Nonlinear Klein-Gordon Models Research Articles

Fifth-order nonlinear Schrödinger equation as Routhian reduction of the nonlinear Klein–Gordon model

We consider the nonlinear Klein–Gordon model with polynomial nonlinearity involving odd and even powers. We elaborate a hybrid Hamiltonian and Lagrangian approach, which is generally referred to as Routh’s procedure, to describe the nonlinear modulation of wave packets assuming the wave field envelope to be a slow function of time and coordinate. An extended fifth-order nonlinear Schrödinger equation is obtained from the Routhian equation for a complex canonical variable that couples the wave field and its momentum. This canonical variable is expressed in terms of the amplitude of fundamental harmonic and its derivatives with respect to coordinate. The resulting fifth-order nonlinear Schrödinger equation is demonstrated to be a Hamiltonian system under a certain constraint imposed on its model parameters. When compared with other techniques used to derive high-order nonlinear Schrödinger models, our approach preserves the Hamiltonian structure of the governing equations for the fundamental harmonic, while the input of higher harmonics is taken into account by means of variational principle applied to the averaged Routhian. In this way, it represents the most general tool for the reduction of nonlinear wave equations to high-order envelope models for slowly modulated wave packets.

Hamiltonian form of extended cubic-quintic nonlinear Schrödinger equation in a nonlinear Klein-Gordon model.

We derive an extended cubic-quintic nonlinear Schrödinger equationwith Hamiltonian structure in a nonlinear Klein-Gordon model with cubic-quintic nonlinearity. We use the nonlinear dispersion relation to properly take into account the input of high-order nonlinear effects in the Hamiltonian perturbation approach to nonlinear modulation. We demonstrate that changing the balance between the cubic and quintic nonlinearities has a significant effect on the stability of unmodulated wave packets to long-wave modulations.

Abundant Bounded and Unbounded Solitary, Periodic, Rogue-Type Wave Solutions and Analysis of Parametric Effect on the Solutions to Nonlinear Klein–Gordon Model

This paper exploits the modified simple equation and dynamical system schemes to integrate the Klein–Gordon (KG) model amid quadratic nonlinearity arising in nonlinear optics, quantum theories, and solid state physics. By implementing the modified simple equation (MSE) technique, we develop some disguise adaptation of analytical solutions in terms of hyperbolic, exponential, and trigonometric functions with some special parameters. We apply the dynamical system to bifurcate the model and draw distinct phase portraits on unlike parametric constraints. Following each orbit of all phase portraits, we originate bounded and unbounded solitary, periodic, and periodic rogue-type wave solutions of the KG model. These two schemes extract widespread classes of solitary, periodic, and periodic rogue-type wave solutions for the KG model jointly due to restrictions on parameters. We also analyze the effect of parameters on the obtained wave solutions and discuss why and when it changes its nature. We illustrate some dynamical features of the acquired solutions via the 3D, 2D, and contour graphics.

Effect of RGUP on the nonlinear Klein-Gordon model with spontaneous symmetry breaking

In the framework of the relativistic generalized uncertainty principle (RGUP), we study the nonlinear Klein-Gordon field with ϕ4 self-interaction. This generalization comes from the quantum gravity theories that predict the existence of a minimal measurable length scale. The quantum gravity effects slightly modify the momentum and change the Lagrangian density of the field. This model, specifically in the context of spontaneous symmetry breaking, is so fruitful and leads us to many interesting phenomena. Due to the complexity of the consequent equations, the model is normally studied via perturbation mechanisms. However, here, we apply a generalized tanh-method which supposed that the field solutions should be sech-functions of segments of space and time. By this extended method, we find some more general solutions with respect to our previous work [1]. Several solutions can be found but we select only the normalizable and bounded solitary fields. The energy spectrum of each field solution is obtained in an analytical form and discussed. The modification parameter of the corresponding RGUP is estimated by using the rest energy of the mass of the Higgs boson.

