Narrow Results

In this paper, we define and explore the Darboux and binary Darboux transformations for a semi-discrete short pulse equation. By iterating the binary Darboux transformation, we obtain quasi-Grammian solutions. Furthermore, we derive explicit matrix solutions for the binary Darboux matrix and reduce them to the elementary Darboux matrix. Finally, we plot the dynamics of bright, dark, double breather, and peak soliton solutions.

In this research, we delve into the exploration of the Kuralay-IIA equation by employing two power full approaches, namely extended hyperbolic function method (EHFM) and the improved modified Sardar-subequation method (IMSSEM). Applications of the Kuralay-IIA equation can be observed in a variety of physical phenomena, particularly in optical fibers, magnetic materials, and nonlinear optics. This equation performs an essential part in understanding gas behavior. Specifically, it offers an image of the relationship between volume, temperature and pressure in a gas system. We proceed through the matter in detail through the use of space curves which demonstrate integrable motion. Using the proposed techniques, our investigation provides an extensive variety of soliton solutions, such as periodic, dark–bright, bright, dark, solitary, and some other solutions. These solutions are in good alignment with previous study findings in this area. Analytical wave solutions are vital as they provide a basic comprehension of the physics or mathematics at play and establish a framework for further studies. The findings obtained from this study may help design models to come. The techniques used in this study are very effective, straightforward, and capable of handling more nonlinear models with high reliability. We use the Mathematica software to validate the accuracy and reliability of our results. Using carefully constructed 2D and 3D graphs generated, the resulting solutions are visually shown.

In this paper, we investigate $Z_2$ symmetric solitonic thick branes constructed by a five-dimensional real scalar field that is coupled to the curvature through a nonminimally derivative coupling term. The effects of the nonminimal derivative coupling parameter on the structure of thick brane, the five-dimensional warp factor and energy density are analyzed. By investigating the tensor perturbation equations of gravity, we also find that the solitonic brane is stable and capable of trapping the graviton zero mode under certain circumstances of the nonminimal derivative coupling constant.

We prove asymptotic stability of trapped solitons in the generalized nonlinear Schrödinger equation with a potential in dimension 1 and for even potential and even initial conditions.

In this paper, we construct a closed form of projectors on the integral noncommutative orbifold T²/Z₆ in terms of elliptic functions by GHS (Gopakumar, Headrick and Spradlin) construction. Thereafter, we give a general solution of projectors on T²/Z₆ and T²/Z₃ with minimal trace and continuous reduced matrix M(k,q₀). The projectors constructed by us possess symmetry and manifestly covariant forms under Z₆ rotation. Since projectors correspond to the soliton solutions of field theory on the noncommutative orbifold, we thus present a series of corresponding manifestly covariant soliton solutions.

We consider the nonlinear Klein–Gordon–Maxwell system derived from the Lagrangian on four-dimensional Minkowski space-time, where ϕ is a complex scalar field and F_μν = ∂_μ𝔸_ν - ∂_ν𝔸_μ is the electromagnetic field. For appropriate nonlinear potentials , the system admits soliton solutions which are gauge invariant generalizations of the non-topological solitons introduced and studied by Lee and collaborators for pure complex scalar fields. In this article, we develop a rigorous dynamical perturbation theory for these solitons in the small e limit, where e is the electromagnetic coupling constant. The main theorems assert the long time stability of the solitons with respect to perturbation by an external electromagnetic field produced by the background current 𝕁^B, and compute their effective dynamics to O(e). The effective dynamical equation is the equation of motion for a relativistic particle acted on by the Lorentz force law familiar from classical electrodynamics. The theorems are valid in a scaling regime in which the external electromagnetic fields are O(1), but vary slowly over space-time scales of , and δ = e^{1 - k} for as e → 0. We work entirely in the energy norm, and the approximation is controlled in this norm for times of .

A formulation of Darboux transformations is proposed for integrable couplings, based on non-semisimple matrix Lie algebras. Applications to a kind of integrable couplings of the AKNS equations are made, along with an explicit formula for the associated Bäcklund transformation. Exact one-soliton-like solutions are computed for the integrable couplings of the second- and third-order AKNS equations, and a type of reduction is created to generate integrable couplings and their one-soliton-like solutions for the NLS and MKdV equations.

In this paper, we study the Ricci–Bourguignon flow of all locally homogenous geometries on closed three-dimensional manifolds. We also consider the evolution of the Yamabe constant under the Ricci–Bourguignon flow. Finally, we prove some results for the Bach-flat shrinking gradient soliton to the Ricci–Bourguignon flow.

The linear and nonlinear properties of the electromagnetic waves are investigated in a magnetized quantum electron–positron (e–p) plasma by employing the quantum hydrodynamic (QHD) model. It is found that the quantum dispersion relation in comparison with the classical version is modified by the quantum corrections through quantum diffraction and statistics. The standard reductive perturbation technique is used to derive the Korteweg–de Vries (KdV) equation. The exact soliton solutions and the existence regions of the solitary waves are also defined precisely. It is also shown that the results are affected by the quantum corrections.

We present extensive results of computer simulations on vibrations in one- and two-dimensional lattices with quartic anharmonicity. The existence of localized mode in ordered lattice is confirmed but the long term stability is questionable. Rather our results indicate an ergodic behavior at large times when several modes are present. Concerning disordered systems we have found that anharmonicity leads to a surprising new phenomenon of anomalous diffusion and thus to the possibility of anharmonically induced transport.

