Φ6 Model Research Articles

Exploring Solitons Solutions of a (3+1)-Dimensional Fractional mKdV-ZK Equation

This study presents the application of the ϕ6 model expansion technique to find exact solutions for the (3+1)-dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation under Jumarie’s modified Riemann–Liouville derivative (JMRLD). The suggested method captures dark, periodic, traveling, and singular soliton solutions, providing deep insights into wave behavior. Clear graphics demonstrate that the solutions are greatly affected by changes in the fractional order, deepening our understanding and revealing the hidden dynamics of wave propagation. The considered equation has several applications in fluid dynamics, plasma physics, and nonlinear optics.

Kink collision in the noncanonical [formula omitted] model: A model with localized inner structures

We study collisions of kinks in the one-space and one-time dimensional noncanonical nonintegrable scalar ϕ6 model. We examine the energy density of the kink, and we find that, as a function of the parameters that control the curvature of the potential, a localized inner structure of the energy density emerges. We also examine the kink excitation spectrum and the dynamics of the kink collisions for a wide range of initial velocities. We find that apart from the resonance windows, the production of two to three oscillons occurs for some values of the principal parameters of the model.

Collisions of kinks in deformed [formula omitted] and [formula omitted] models

Two hyperbolic-deformed field theoretic models are discussed. In both of them, due to the effect of specific deformation function on the well known φ4 and φ6 models, their internal structure may change significantly. Unlike the φ4 kinks solutions, which has only one internal mode in addition to its translational mode, the kinks of the hyperbolic-deformed φ4 model can have several internal modes. Moreover, the deformation on the φ6 model has other interesting features, because the kinks of the φ6 model have only a zero mode and the deformation may cause the appearance of internal mode for both kink and antikink. The presence of the new internal modes modify the collisions which we explore in the present work. The deformation relies on a real parameter, which controls the number of internal modes, and we also study how the deformation parameter alter the mass of the kinks and the critical velocities involved in the collisions.

Exact solution of paraxial wave dynamical model with Kerr Media by using [formula omitted] model expansion technique

This paper discusses optical soliton solutions for the paraxial wave model (PWM) in Kerr law media. Mathematically, the PWM for a monochromatic beam is equivalent to the Schrödinger equation of free quantum particle. The ϕ6 model expansion technique is used to obtain dark, bright, singular, bright-dark combined and periodic solitons. The solutions obtained by this method are in the form of trigonometric, hyperbolic, and exponential functions. To prompt the important propagated structures, some scrutinized solutions are demonstrated in the form of 3D, contour, and 2D plots using specific values to the parameters with constrained conditions. It is observed that this method is an efficient, secure, and stable tool for new precise solitons for various NLPDEs in engineering and applied sciences.

Kink Collision in the Noncanonical Φ⁶ Model: A Model with Localized Inner Structures

Kink Collision in the Noncanonical Φ⁶ Model: A Model with Localized Inner Structures

Renormalization functions of the tricritical O(N)-symmetric Φ6 model beyond the next-to-leading order in 1/N

We investigate higher-order corrections to the effective potential of the tricritical O(N)-symmetric model in 3-2ε dimensions in its phase exhibiting spontaneous breaking of its scale symmetry. The renormalization group β-function and the anomalous dimension γ of this model are computed up to the next-to-next-to-leading order in the1/N expansion technique and using a dimensional regularization in a minimal subtraction scheme.

Solitons in magneto–optic waveguides with dual–power law nonlinearity

This paper applies new ϕ6 model expansion scheme to obtain solitons in magneto–optic waveguides that come with dual–power law nonlinearity. Initially, the solutions are in terms of Jacobi's elliptic functions. Upon approaching the limiting values of the modulus of ellipticity, bright, dark and singular solitons emerge. The existence criteria of such solitons are also presented.

Staccato radiation from the decay of large amplitude oscillons

We study the decay of large amplitude, almost periodic breather-like states in a deformed sine-Gordon model in one spatial dimension. We discover that these objects decay in a staggered fashion via a series of transitions, during which higher harmonics are released as short, staccato bursts of radiation. Further, we argue that this phenomenon is not restricted to one particular model, and that similar mechanisms of radiative decay of long-lived oscillating states can be observed for a wide class of physical systems, including the ϕ6 model.

