Terms Of Solitons Research Articles

A nontopological soliton in an [formula omitted] supersymmetric Abelian gauge model

A version of N=1 supersymmetric scalar electrodynamics is considered here, and it is shown that an electrically charged nontopological soliton exists in this model. In addition to the long-range electric field, the soliton also possesses a long-range scalar field, which leads to a modification of the intersoliton interaction potential at large distances. The supersymmetry of the model makes it possible to express fermionic zero modes of the soliton in terms of bosonic fields. The properties of the nontopological soliton are investigated using analytical and numerical methods.

Parametric effects on paraxial nonlinear Schrödinger equation in Kerr media

• The mentioned techniques are applied to the paraxial NLS equation in Kerr media. • Optical soliton solutions are obtained and has parameters connected to the model. • Analyzed the nature of the obtained solutions by 3D and 2D wave profiles. • Nonlinear and free parameters effects on the paraxial NLS equation have been analyzed. • Wave profiles streamline pattern and immediate local directions have been analyzed. In this study, we have considered the (2+1)-dimensional paraxial nonlinear Schrödinger (NLS) equation in Kerr media and used the ( w / g ) - expansion method. The g ′ and ( g ′ / g 2 ) -expansion techniques have been customized from the ( w / g ) - expansion method. We applied these two techniques to the paraxial NLS equation and found the optical soliton solutions. The optical soliton solutions are attained as the flat kink, kink, singular kink, peakon, anti-parabolic, W-shape, M-shape, bell, and periodic wave solitons in terms of free parameters. We have presented three-dimensional (3D), two-dimensional (2D) and contour plots of the obtained results and discussed the effect of the free parameters and nonlinearity of the equation by determining different parametric values, which have not been discussed in the previous literature. We have studied the impact of the Kerr nonlinearity and wavenumber on the travelling wave solutions. Moreover, we also analyze the streamlines pattern and instantaneous local directions of the wave profile. All wave phenomena are applied to signal transmission, magneto-acoustic waves in plasma, optical fiber art, coastal engineering, quantum mechanics, hydro-magnetic waves, nonlinear optics and so on. The achieved solutions prove that the proposed methods are very powerful and effective for modern science and engineering for scrutinizing nonlinear evolutionary equations.

Applications of some types of solitons within the framework of Kählerian spacetime manifolds

In this paper, we study applications of some certain types of solitons such as conformal Ricci soliton, conformal [Formula: see text]-Ricci–Yamabe soliton and [Formula: see text]-Ricci soliton on Kählerian spacetime manifolds. Further, we have developed the characteristics of conformal Ricci soliton and conformal [Formula: see text]-Ricci–Yamabe soliton on almost pseudo-symmetric Kählerian spacetime manifolds. Here, we have signalized the nature of solitons in terms of shrinking, steady or expanding and we have also presented the relationship between [Formula: see text] and [Formula: see text] in terms of conformal [Formula: see text]-Ricci–Yamabe soliton. Finally, we have embellished the classification of the potential function with respect to gradient [Formula: see text]-Ricci soliton on Kählerian spacetime manifolds.

Quantum decoherence of dark pulses in optical microresonators

Quantum fluctuations disrupt the cyclic motions of dissipative Kerr solitons (DKSs) in nonlinear optical microresonators and consequently cause timing jitter of the emitted pulse trains. This problem is translated to the performance of several applications that employ DKSs as compact frequency comb sources. Recently, device manufacturing and noise reduction technologies have advanced to unveil the quantum properties of DKSs. Here we investigate the quantum decoherence of DKSs existing in normal-dispersion microresonators known as dark pulses. By virtue of the very large material nonlinearity, we directly observe the quantum decoherence of dark pulses in an AlGaAs-on-insulator microresonator, and the underlying dynamical processes are resolved by injecting stochastic photons into the microresonators. Moreover, phase correlation measurements show that the uniformity of comb spacing of quantum-limited dark pulses is better than 1.2 × 10−16 and 2.5 × 10−13 when normalized to the optical carrier frequencies and repetition frequencies, respectively. Comparing DKSs generated in different material platforms explicitly confirms the advantages of dark pulses over bright solitons in terms of quantum-limited coherence. Our work establishes a critical performance assessment of DKSs, providing guidelines for coherence engineering of chip-scale optical frequency combs.