Envelope solitons of the Nonlinear Schrödinger equation (NLS) under quantum potential's influence are studied. Corresponding problem is found to be integrable for an arbitrary strength, s ≠ 1, of the quantum potential. For s < 1, the model is equivalent to the usual NLS with rescaled coupling constant, while for s > 1, to the reaction–diffusion system. The last one is related to the anti-de Sitter (AdS) space valued Heisenberg model, realizing a particular gauge fixing condition of the (1+1)-dimensional Jackiw–Teitelboim gravity. For this gravity model, by the Madelung fluid representation we derive the acoustic form of the space–time metric. The space–time points, where dispersion changes the sign, correspond to the event horizon, while the soliton solution to the AdS black hole. Moving with the above bounded velocity, it describes evolution on the one sheet hyperboloid with nontrivial winding number, and creates under collision, the resonance states which we study by the Hirota bilinear method.

A derivative nonlinear Schrödinger model is shown to support localized N-body bound states for several ranges (called bands) of the coupling constant η. The ranges of η within each band can be completely determined using number theoretic concepts such as Farey sequences and continued fractions. For N≥3, the N-body bound states can have both positive and negative momenta. For η>0, bound states with positive momentum have positive binding energy, while states with negative momentum have negative binding energy.

The large-N limit of fermionic vectors models is studied using bilocal variables in the framework of a collective field theory approach. The large-N configuration is determined completely using only classical solutions of the model. Further, the Bethe–Salpeter equations of the model are cast as a Green's function problem. One of the main results of this work is to show that this Green's function is in fact the large-N bilocal itself.

An unified approximated solution for symmetric Skyrmions was proposed for the SU(2) Skyrme model for baryon numbers up to 8, which take the hybrid form of a kink-like solution and that given by the instanton method. The Skyrmion profiles are examined by computing lowest soliton energy as well as the static properties of nucleons within the framework of collective quantization, with a good agreement with the exact numerical results. The comparisons with the previous computations as well as the experimental data are also presented.

Light chemical elements are, for instance, produced through ion collisions taking place in the core of stars, where fusion is particularly important to the synthesis of chemical elements. Meanwhile soliton provides transparency leading to the hindrance of fusion cross-section. In order to explain high fusion cross-section actually observed in low incident energies, it is necessary to discover the suppression mechanism of soliton propagation. In this paper, based on a systematic three-dimensional time-dependent density functional calculation, the existence of soliton is examined for ion collisions with some incident energies, impact parameters, and nuclear force parameter sets. As a result, solitons are suggested to exist highly depending on the energy. The suppression of soliton is consequently due to the spin-orbit force and the momentum-dependent components of the nuclear force.

We investigate a two-dimensional N = 2 supersymmetric model which consists of n chiral superfields with Kähler potential. When we define quantum observables, we are always plagued by operator ordering problem. Among various ways to fix the operator order, we rely upon the supersymmetry. We demonstrate that the correct operator order is given by requiring the super-Poincaré algebra by carrying out the canonical Dirac bracket quantization. This is shown to be also true when the supersymmetry algebra has a central extension by the presence of topological soliton. It is also shown that the path of soliton is a straight line in the complex plane of superpotential W and triangular mass inequality holds. One half of supersymmetry is broken by the presence of soliton.

Domain wall solitons are the simplest topological objects in field theories. The conventional translational symmetry in a field theory is the generator of a one-parameter family of domain wall solutions, and induces a massless moduli field which propagates along a domain wall. We study similar issues in braided noncommutative field theories possessing Hopf algebraic translational symmetries. As a concrete example, we discuss a domain wall soliton in the scalar braided noncommutative field theory in Lie-algebraic noncommutative spacetime, which has a Hopf algebraic translational symmetry. We construct explicitly a one-parameter family of solutions in perturbation of the noncommutativity parameter. We then find the massless moduli field which propagates on the domain wall soliton. This work is based on arXiv:0711.3059.

Soliton solutions in a scalar field theory defined on an AdS₁₊₁ background space-time are investigated. An analytic soliton solution is obtained in a polynomial model, and the classical soliton mass is calculated. The fluctuation spectrum around the soliton solution is determined, and the one-loop quantum correction to the soliton mass is computed in the semi-classical approximation.

It is known that gravitational and electromagnetic fields of an electron are described by the ultra-extreme Kerr-Newman (KN) black hole solution with extremely high spin/mass ratio. This solution is singular and has a topological defect, the Kerr singular ring, which may be regularized by introducing the solitonic source based on the Higgs mechanism of symmetry breaking. The source represents a domain wall bubble interpolating between the flat region inside the bubble and external KN solution. It was shown recently that the source represents a supersymmetric bag model, and its structure is unambiguously determined by Bogomolnyi equations. The Dirac equation is embedded inside the bag consistently with twistor structure of the Kerr geometry, and acquires the mass from the Yukawa coupling with Higgs field. The KN bag turns out to be flexible, and for parameters of an electron, it takes the form of very thin disk with a circular string placed along sharp boundary of the disk. Excitation of this string by a traveling wave creates a circulating singular pole, indicating that the bag-like source of KN solution unifies the dressed and point-like electron in a single bag-string-quark system.

We review recent progress in the computation of leading quantum corrections to the energies of classical solitons with topological structure, including multi-soliton models in one space dimension and string configurations in three space dimensions. Taking advantage of analytic continuation techniques to efficiently organize the calculations, we show how quantum corrections affect the stability of solitons in the Shifman–Voloshin model, stabilize charged electroweak strings coupled to a heavy fermion doublet, and bind Nielsen–Olesen vortices at the classical transition between type I and type II superconductors.