Kink Solutions Research Articles

The nonlocal potential transformation method and solitary wave solutions for higher dimensions in shallow water waves

In this paper, the integrable (4 + 1)-dimensional Fokas equation is investigated. Exploiting a set of non-singular local multipliers, we present a set of local conservation laws for the equation. The nonlocally related partial differential equation (PDE) systems are found. Nine nonlocally related systems are discussed reveal thirty five interesting closed form solutions of the equation. These solutions contain different types of wave solutions, double soliton, multi-solitons, kink and periodic wave solutions. Some of the resulting solutions are graphically illustrated. Furthermore, we apply the sine-cosine method to find other traveling wave solutions for this equation.

Lie symmetries and similarity solutions for a family of 1+1 fifth-order partial differential equations

We apply the theory of infinitesimal transformations for the study of a family of 1+1 fifth-order partial differential equations which have been proposed before for the description of multiple kink solutions. In this analysis we perform a complete classification of the Lie symmetries and of the one-dimensional optimal system. The results are applied for the derivation of similarity solutions and in particular we find travel-wave and scaling solutions. We show that the kink-solution of these equations can be recovered by the use of the Lie symmetries, while new solutions are also derived.

Nonlinear oscillations of topological structures in the sine-Gordon systems

The nonlinear effect of the energy localization on topological inhomogeneities is investigated in the sine-Gordon systems. The regimes of nonlinear oscillations of nonequilibrium configurations of domain walls in the quasi-one-dimensional ferromagnet are described in terms of kink and breather solutions of the sine-Gordon equation. The conditions of the energy localization, i.e., the formation of breather excitations on these topological inhomogeneities, are found for the initial configurations of the dilated double kink structures. The results are obtained in the framework of the Schrödinger-type equation of the direct scattering problem associated with the sine-Gordon equation. It is shown that the final state of the evolution of the nonequilibrium topological spin structure represents the multi-frequency precessing domain wall in the ferromagnet, which radiates the continuous spectrum waves.

Asymptotics for 1D Klein-Gordon Equations with Variable Coefficient Quadratic Nonlinearities

We initiate the study of the asymptotic behavior of small solutions to one-dimensional Klein-Gordon equations with variable coefficient quadratic nonlinearities. The main discovery in this work is a striking resonant interaction between specific spatial frequencies of the variable coefficient and the temporal oscillations of the solutions. In the resonant case a novel type of modified scattering behavior occurs that exhibits a logarithmic slow-down of the decay rate along certain rays. In the non-resonant case we introduce a new variable coefficient quadratic normal form and establish sharp decay estimates and asymptotics in the presence of a critically dispersing constant coefficient cubic nonlinearity. The Klein-Gordon models considered in this paper are motivated by the study of the asymptotic stability of kink solutions to classical nonlinear scalar field equations on the real line.

Abundant interaction between lump and k-kink, periodic and other analytical solutions for the (3+1)-D Burger system by bilinear analysis

In this paper, we study the (3+1)-dimensional Burger system which is considered in soliton theory and generated by considering the Hirota bilinear operators. The bilinear frame to the Burger system by using the multi-dimensional Bell polynomials is constructed. Also, based on the binary Bäcklund transformations, the generalized Bell polynomials are written. We retrieve some novel exact analytical solutions, containing interaction between lump and two kink wave solutions, interaction between lump and periodic wave solutions, interaction between stripe and periodic solutions, breather wave solutions, cross-kink wave solutions, interaction between kink and periodic wave solutions, multi-wave solutions, and finally solitary wave solutions for the (3+1)-dimensional Burger system by Maple symbolic computations. The required conditions of the analyticity and positivity of the solutions can be easily achieved by taking special choices of the involved parameters. The main ingredients for this scheme are to recover the Hirota trilinear forms and their generalized equivalences.

Interaction between kinks and antikinks with double long-range tails

We explore a class of ϕ4n models with kink and antikink solutions that have long-range tails on both sides, specializing to the cases with n=2 and n=3. A recently developed method of an accelerating kink ansatz is used to estimate the force between the kink and the antikink. We use state-of-the-art numerical methods to initialize the system in a kink-antikink configuration with a finite initial velocity and to evolve the system according to the equations of motion. Among these methods, we propose a computationally efficient way to initialize the velocity field of the system. Interestingly, we discover that, for this class of models, ϕ4n with n>1, the kink-antikink annihilation behaves differently from the archetypal ϕ4 model or even the kinks with one long-range tail because there is neither long-lived bion formation nor resonance windows and the critical velocity is ultrarelativistic.

Bifurcations and Exact Traveling Wave Solutions in Two Nonlinear Wave Systems

This paper studies two nonlinear wave models. From the dynamical systems approach and using the singular traveling wave theory developed by [Li & Chen, 2007], all possible bounded solutions (solitary wave solutions, periodic wave solutions, as well as kink and anti-kink wave solutions) are obtained under different parameter conditions. More than 27 explicit exact parametric representations are derived. The new results complete the recent studies of [Rogers et al., 2020; Zayeda et al., 2020].

