wave-short wave equations on theta-function backgrounds

Based on the Baker-Akhiezer functions on Riemann surfaces, a general method is proposed to construct rogue-wave solutions and breather solutions of integrable nonlinear evolution equations associated with higher-order matrix spectral problems on theta-function periodic backgrounds. The derivative Yajima-Oikawa long wave-short wave equation associated with a 3×3 matrix Lax pair is chosen as an illustrative example of our method. First, a theta-function seed solution for the derivative Yajima-Oikawa long wave-short wave equation is derived by using the Weierstrass functions. Then, the general solution of the 3×3 matrix spectral problem and its derivatives with respect to the spectral parameter are obtained with the help of the Baker-Akhiezer functions. Finally, by means of Darboux transformations, explicit exact solutions such as rogue waves and breathers on theta-function backgrounds for the derivative Yajima-Oikawa long wave-short wave equation are constructed. Taking the advantage of the Jacobi θ-functions, these new solutions are numerically calculated and illustrated. In addition, by observing the explicit solutions, it is found that two rogue waves of different shapes may appear in one solution, which is a new phenomenon different from the solutions on constant backgrounds.

Homeostasis at different backgrounds: The roles of overlayed feedback structures in vertebrate photoadaptation.

We have studied the resetting behavior of eight basic integral controller motifs with respect to different but constant backgrounds. We found that the controllers split symmetrically into two classes: one class, based on derepression of the compensatory flux, leads to more rapid resetting kinetics as backgrounds increase. The other class, which directly activates the compensatory flux, shows a slowing down in the resetting at increased backgrounds. We found a striking analogy between the resetting kinetics of vertebrate photoreceptors and controllers based on derepression, i.e. vertebrate rod or cone cells show decreased sensitivities and accelerated response kinetics as background illuminations increase. The central molecular model of vertebrate photoadaptation consists of an overlay of three negative feedback loops with cytosolic calcium ([Formula: see text]), cyclic guanosine monophosphate (cGMP) and cyclic nucleotide-gated (CNG) channels as components. While in one of the feedback loops the extrusion of [Formula: see text] by potassium-dependent sodium-calcium exchangers (NCKX) can lead to integral control with cGMP as the controlled variable, the expected robust perfect adaptation of cGMP is lost, because of the two other feedback loops. They avoid that [Formula: see text] levels become too high and toxic. Looking at psychophysical laws, we found that in all of the above mentioned basic controllers Weber's law is followed when a "just noticeable difference" (threshold) of 1% of the controlled variable's set-point was considered. Applying comparable threshold pulses or steps to the photoadaptation model we find, in agreement with experimental results, that Weber's law is followed for relatively high backgrounds, while Stephens' power law gives a better description when backgrounds are low. Limitations of our photoadaption model, in particular with respect to potassium/sodium homeostasis, are discussed. Finally, we discuss possible implication of background perturbations in biological controllers when compensatory fluxes are based on activation.

Nonautonomous Breather and Rogue Wave in Spinor Bose–Einstein Condensates with Space-Time Modulated Potentials

To study controlled evolution of nonautonomous matter-wave breathers and rogue waves in spinor Bose–Einstein condensates with spatiotemporal modulation, we focus on a system of three coupled Gross–Pitaevskii equations with spacetime-dependent external potentials and temporally modulated gain-loss distributions. With different external potentials and gain-loss distributions, various solutions for controlled nonautonomous matter-wave breathers and rogue waves are derived by the Darboux transformation method, such as breathers and rogue waves on arched and constant backgrounds which have the periodic and parabolic trajectories. Effects of the gain-loss distribution and linear potential on the breathers and rogue waves are studied. Nonautonomous two-breathers on the arched and constant backgrounds are also derived.

Nonlinear electrodynamics without birefringence

All solutions of the no-birefringence conditions for nonlinear electrodynamics are found. In addition to the known Born-Infeld and Plebanski cases, we find a “reverse Born-Infeld” case, which has a limit to Plebanski, and an “extreme-Born-Infeld” case, which arises as a Lagrangian constraint. Only Born-Infeld has a weak-field limit, and only Born-Infeld and extreme-Born-Infeld avoid superluminal propagation in constant electromagnetic backgrounds, but all cases have a conformal strong-field limit that coincides with the strong-field limit of Born-Infeld found by Bialynicki-Birula.