The Enhancement of Energy-Carrying Capacity in Liquid with Gas Bubbles, in Terms of Solitons

A generalized (3 + 1)-dimensional nonlinear wave is investigated, which defines many nonlinear phenomena in liquid containing gas bubbles. Basic theories of the natural phenomenons are usually described by nonlinear evolution equations, for example, nonlinear sciences, marine engineering, fluid dynamics, scientific applications, and ocean plasma physics. The new extended algebraic method is applied to solve the model under consideration. Furthermore, the nonlinear model is converted into an ordinary differential equation through the next wave transformation. A well-known analytical approach is used to obtain more general solutions of different types with the help of Mathematica. Shock, singular, mixed-complex solitary-shock, mixed-singular, mixed-shock singular, mixed trigonometric, periodic, mixed-periodic, mixed-hyperbolic solutions are obtained. As a result, it is found that the energy-carrying capacity of liquid with gas bubbles and its propagation can be increased. The stability of the considered model is ensured by the modulation instability gain spectrum generated and proposed with acceptable constant values. Two-dimensional, three-dimensional, and contour surfaces are plotted to see the physical properties of the obtained solutions.

A Mathematical Study of the (3+1)-D Variable Coefficients Generalized Shallow Water Wave Equation with Its Application in the Interaction between the Lump and Soliton Solutions

In this paper, the Hirota bilinear method, which is an important scheme, is used. The equation of the shallow water wave in oceanography and atmospheric science is extended to (3+1) dimensions, which is a well-known equation. A lot of classes of rational solutions by selecting the interaction between a lump and one- or two-soliton solutions are obtained. The bilinear form is considered in terms of Hirota derivatives. Accordingly, the logarithm algorithm to obtain the exact solutions of a (3+1)-dimensional variable-coefficient (VC) generalized shallow water wave equation is utilized. The analytical treatment of extended homoclinic breather wave solutions is studied and plotted in three forms 3D, 2D, and density plots. Using suitable mathematical assumptions, the established solutions are included in view of a combination of two periodic and two solitons in terms of two trigonometric and two hyperbolic functions for the governing equation. Maple software for computing the complicated calculations of nonlinear algebra equations is used. The effect of the free parameters on the behavior of acquired figures to a few obtained solutions for two nonlinear rational exact cases was also discussed.

The nonlinear vibration and dispersive wave systems with extended homoclinic breather wave solutions

Abstract This article investigates the extended homoclinic (heteroclinic) breather wave solutions and interaction periodic and dark soliton solutions to the nonlinear vibration and dispersive wave systems. The solutions include periodic, breather, and soliton solutions. The bilinear form is considered in terms of Hirota derivatives. Accordingly, we utilize the Cole–Hopf algorithm to obtain the exact solutions of the ( 2 + 1 2+1 )-dimensional modified dispersive water-wave system. The analytical treatment of extended homoclinic breather wave solutions and interaction periodic and dark soliton solutions are studied and plotted in four forms of density plots. A nonlinear vibration system will be studied. Employing appropriate mathematical assumptions, the novel kinds of the extended homoclinic breather wave solutions and interaction periodic and dark soliton solutions are derived and constructed in view of the combination of kink, periodic, and soliton for an extended homoclinic breather and also a combination of two kinks, periodic and dark soliton in terms of exponential, trigonometric, hyperbolic functions for interaction periodic and dark soliton of the governing equation. To achieve this, the illustrative example of the ( 2 + 1 2+1 )-D modified dispersive water-wave system is furnished to demonstrate the feasibility and reliability of the procedure applied in this research. The trajectory solutions of the traveling waves are offered explicitly and graphically. The effect of the free parameters on the behavior of designed figures of a few obtained solutions for two nonlinear rational exact cases was also considered. By comparing the suggested scheme with the other existing methods, the results state that the execution of this technique is succinct, extensive, and straightforward.