Bright, dark, kink, singular and periodic soliton solutions of Lakshmanan–Porsezian–Daniel model by generalized projective Riccati equations method

In this article, the soliton solutions of the nonlinear Lakshmanan–Porsezian–Daniel (LPD) model are extracted, using generalized projective Riccati equations method. The proposed model is discussed for Kerr law, parabolic law and anti-cubic law of nonlinearity. As a result, some new solitary wave solutions are obtained in a unified way. These solitary wave solutions include kink-shaped, bell-shaped, singular solitons and periodic solutions. The constraint conditions for the validity of the constructed solutions are also provided. In order to understand the physical dynamics of the LPD model, some obtained solitons are presented graphically.

Thirty traveling wave solutions to the systems of ion sound and Langmuir waves

Nine kinds of general solutions of certain popular first-order ordinary differential equation are obtained by direct calculations. Compared with complete discrimination system for polynomials, our classifications of solutions straightly depend on the coefficients of the ordinary differential equation and hence are easier to be employed. According to the nine kinds of solutions, we establish thirty traveling wave solutions to the coupled systems of ion sound and Langmuir waves, including rational function solution, exponential function solution, trigonometric function solutions, hyperbolic function solutions and Jacobi elliptic function solutions. To the best of our knowledge, many of them are new. Solutions discussed in two recent articles are showed to be equivalent to some special cases of our thirty traveling wave solutions. Our solutions include the kink and anti-kink soliton solutions, dark and bright soliton solutions as well as periodic soliton solutions and solitary wave solutions. We believe that these solutions will be of great use to researchers concerning with nonlinear physical phenomena. In addition, the famous (1+1)-dimensional dispersive long wave equations is taken to illustrate the applicability of these thirty solutions.

Interaction among lump, periodic, and kink solutions with dynamical analysis to the conformable time-fractional Phi-four equation

This study’s prime goal is to explore new and general analytic solutions to the conformable time-fractional Phi-four equation by utilizing the advanced exp ( − φ ( ξ ) ) -expansion approach with the conformable derivative principle’s help. This approach is essentially a modified particular case of the generalized form of exp ( − φ ( ξ ) ) -expansion method. Concerning hyperbolic and trigonometric function solutions, an enormous class of modern precise moving wave solutions is revealed. Some interaction solutions obtained, such as lump-periodic and lump-periodic-kink structure and periodic wave, are seen by showing their three-dimensional (3D) and two-dimensional (2D) combination plots with the aid of computational software Maple. We have portrayed the figures of the estimated solutions to understand the physical phenomena.

Kink and anti-kink wave solutions for the generalized KdV equation with Fisher-type nonlinearity

This paper proposes a new dispersion-convection-reaction model, which is called the gKdV-Fisher equation, to obtain the travelling wave solutions by using the Riccati equation method. The proposed equation is a third-order dispersive partial differential equation combining the purely nonlinear convective term with the purely nonlinear reactive term. The obtained global and blow-up solutions, which might be used in the further numerical and analytical analyses of such models, are illustrated with suitable parameters.

Revisit on two-dimensional self-gravitating kinks: superpotential formalism and linear stability

Self-gravitating kink solutions of a two-dimensional dilaton gravity are revisited in this work. Analytical kink solutions are derived from a concise superpotential formalism of the dynamical equations. A general analysis on the linear stability is conducted for an arbitrary static solution of the model. After gauge fixing, a Schrödinger-like equation with factorizable Hamiltonian operator is obtained, which ensures the linear stability of the solution.

A minimal nonlinearity logarithmic potential: Kinks with super-exponential profiles

We study a [Formula: see text]-dimensional field theory based on the [Formula: see text] potential which represents minimal nonlinearity in the context of phase transitions. There are three degenerate minima at [Formula: see text] and [Formula: see text]. There are novel, asymmetric kink solutions of the form [Formula: see text] connecting the minima at [Formula: see text] and [Formula: see text]. The domains with [Formula: see text] repel the linear excitations, the waves (e.g., phonons). Topology restricts the domain sequences and therefore the ordering of the domain walls. Collisions between domain walls are rich for properties such as transmission of kinks and particle conversion, etc. For illustrative purposes we provide a comparison of these results with the [Formula: see text] model and its half-kink solution, which has an exponential tail in contrast to the super-exponential tail for the [Formula: see text] potential. Finally, we place the results in the context of other logarithmic models.

Diverse solitary and Jacobian solutions in a continually laminated fluid with respect to shear flows through the Ostrovsky equation

In this paper, the generalized Jacobi elliptical functional (JEF) and modified Khater (MK) methods are employed to find the soliton, breather, kink, periodic kink, and lump wave solutions of the Ostrovsky equation. This model is considered as a mathematical modification model of the Korteweg-de Vries (KdV) equation with respect to the effects of background rotation. The solitary solutions of the well-known mathematical model (KdV equation) usually decay and are replaced by radiating inertia gravity waves. The obtained solitary solutions emerge the localized wave packet as a persistent and dominant feature. Many distinct solutions are obtained through the employed computational schemes. Moreover, some solutions are sketched in 2D, 3D, and contour plots. The effective and powerful of the two used computational schemes are tested. Furthermore, the accuracy of the obtained solutions is examined through a comparison between them and that had been obtained in previously published research.