Hamiltonian form of an Extended Nonlinear Schrödinger Equation for Modelling the Wave field in a System with Quadratic and Cubic Nonlinearities

We derive a Hamiltonian form of the fourth-order (extended) nonlinear Schrödinger equation (NLSE) in a nonlinear Klein–Gordon model with quadratic and cubic nonlinearities. This equation describes the propagation of the envelope of slowly modulated wave packets approximated by a superposition of the fundamental, second, and zeroth harmonics. Although extended NLSEs are not generally Hamiltonian PDEs, the equation derived here is a Hamiltonian PDE that preserves the Hamiltonian structure of the original nonlinear Klein–Gordon equation. This could be achieved by expressing the fundamental harmonic and its first derivative in symplectic form, with the second and zeroth harmonics calculated from the variational principle. We demonstrate that the non-Hamiltonian form of the extended NLSE under discussion can be retrieved by a simple transformation of variables.

Kink dynamics in a nonlinear beam model

In this paper, we study the single kink and the kink-antikink collisions of a nonlinear beam equation bearing a fourth-derivative term. We numerically explore some of the key characteristics of the single kink both in its standing wave and in its traveling wave form. A point of emphasis is the study of kink-antikink collisions, exploring the critical velocity for single-bounce (and separation) and infinite-bounce (where the kink and antikink trap each other) windows. The relevant phenomenology turns out to be dramatically different than that of the corresponding nonlinear Klein-Gordon (i.e., ϕ4) model. Our computations show that for small initial velocities, the kink and antikink reflect nearly elastically without colliding. For an intermediate interval of velocities, the two waves trap each other, while for large speeds a single inelastic collision between them takes place. Lastly, we briefly touch upon the use of collective coordinates (CC) method and their predictions of the relevant phenomenology. When one degree of freedom is used in the CC approach, the results match well the numerical ones for small values of initial velocity. However, for bigger values of initial velocity, it is inferred that more degrees of freedom need to be self-consistently included in order to capture the collision phenomenology.

Numerical investigation of fractional nonlinear sine-Gordon and Klein-Gordon models arising in relativistic quantum mechanics

This paper presents a method for the approximate solution of the time-fractional nonlinear sine-Gordon and the Klein-Gordon models described in Caputo sense and with the order 1 < γ < 2. This method discretizes the unknown solution in two main steps. First, the temporal discretization of the governing problems is obtained by means of the finite difference scheme. Second, the spatial terms are expanded using local radial basis functions, where each basis function is approximated by a weighted linear summation of function values. The spatial discretization achieved through the local radial basis functions and finite difference (LRBF-FD) is highly accurate, since in local collocation techniques we only consider the nodes located in the subdomain around the collocation node surrounding the local collocation point. In fact, the new approach avoids the ill-conditioning problem resulting from the adoption of the global collocation and leads to sparse systems. The numerical stability and convergence are examined and confirmed numerically to support the theoretical formulation. Numerical experiments assess the effectiveness and capability of the algorithm and demonstrate its good computational performance on irregular domains.

Decay of small odd solutions for long range Schrödinger and Hartree equations in one dimension

We consider the long time asymptotics of (not necessarily small) odd solutions to the nonlinear Schrödinger equation with semi-linear and nonlocal Hartree nonlinearities, in one dimension of space. We assume data in the energy space only, and we prove decay to zero in compact regions of space as time tends to infinity. We give three different results where decay holds: semilinear NLS, NLS with a suitable potential, and defocusing Hartree. The proof is based on the use of suitable virial identities, in the spirit of nonlinear Klein–Gordon models (Kowalczyk et al 2017 Lett. Math. Phys. 107 921–31), and covers scattering sub, critical and supercritical (long range) nonlinearities. No spectral assumptions on the NLS with potential are needed.