Quantum dynamics of dissipative Kerr solitons

Dissipative Kerr solitons arising from parametric gain in ring microresonators are usually described within a classical mean-field framework. Here, we develop a quantum-mechanical model of dissipative Kerr solitons in terms of the Lindblad master equation and study the model via the truncated Wigner method, which accounts for quantum effects to leading order. We show that, within this open quantum system framework, the soliton experiences a finite coherence time due to quantum fluctuations originating from losses. Reading the results in terms of the theory of open quantum systems allows us to estimate the Liouvillian spectrum of the system. It is characterized by a set of eigenvalues with a finite imaginary part and a vanishing real part in the limit of vanishing quantum fluctuations. This feature shows that dissipative Kerr solitons are a specific class of dissipative time crystals.

Soliton solution of coupled Korteweg-de Vries equation by quintic UAH tension B-spline differential quadrature method

The concern of this paper is to acquire the numerical solution of Coupled Korteweg-de Vries (CKdV) equation describing the behaviour of shallow water surfaces in terms of solitons. A new regime quintic Uniform Algebraic Hyperbolic (QUAH) tension B-spline with a very well-known method called Differential Quadrature Method (DQM) is used. Over the past few years, the idea of consuming DQM to solve partial differential equations numerically has acknowledged much consideration all over the science community. Initially, the QUAH tension B-spline is used as a basis function in DQM to solve the weighting coefficients. After calculating weighting coefficients, the basis function converts the Initial boundary value problem into an ordinary differential equation, which is further solved by using a strong preserving Runge-Kutta 43 regime. Then, to check the accuracy and amendment of the used method, different test problems are solved numerically as well as error norms L2 and L∞ are calculated. Calculated results are shown graphically and in tabular form for quick and easy access. An obtained result gives the insight that the proposed technique is effective and efficient for elucidating the variety of complex partial differential equations.

Head-on Collisional Effects of Ion Acoustic Waves in Magnetized Plasma: Linear and Nonlinear Analyses

Head-on collisional phenomena of ion acoustic (IA) waves in electron–positron–ion magnetized plasmas are investigated through linear and nonlinear analyses. The plasma system is considered collisionless and comprising cold ions, kappa distributed cold electrons, and positrons. Dispersion relation is derived with the aid of the linear perturbation method. The two sided nonlinear Korteweg-de Vries (KdV) equations are derived employing extended Poincare-Lighthill-Kuo (ePLK) method. The effects of wave frequency on linear group velocity with species density ratio are studied. Further, the production of ion acoustic solitons in terms of ion gyrofrequency ( $${\omega }_{\mathrm{g}}$$ ), superthermality parameters for electrons ( $${\kappa }_{\mathrm{e}}$$ ), and positrons ( $${\kappa }_{\mathrm{p}}$$ ) is investigated. Additionally, the effects of obliqueness (angle between magnetic field and wave propagation vector) on nonlinearity coefficient and phase shifts with density ratios (positron to electron and ion to electron) are demonstrated.

Sharp upper diameter bounds for compact shrinking Ricci solitons

We give a sharp upper diameter bound for a compact shrinking Ricci soliton in terms of its scalar curvature integral and the Perelman’s entropy functional. The sharp cases could occur at round spheres. The proof mainly relies on a sharp logarithmic Sobolev inequality of gradient shrinking Ricci solitons and a Vitali-type covering argument.