Abundant lump-type solutions for the extended (3+1)-dimensional Jimbo–Miwa equation

In this work, we research the mixed lump–kink solutions and their dynamic properties of the extended (3+1)-dimensional Jimbo–Miwa equation. Mixed lump–kink solutions are triggered by the interaction between lump soliton and exponential function, the interaction between lump soliton and hyperbolic function, and the interaction between lump soliton and periodic wave. Their dynamics properties are shown by means of many graphics and corresponding density plots.

Reaction–diffusion–convection equations in two spatial dimensions; continuous and discrete dynamics

In this paper, we investigate some two-dimensional (with respect to spatial independent variables) reaction–diffusion–convection equations with various nonlinear (reaction) terms. Using Hirota bilinear formalism with a free auxiliary function, we obtain kink solutions and many spatio-temporal discretizations having birational form.

Dynamical Behavior of Traveling Wave Solutions for a (2+1)-Dimensional Bogoyavlenskii Coupled System

In this paper, we applied some computational tools, namely the modified extended tanh method via a Riccati equation, the general Exp $$_{a}$$ -function method and the bifurcation methods to study a nonlinear (2+1)-dimensional Bogoyavlenskii coupled system in thin-film ferroelectric medium to construct exact traveling wave solutions. By applying a classical wave transformation we obtained an ordinary differential equations. As a result, some new traveling wave solutions are obtained including hyperbolic, trigonometric, exponential functions and rational forms. If the parameters take specific values, then the periodic wave, solitary waves, kink and anti-kink wave solutions are derived from the traveling waves. Also, we draw 2D and 3D graphics of exact solutions for the special cases of these nonlinear equations by the help of programming language Maple.

Bilinear Bäcklund transformation, kink and breather-wave solutions for a generalized (2+1)-dimensional Hirota–Satsuma–Ito equation in fluid mechanics

Fluid mechanics studies are applied in the turbines, airplanes, ships, rivers, windmills, pipes, etc. Under investigation is a generalized (2+1)-dimensional Hirota–Satsuma–Ito equation in fluid mechanics. Via the Hirota bilinear method, bilinear Backlund transformation is obtained, based on which the kink solutions are constructed. Breather-wave solutions are constructed via the extended homoclinic test approach. When the periods of the breather-wave solutions tend to infinity, the lump solutions are derived. It is found that the amplitudes and shapes of the breather waves keep unchanged during the propagation. Effects of the coefficients in the equation on the amplitudes and velocities for the breather waves are analyzed graphically.

Kink solutions in a generalized scalar φ 4 G field model

We study a scalar field model in a two dimensional space-time with a generalized potential which has four minima, obtaining novel kink solutions with well defined properties although the potential is non-analytical at the origin. The model contains a control parameter δ that breaks the degeneracy of the potential minima, giving rise to two different phases for the system. The δ < 0 phases do not possess solitary wave solutions. At the transition point δ = 0 all the potential minima are degenerate and three different kink solutions result. As the transition to the δ > 0 phase takes place, the minima of the potential are no longer degenerate and a unique kink ϕ δ solution is produced. Remarkably, this kink is a coherent structure that results from the merge of three kinks that can be identified with those observed at the transition point. To support the interpretation of ϕ δ as a bound state of three kinks, we calculate the force between the kink-kink pair components of ϕ δ , obtaining an expression that has both exponentially repulsive and constant attractive contributions that yields an equilibrium configuration, explaining the formation of the ϕ δ multi-kink state. We further investigate kink properties including their stability guaranteed by the positive defined spectrum of small fluctuations around the kink configurations. The findings of our work together with a semiclassical WKB quantization, including the one loop mass renormalization, enable computing quantum corrections to the kink masses. The general results could be relevant to the development of effective theories for non-equilibrium steady states and for the understanding of the formation of coherent structures.

Generation of wave packets and breathers by oscillating kinks in the sine-Gordon system

Evolution of the nonequilibrium inhomogeneities and topological defects is studied in terms of complex kink solutions of the sine–Gordon equation. The weakly damped oscillation of the sine–Gordon kink, named as the kink quasimode, is described explicitly. It is shown that the oscillatory kink behavior and the wave packet generation depend significantly on the initial nonequilibrium kink profile. In order to specify conditions of the generation of wobbling kinks with a multibreather structure we reformulate the direct scattering problem associated with the sine-Gordon equation as the spectral problem of the Schrödinger operator. We obtain the dependence of the radiation energy, which is emitted during formation of the multi-frequency wobbling kink, on the effective dimension of its initial profile.