Rational, semi-rational solution and self-consistent sources extension of the variable-coefficient extended modified Kadomtsev-Petviashvili equation

In this paper, we apply Hirota bilinear method and determinant technique to derive the Nth-order rational solution expressed compactly in terms of Matsuno determinants for the variable-coefficient extended modified Kadomtsev-Petviashvili (mKP) equation. As a special case, we obtain the M-lump solution expressed in terms of 2M × 2M determinants for the mKPI equation and investigate the dynamical behaviors of 1- and 2-lump solutions. Furthermore, we present the Wronskian and Grammian solution for the variable-coefficient extended mKP equation. Based on the Grammian solution, we construct the line soliton, the line breather and the semi-rational solution on constant and periodic backgrounds for the mKPI equation. Through the asymptotic analysis, we show that the semi-rational solutions describe the fission and fusion of lumps and line solitons. In addition, we construct the variable-coefficient extended mKP equation with self-consistent sources via the source generation procedure and derive its N-soliton solution in the compact form of Grammian and Wronskian.

Studies on the rogue waves of a (3 + 1)-dimensional Gross–Pitaevskii equation in the Bose–Einstein condensation

On the basis of the application in the Bose–Einstein condensation, we investigate a (3 + 1)-dimensional Gross–Pitaevskii equation with distributed time-dependent coefficients. With the aid of the Kadomtsev–Petviashvili hierarchy reduction method, we construct the [Formula: see text]th-order rogue-wave solutions in terms of the Gram determinant by introducing appropriate constraints. Using different coefficients for the diffraction [Formula: see text] and gain/loss [Formula: see text], we demonstrate the behaviors of the first- and second-order rogue waves by analytical and graphical means. We find that only if [Formula: see text], the rogue waves appear on the constant backgrounds; otherwise, the heights of the backgrounds change as time goes on. With the different choices of [Formula: see text] and [Formula: see text], the long-live, rapid-reducing and periodic rogue waves are discussed. The separated and aggregated second-order rogue waves are also shown on the constant and periodical backgrounds.

Plane Wave Backgrounds in the Worldline Formalism

Abstract Plane-wave backgrounds play a special role in strong-field QED as non-trivial field configuration simple enough to be treated analytically whilst still leading to rich physical consequences. In vacuum and in constant backgrounds, the first quantised, string-inspired “Worldline Approach” to field theoryoffers substantial simplifications and calculational efficiency. We present a new, general approach to incorporating plane wave backgrounds into the Worldline Formalism extending initial work by Ilderton and Torgrimsson. The method uses resummation techniques to take the background into account non-perturbatively and yields “Master Formulae” for scattering amplitudes in the background that may offer an alternative tool to studying QED in plane waves as has been achieved in the constant field case.

Solitons, breathers and rational solutions for a (2+1)-dimensional dispersive long wave system

In this paper, we investigate a (2+1)-dimensional dispersive long wave system with an arbitrary constant a. Solitons, line breathers, rational solutions and algebraic solitons of the system are constructed based on Hirota’s bilinear method and Kadomtsev–Petviashvili (KP) hierarchy reduction technique. These solutions are expressed in terms of N×N determinant. When the size of the determinant N is odd, solutions on periodic backgrounds are generated. Solutions on constant backgrounds are derived when N is even. By using asymptotic analysis, we elucidate explicit expressions of asymptotic algebraic solitons localized in the straight for the algebraic soliton solutions. As a byproduct, we find lump solutions for an intermediate potential of the system. The effects of constant a are discussed and it is shown that the constant can affect collision of solitons. Dynamics of the obtained solutions are discussed with plots.