Cubic–quartic nonlinear Schrödinger equation in birefringent fibers with the presence of perturbation terms

ABSTRACT In this paper, we derive the solitons solutions to the cubic–quartic nonlinear Schrödinger equation (CQ-NLSE) in birefringent fibers with three laws of nonlinearity. These laws are parabolic law, quadratic-cubic law and non-local law. The new extended generalized Kudryashov method and our new method proposed for the first time have been applied. Many solutions have been found. Dark, bright and singular soliton solutions existed under constraint conditions. The comparative analysis of the two proposed methods shows that in the two cases of parabolic and non-local laws, the first method gives the dark and singular solitons in terms of tanh–coth hyperbolic functions, respectively, while the second method gives the bright and singular solitons in terms of sech–csch hyperbolic functions, respectively. But in the case of quadratic-cubic law, both methods give the bright-singular solitons in terms of sech–csch hyperbolic functions, respectively, with different arguments.

Almost η-Ricci and almost η-Yamabe solitons with torse-forming potential vector field

We provide properties of almost η-Ricci and almost η-Yamabe solitons on submanifolds isometrically immersed into a Riemannian manifold whose potential vector field is the tangential component of a torse-forming vector field on , treating also the case of a minimal or pseudo quasi-umbilical hypersurface. Moreover, we give necessary and sufficient conditions for an orientable hypersurface of the unit sphere to be an almost η-Ricci or an almost η-Yamabe soliton in terms of the second fundamental tensor field.

Large amplitude ion-acoustic solitary waves in a warm negative ion plasma with superthermal electrons: The fast mode revisited

Large amplitude ion-acoustic fast mode solitary waves in a negative ion plasma with kappa electrons are revisited, using the Sagdeev pseudopotential approach. As is well known, this plasma supports the propagation of both compressive and rarefactive solitons, and there exist a range of parameter values in which the two types of structures coexist. This is confirmed by the present study, which is based on well-established soliton existence domains. After investigating the existence of solitons in terms of the lower and upper Mach number limits for broader regions in the parameter space, we have found that as a result of the ion thermal effects, the range of the allowed Mach numbers is reduced and only small amplitude rarefactive solitons propagate in this plasma, an effect that is enhanced by the superthermal behavior of the electrons. Rearranging our analytical work so as to get a two-positive ion plasma, our results show the presence of stopbands in the soliton existence domains, as reported by Nsengiyumva et al. [Phys. Plasmas 21, 102301 (2014)], despite the use of different normalization and different parameter space. This suggests that the observed stopbands are a real phenomenon, which needs consideration when studying plasma waves.

The β Fermi-Pasta-Ulam-Tsingou recurrence problem.

We perform a thorough investigation of the first Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence in the β-FPUT chain for both positive and negative β. We show numerically that the rescaled FPUT recurrence time Tr=tr/(N+1)3 depends, for large N, only on the parameter S≡Eβ(N+1). Our numerics also reveal that for small |S|, Tr is linear in S with positive slope for both positive and negative β. For large |S|, Tr is proportional to |S|-1/2 for both positive and negative β but with different multiplicative constants. We numerically study the continuum limit and find that the recurrence time closely follows the |S|-1/2 scaling and can be interpreted in terms of solitons, as in the case of the KdV equation for the α chain. The difference in the multiplicative factors between positive and negative β arises from soliton-kink interactions that exist only in the negative β case. We complement our numerical results with analytical considerations in the nearly linear regime (small |S|) and in the highly nonlinear regime (large |S|). For the former, we extend previous results using a shifted-frequency perturbation theory and find a closed form for Tr that depends only on S. In the latter regime, we show that Tr∝|S|-1/2 is predicted by the soliton theory in the continuum limit. We then investigate the existence of the FPUT recurrences and show that their disappearance surprisingly depends only on Eβ for large N, not S. Finally, we end by discussing the striking differences in the amount of energy mixing between positive and negative β and offer some remarks on the thermodynamic limit.