Planar and radial kinks in nonlinear Klein-Gordon models: Existence, stability, and dynamics

We consider effectively one-dimensional planar and radial kinks in two-dimensional nonlinear Klein-Gordon models and focus on the sine-Gordon model and the $\phi^4$ variants thereof. We adapt an adiabatic invariant formulation recently developed for nonlinear Schr{\"o}dinger equations, and we study the transverse stability of these kinks. This enables us to characterize one-dimensional planar kinks as solitonic filaments, whose stationary states and corresponding spectral stability can be characterized not only in the homogeneous case, but also in the presence of external potentials. Beyond that, the full nonlinear (transverse) dynamics of such filaments are described using the reduced, one-dimensional, adiabatic invariant formulation. For radial kinks, this approach confirms their azimuthal stability. It also predicts the possibility of creating stationary and stable ring-like kinks. In all cases we corroborate the results of our methodology with full numerics on the original sine-Gordon and $\phi^4$ models.

A compact fourth-order in space energy-preserving method for Riesz space-fractional nonlinear wave equations

In this work, we investigate numerically a nonlinear hyperbolic partial differential equation with space fractional derivatives of the Riesz type. The model under consideration generalizes various nonlinear wave equations, including the sine-Gordon and the nonlinear Klein–Gordon models. The system considered in this work is conservative when homogeneous Dirichlet boundary conditions are imposed. Motivated by this fact, we propose a finite-difference method based on fractional centered differences that is capable of preserving the discrete energy of the system. The method under consideration is a nonlinear implicit scheme which has various numerical properties. Among the most interesting numerical features, we show that the methodology is consistent of second order in time and fourth order in space. Moreover, we show that the technique is stable and convergent. Some numerical simulations show that the method is capable of preserving the energy of the discrete system. This characteristic of the technique is in obvious agreement with the properties of its continuous counterpart.

Behavior of gap solitons in anharmonic lattices.

Using the theory of bifurcation, we provide and find gap soliton dynamics in a nonlinear Klein-Gordon model with anharmonic, cubic, and quartic interactions immersed in a parametrized on-site substrate potential. The case of a deformable substrate potential allows theoretical adaptation of the model to various physical situations. Nonconvex interactions in lattice systems lead to a number of interesting phenomena that cannot be produced with linear coupling alone. By investigating the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we derive a variety of exotic solutions corresponding to the phase trajectories under different parameter conditions. Moreover, we demonstrate how and why traveling waves lose their smoothness and develop into solutions with compact support or breaking.

Kink topology control by high-frequency external forces in nonlinear Klein-Gordon models.

A method of averaging is applied to study the dynamics of a kink in the damped double sine-Gordon equation driven by both external (nonparametric) and parametric periodic forces at high frequencies. This theoretical approach leads to the study of a double sine-Gordon equation with an effective potential and an effective additive force. Direct numerical simulations show how the appearance of two connected π kinks and of an individual π kink can be controlled via the frequency. An anomalous negative mobility phenomenon is also predicted by theory and confirmed by simulations of the original equation.

Effects of parity-time symmetry in nonlinear Klein-Gordon models and their stationary kinks

In this work, we introduce some basic principles of PT-symmetric Klein-Gordon nonlinear field theories. By formulating a particular antisymmetric gain and loss profile, we illustrate that the stationary states of the model do not change. However, the stability critically depends on the gain and loss profile. For a symmetrically placed solitary wave (in either the continuum model or a discrete analog of the nonlinear Klein-Gordon type), there is no effect on the steady state spectrum. However, for asymmetrically placed solutions, there exists a measurable effect of which a perturbative mathematical characterization is offered. It is generally found that asymmetry towards the lossy side leads towards stability, while towards the gain side produces instability. Furthermore, a host of finite size effects, which disappear in the infinite domain limit, are illustrated in connection to the continuous spectrum of the problem.

The effects of horizontal singular straight line in a generalized nonlinear Klein–Gordon model equation

In this paper, we investigate bounded traveling waves of the generalized nonlinear Klein–Gordon model equations by using bifurcation theory of planar dynamical systems to study the effects of horizontal singular straight lines in nonlinear wave equations. Besides the well-known smooth traveling wave solutions and the non-smooth ones, four kinds of new bounded singular traveling wave solution are found for the first time. These singular traveling wave solutions are characterized by discontinuous second-order derivatives at some points, even though their first-order derivatives are continuous. Obviously, they are different from the singular traveling wave solutions such as compactons, cuspons, peakons. Their implicit expressions are also studied in this paper. These new interesting singular solutions, which are firstly founded, enrich the results on the traveling wave solutions of nonlinear equations. It is worth mentioning that the nonlinear equations with horizontal singular straight lines may have abundant and interesting new kinds of traveling wave solution.