De Sitter vacua in the string landscape

The late-time behavior of our universe is one of accelerated expansion, or that of a de Sitter space, and therefore motivates us to look for time-dependent backgrounds. Finding such backgrounds in string theory has always been a challenging problem. An even harder problem is to find time-dependent backgrounds that allow positive dark energies. As a first step to handle such scenarios, we study a time-dependent background in type IIB theory, with four-dimensional de Sitter isometries, by uplifting it to M-theory and then realizing it as a coherent, or squeezed-coherent, state over an appropriate solitonic configuration. While classically such a background does not solve the equations of motion, the corresponding Schwinger-Dyson equations reveal that there are deeper issues that may even prohibit a solution to exist at the quantum level, as long as the internal space remains time-independent. A more generic analysis is then called for, where both the effective four-dimensional space-time, the internal space, and the background fluxes are all time-dependent. We study in details such a background by including perturbative and non-perturbative as well as local and non-local quantum terms. Our analysis reveals a distinct possibility of the emergence of a four-dimensional positive curvature space-time with de Sitter isometries and time-independent Newton's constant in the landscape of type IIB string theory. We argue how the no-go and the swampland criteria are avoided in generating such a background, and compare it with other possibilities involving backgrounds with time-dependent Newton constants. These time-varying Newton constant backgrounds typically lead to unavoidable late time singularities, amongst other issues.

Tau functions of (n, 1) curves and soliton solutions on nonzero constant backgrounds

We study the tau function of the KP-hierarchy associated with an (n,1) curve $y^n=x-\alpha$. If $\alpha=0$ the corresponding tau function is 1. On the other hand if $\alpha\neq 0$ the tau function becomes the exponential of a quadratic function of the time variables. By applying vertex opertaors to the latter we obtain soliton solutions on non-zero constant backgrounds.

Soliton solutions to the reverse-time nonlocal Davey–Stewartson III equation

This study derives solitons solutions of the reverse-time nonlocal Davey–Stewartson III equation by the Kadomtsev–Petviashvili hierarchy reduction method and Hirota’s bilinear method. The solutions are expressed as N×N Gram-type determinants with different parametric reduction conditions. N-soliton and line breather solutions on both constant and periodic backgrounds are derived. The dynamics of these solutions are discussed. All possible configurations of these solutions are illustrated for 1≤N≤4. Both intersecting and parallel solitons are presented. In particular, the elastic and inelastic collisions of the two parallel-soliton solutions are determined. In the inelastic case, the amplitudes of the solitons change after collision. Moreover, the parametric conditions for determining the inelastic collisions are derived, and all possible types of inelastic behaviors are obtained and displayed.

Non-Riemannian isometries from double field theory

We explore the notion of isometries in non-Riemannian geometries. Such geometries include and generalise the backgrounds of non-relativistic string theory, and they can be naturally described using the formalism of double field theory. Adopting this approach, we first solve the corresponding Killing equations for constant flat non-Riemannian backgrounds and show that they admit an infinite-dimensional algebra of isometries which includes a particular type of supertranslations. These symmetries correspond to known worldsheet Noether symmetries of the Gomis-Ooguri non-relativistic string, which we now interpret as isometries of its non-Riemannian doubled background. We further consider the extension to supersymmetric double field theory and show that the corresponding Killing spinors can depend arbitrarily on the non-Riemannian directions, leading to “supersupersymmetries” that square to supertranslations.

R2 corrected AdS2 holography

We approach the problem of constructing an explicit holographic dictionary for the AdS2/CFT1 correspondence in the context of higher derivative gravitational actions in AdS2 space-times. These actions are obtained by an S2 reduction of four-dimensional mathcal{N} = 2 Wilsonian effective actions with Weyl squared interactions restricted to constant scalar backgrounds. BPS black hole near-horizon space-times fall into this class of backgrounds, and by identifying the boundary operators dual to the bulk fields, we explicitly show how the Wald entropy of the BPS black hole is holographically encoded in the anomalous transformation of the operator dual to a composite bulk field. Additionally, using a 2d/3d lift, we show that the CFT holographically dual to AdS2 is naturally embedded in the chiral half of the CFT2 dual to the AdS3 space-time, and we identify the specific operator in CFT1 that encodes the chiral central charge of the CFT2.