Optical solitons in birefringent fibers with quadratic–cubic refractive index by ϕ6–model expansion

This paper employes the new ϕ6–model expansion to derive soliton solutions in birefringent fibers with quadratic-cubic law of refractive index. This scheme reveals bright, dark, singular and combo optical solitons to the model. Their respective existence criteria are also included.

Kink scattering in a hybrid model

In this work we consider a model where the potential has two topological sectors connecting three adjacent minima, as occurs with the ϕ6 model. In each topological sector, the potential is symmetric around the local maximum. For ϕ>0 there is a linear map between the model and the λϕ4 model. For ϕ<0 the potential is reflected. Linear stability analysis of kink and antikink lead to discrete and continuum modes related by a linear coordinate transformation to those known analytically for the λϕ4 model. Fixing one topological sector, the structure of antikink-kink scattering is related to the observed in the λϕ4 model. For kink-antikink collisions a new structure of bounce windows appear. Depending on the initial velocity, one can have oscillations of the scalar field at the center of mass even for one bounce, or a change of topological sector. We also found a structure of one-bounce, with secondary windows corresponding to the changing of the topological sector accumulating close to each one-bounce windows. The kink-kink collisions are characterized by a repulsive interaction and there is no possibility of forming a bound state.

Kink dynamics in a parametric \u03d56 system: a model with controllably many internal modes

We explore a variant of the ϕ6 model originally proposed in Phys. Rev.D 12 (1975) 1606 as a prototypical, so-called, “bag” model in which domain walls play the role of quarks within hadrons. We examine the steady state of the model, namely an apparent bound state of two kink structures. We explore its linearization, and we find that, as a function of a parameter controlling the curvature of the potential, an effectively arbitrary number of internal modes may arise in the point spectrum of the linearization about the domain wall profile. We explore some of the key characteristics of kink-antikink collisions, such as the critical velocity and the multi-bounce windows, and how they depend on the principal parameter of the model. We find that the critical velocity exhibits a non-monotonic dependence on the parameter controlling the curvature of the potential. For the multi-bounce windows, we find that their range and complexity decrease as the relevant parameter decreases (and as the number of internal modes in the model increases). We use a modified collective coordinates method [in the spirit of recent works such as Phys. Rev.D 94 (2016) 085008] in order to capture the relevant phenomenology in a semi-analytical manner.

Could the primordial radiation be responsible for vanishing of topological defects?

We study the motion of domain walls in 1+1, 2+1 and briefly 3+1 d relativistic ϕ6 model with three equal vacua in the presence of radiation. We show that even small fluctuations can trigger a chain reaction leading to vanishing of the domain walls. Only one vacuum remains stable and domains containing other vacua vanish. We explain this phenomenon in terms of radiation pressure (both positive and negative). We construct an effective model which translates the fluctuations into additional term in the field theory potential. In case of two dimensional model we find a relation between the critical size of the bulk and amplitude of the perturbation.

Multi-kink collisions in the \u03d56 model

We study simultaneous collisions of two, three, and four kinks and antikinks of the ϕ6 model at the same spatial point. Unlike the ϕ4 kinks, the ϕ6 kinks are asymmetric and this enriches the variety of the collision scenarios. In our numerical simulations we observe both reflection and bound state formation depending on the number of kinks and on their spatial ordering in the initial configuration. We also analyze the extreme values of the energy densities and the field gradient observed during the collisions. Our results suggest that very high energy densities can be produced in multi-kink collisions in a controllable manner. Appearance of high energy density spots in multi-kink collisions can be important in various physical applications of the Klein-Gordon model.

Emerging Translational Variance: Vacuum Polarization Energy of the ϕ6 Kink

We propose an efficient method to compute the vacuum polarization energy of static field configurations that do not allow decomposition into symmetric and antisymmetric channels in one space dimension. In particular, we compute the vacuum polarization energy of the kink soliton in the ϕ6 model. We link the dependence of this energy on the position of the center of the soliton to the different masses of the quantum fluctuations at negative and positive spatial infinity.