Collective-Coordinate Analysis of Inhomogeneous Nonlinear Klein–Gordon Field Theory

Two different sets of collective-coordinate equations for solitary solutions of Nonlinear Klein-Gordon (NKG) model is introduced. The collective-coordinate equations are derived using different approaches for adding the inhomogeneities as exrernal potentials to the soliton equation of motion. Interaction of the NKG field with a local inhomogeneity like a delta function potential wall and also delta function potential well is investigated using the presented collective-coordinate equations and the results of two different models are compared. Most of the characters of the interaction are derived analytically. Analytical results are also compared with the results of numerical simulations.

Soliton–potential interaction in the nonlinear Klein–Gordon model

The interaction of solitons with external potentials in the nonlinear Klein–Gordon field theory is investigated using an improved model. The presented model has been constructed with a better approximation for adding the potential to the Lagrangian through the metric of background space–time. The results of the model are compared with another model and the differences are discussed.

Parameter Region for Existence of Non-classical Solitons

We study a pure nonlinear model of a chain formed by particles that are linked each other by an enharmonic type. This lattice nonlinear Klein–Gordon model is subsequently studied in its continuum version. We use the dynamical systems approach for analyzing the properties of the non-classical structures that support the model. Several non-classical structures like peakons, kink compactons and crodwon or bubble compactons are generated along the chain for the specific region of the parameter space. It is shown that the phase space trajectories are nonclassical curves and show unexpected behaviors. The first type of phase transition in the parametric space occurs when the number of centers and saddles changes while the main phase state parameter becomes critical.

Threshold field in nucleation process for an inhomogeneous deformable nonlinear Klein-Gordon system

We investigate the effect of local inhomogeneity on the nucleation process of kink-antikink pairs in the driven nonlinear Klein-Gordon model with the Remoissenet-Peyrard substrate potential, whose shape can be varied as a function of the shape parameter and which has the sine-Gordon shape as a particular case. From numerical integrations of the model equations under defined conditions, the configuration of the critical nucleus in the presence of a localized inhomogeneity is determined and shown to be affected by the shape parameter of the substrate potential. The expression of the depinning threshold field of kink-antikink pairs, which is the value of the applied field at which the process of the nucleation of kink-antikink pairs takes place, is also obtained. The dependence of this depinning threshold field on the shape parameter r shows that it strongly increases, for small intensity of the impurity potential when the shape of the substrate potential deviates from the sinusoidal one.

Bifurcation analysis of travelling wave solutions in the nonlinear Klein–Gordon model with anharmonic coupling

By using bifurcation method of phase portraits and theory of dynamical systems to the nonlinear Klein–Gordon model, in the range 0 ⩽ r < 1 , numbers of kink compactons, solitary wave, kink and anti-kink waves are given for each parameter condition. The parameter condition when the type of travelling wave changes is obtained.

Thermal nucleation of kink-antikink pairs in the presence of impurities: The case of a Remoissenet-Peyrard substrate potential

Thermal nucleation of kink-antikink pairs in a nonlinear Klein-Gordon (NKG) model with a Remoissenet-Peyrard (RP) substrate potential in the presence of impurities and coupled to an applied field is analyzed in the limits of moderate temperature and strong damping. Using the Kolmogorov method, the average velocity of particles of the lattice is calculated and its dependence on the intensity of impurities is discussed in connection with the deformability parameter or the shape of the RP substrate potential. Numerical values are carried out by making use of parameters of the hydrogen atom adsorbed in the tungsten and ruthenium substrates. We show that, for large values of the applied field, the presence of impurities in the system makes the nucleation process of kink-antikink pairs more favorable in the high-temperature regime while they contribute to make it less favorable in the low-temperature regime.