Compact $$\lambda $$λ-translating Solitons with Boundary

A $$\lambda $$ -translating soliton with density vector $$\mathbf {v}$$ is a surface $$\varSigma $$ in Euclidean space $$\mathbb {R}^3$$ whose mean curvature H satisfies $$2H=2\lambda +\langle N,\mathbf {v}\rangle $$ , where N is the Gauss map of $$\varSigma $$ . In this article, we study the shape of a compact $$\lambda $$ -translating soliton in terms of its boundary. If $$\varGamma $$ is a given closed curve, we deduce under what conditions on $$\lambda $$ there exists a compact $$\lambda $$ -translating soliton $$\varSigma $$ with boundary $$\varGamma $$ and we provide estimates of the surface area depending on the height of $$\varSigma $$ . Finally, we study the shape of $$\varSigma $$ related with the geometry of $$\varGamma $$ , in particular, we give conditions that assert that $$\varSigma $$ inherits the symmetries of its boundary $$\varGamma $$ .

An upper diameter bound for compact Ricci solitons with application to the Hitchin–Thorpe inequality

In this article, stimulated by Fernández-López and García-Río, we shall give an upper diameter bound for compact shrinking Ricci solitons in terms of the range of the scalar curvature, as well as in terms of the range of the potential function. As an application, we shall provide some new sufficient conditions for compact four-dimensional shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality.

Nonautonomous solitons in terms of the double Wronskian determinant for a variable-coefficient Gross–Pitaevskii equation in the Bose–Einstein condensate

Under investigation in this paper is a variable-coefficient Gross–Pitaevskii equation which describes the Bose–Einstein condensate. Lax pair, bilinear forms and bilinear Bäcklund transformation for the equation under some integrable conditions are derived. Based on the Lax pair and bilinear forms, double Wronskian solutions are constructed and verified. The [Formula: see text]th-order nonautonomous solitons in terms of the double Wronskian determinant are given. Propagation and interaction for the first- and second-order nonautonomous solitons are discussed from three cases. Amplitudes of the first- and second-order nonautonomous solitons are affected by a real parameter related to the variable coefficients, but independent of the gain-or-loss coefficient [Formula: see text] and linear external potential coefficient [Formula: see text]. For Case 1 [Formula: see text], [Formula: see text] leads to the accelerated propagation of nonautonomous solitons. Parabolic-, cubic-, exponential- and cosine-type nonautonomous solitons are exhibited due to the different choices of [Formula: see text]. For Case 2 [Formula: see text], if the real part of the spectral parameter equals 0, stationary soliton can be formed. If we take the harmonic external potential coefficient [Formula: see text] as a positive constant and let the real parts of the two spectral parameters be the same, bound-state-like structures can be formed, but there are only one attractive and two repulsive procedures. For Case 3 [[Formula: see text] and [Formula: see text] are taken as nonzero constants], head-on interaction, overtaking interaction and bound-state structure can be formed based on the signs of the two spectral parameters.

Remark on a diameter bound for complete Riemannian manifolds with positive Bakry–Émery Ricci curvature

In this paper, we shall give a new upper diameter estimate for complete Riemannian manifolds in the case that the Bakry–Émery Ricci curvature has a positive lower bound and the norm of the potential function has an upper bound. Our diameter estimate improves previous ones obtained by Wei and Wylie (2009) [12] and Limoncu (2012) [8]. As an application, we shall give an upper diameter bound for compact shrinking Ricci solitons in terms of the maximum value of the scalar curvature. By using such a diameter bound, we shall provide some new sufficient conditions for four-dimensional compact shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality.

On Calabi extremal Kähler-Ricci solitons

In this note we give a characterization of Kaehler metrics which are both Calabi extremal and Kaehler-Ricci solitons in terms of complex Hessians and the Riemann curvature tensor. We apply it to prove that, under the assumption of positivity of the holomorphic sectional curvature, these metrics are Einstein.