Hyper-Riccati equations and integrable reductions permitting stationary solutions for complex hyperbolic field equations

We consider the hyper-Riccati differential equation y′2=P(y) where the degree of P is at least five. Unlike their degree-four counterparts, such equations are seldom exactly integrable. However, for appropriate parameter regimes, we can obtain perturbation solutions centered around solutions to lower-degree problems. We consider explicit perturbation solutions for the degree-five and six models. In the second half of the paper, we consider a class of stationary solutions for complex scalar fields. Such solutions are governed by hyper-Riccati differential equations of degree at least four. While the equation considered is sufficiently general to hold a number of models as special cases, particular attention is paid to examples related to (i) complex nonlinear self-interaction models (such as the φ4 model), and (ii) complex Rayleigh wave equations, where solutions may be expressed in terms of elliptic functions. For all examples considered, we provide a large number of exact stationary solutions of the form u(x,t)=e-ωtψ(x), including degenerate limits. In contrast to these models, the φ6 interaction model corresponds to a hyper-Riccati differential equation of degree six, and the qualitative behavior of solutions to this model depend greatly on the coefficient of the degree-six term in P(y). While this model is, in general, not integrable for ψ(x), we find many exactly solvable reductions. For this model, exact solutions are obtained when possible, and perturbation solutions are obtained for weak-coupling (which is an essential, physically relevant degenerate case). This weak coupling relates solutions of the φ4 model to those of the φ6 model. The mathematical results obtained are then related to the physics of these models. We are able to classify the physically significant solutions in terms of elliptic functions. We also demonstrate the existence of singular degenerate solutions. The specific class of solution obtained depends strongly on the spectrum ω, and for the φ6 model we are able to classify the existence of periodic solutions in terms of the spectrum and the sixth-order coupling parameter. Applications to the Goldstone boson and kink solutions for the φ6 model are discussed.

Higher order periodic solutions of coupled ϕ4 and ϕ6 models

We obtain several higher order periodic solutions of (i) a coupled symmetric ϕ4 model in an external field, (ii) a coupled asymmetric ϕ4 model, (iii) a coupled symmetric–asymmetric ϕ4 model and (iv) a coupled ϕ6 model in terms of Lamé polynomials and obtain the corresponding hyperbolic solutions in the appropriate limit. All these solutions are unusual in the sense that while they are the solutions of the coupled problems, they are not the solutions of the corresponding uncoupled problems. Possible physical applications of these solutions include periodic domain walls in magnetic and structural phase transitions as well as in field theory.

Radiation of scalar oscillons in 2 and 3 dimensions

The radiation loss of small-amplitude radially symmetric oscillons (long-living, spatially localized, time-dependent solutions) in two- and three-dimensional scalar field theories is computed analytically in the small-amplitude expansion. The amplitude of the radiation is beyond all orders in perturbation theory and it is determined using matched asymptotic series expansions and Borel summation. The general results are illustrated on the case of the two- and three-dimensional sine-Gordon theory and a two-dimensional ϕ6 model. The analytic predictions are found to be in good agreement with the results of numerical simulations of oscillons.

Domain wall and periodic solutions of a coupled ϕ6 model

Coupled triple well (ϕ6) one-dimensional potentials occur in both condensed matter physics and field theory. Here we provide a set of exact periodic solutions in terms of elliptic functions (domain wall arrays) and obtain single domain wall solutions in specific limits. Topological, nontopological (e.g., some pulselike solutions), as well as mixed domain walls are obtained. We relate these solutions to structural phase transitions in materials with polarization, shuffle modes, and strain. We calculate the energy and the asymptotic interaction between solitons for various solutions.

Dynamics of first-order phase transitions in the φ4– φ6 model caused by the parametric instability of the metastable vacuum

We suggest that the parametric resonance underlies the mechanism responsible for first-order phase transitions in non-dissipative systems. This conjecture is examined with the nonlinear Klein–Gordon equation describing a massive real scalar field with φ 4– φ 6 selfcoupling. The solutions for the unstable modes of the false vacuum fluctuations are constructed and the characteristic exponents are founded in both instability domains on the amplitude–wavevector chart. The growth of the vacuum oscillations, as well as the all stages of the subsequent phase transition, including the bubble formation and the appearance of a quasi-turbulent state of a new phase, were observed by us in the numerical experiment.

The Soliton-Like Solutions of ϕ6 and ϕ4 + ϕ3 Models with and Without Dissipation11The project supported by National Natural Science Foundation of China

Besides the ϕ4 theory, the ϕ6 model and the ϕ4 + ϕ3 model are also useful in physics. Some specific types of solutions of them can be obtained from the solutions of ϕ4 model. The important solutions previously known are the special examples of ours. The dissipation equations of these models are also studied. The results indicate that the friction effects spread the singular solutions with velocity v = 1 (Heaviside function) of the undamped equations such that they are nonsingular and their velocities are lower than one.