Kink Solutions Research Articles

Correction to: Breather, kink and rogue wave solutions of Sharma-Tasso-Olver-like equation

Correction to: Breather, kink and rogue wave solutions of Sharma-Tasso-Olver-like equation

Spontaneous emission from excited quantum kinks

Many kink solutions enjoy internal excitations, called shape modes. In some 1+1d scalar models, such as the ϕ4 double-well model, when a kink’s shape mode is excited twice it may decay to a ground state kink plus a meson. We analytically calculate the decay rates of both the twice-excited shape mode and also a coherent state corresponding to the classically excited shape mode. In the case of the ϕ4 model, we find that the latter agrees with the classical result of Manton and Merabet.

Study of solitary and kink waves, stability analysis, and fractional effect in magnetized plasma

The (3 + 1)-dimensional fractional modified Zakharov Kuznetsov (mZK) equation is one of the nonlinear models to indicate the impact of magnetic fields on weak ion-acoustic waves in plasma; made up of cool and hot electrons. The primary goal of the present study is to use the (m+G′G)-expansion technique to seek the solutions of mZK equation. The solutions are gained in the form of kink, dark, singular periodic and W-type soliton solutions. The influence of the fractional parameter on waveforms has also been examined by representing 2D and 3D graphs for distinct values of fractional-order β. Moreover, we utilize Hamiltonian system properties to confirm the stability of the solution. The (m+G′G)-expansion technique can also be used to examine the nonlinear evolution models being developed in various scientific and technological fields, such as mathematical physics and plasma physics. The soliton solutions attained by using the above technique have not been derived yet.

Novel superposed kinklike and pulselike solutions for several nonlocal nonlinear equations

We show that a number of nonlocal nonlinear equations, including the Ablowitz–Musslimani and Yang variant of the nonlocal nonlinear Schrödinger (NLS) equation, the nonlocal modified Korteweg de Vries (mKdV) equation, and the nonlocal Hirota equation, admit novel kinklike and pulselike superposed periodic solutions. Furthermore, we show that the nonlocal mKdV equation also admits the superposed (hyperbolic) kink–antikink solution. In addition, we show that while the nonlocal Ablowitz–Musslimani variant of the NLS admits complex parity-time reversal-invariant kink and pulse solutions, neither the local NLS nor the Yang variant of the nonlocal NLS admits such solutions. Finally, except for the Yang variant of the nonlocal NLS, we show that the other three nonlocal equations admit both the kink and pulse solutions in the same model.

OPE coefficients and the mass-gap from the integrable scattering description of 2D CFT’s

Many two-dimensional conformal field theories have an alternative integrable scattering description, which reproduces their spectrum of conformal weights. Taking as an example the case of the Lee-Yang nonunitary CFT and the 3-state Potts minimal model, we derive formulas, in terms of their integrable description, for the OPE coefficients of a certain specific primary operator and two identical but otherwise essentially arbitrary operators. As a side result we also obtain a novel formula for the mass-gap relation for the integrable massive deformation of the CFT. These results are obtained through expressing the first nontrivial coefficient in the UV expansion of the energy in terms of the integrable CFT data, i.e. the kink and anti-kink TBA solutions.

Impacts of nonlinearity and wave dispersion parameters on the soliton pulses of the (2+1)-dimensional Kundu–Mukherjee–Naskar equation

In this study, we explain the impacts of nonlinearity and wave dispersion parameters on the soliton pulses of the (2+1)-dimensional Kundu–Mukherjee–Naskar equation (KMNE). In this regard, some new optical solitons are received via the unified method to the aforesaid equation to explain such impacts of the soliton pulses. The presenting optical solitons are expressed by the dark, periodic, bell, bright, kink, and singular soliton solutions. Taking into account the two impacts help stabilize the soliton pulses during their propagation by generating new dynamics depending upon the nonlinearity and the wave dispersion parameters of the studied equation. All the characteristics of the soliton pulses are exhibited graphically. It is found from the graphical outputs that the soliton profiles are decreasing and increasing with the increase of nonlinearity and dispersion parameters, respectively. The outcomes reveal that the soliton pulses are balanced due to the influences of nonlinearity and wave dispersion parameters of the aforementioned equation. It is mentioned that the impact of wave dispersion and nonlinearity parameters on the soliton pulses has not been discussed in the past. Therefore, the applied method permits the explanation of the various wave dynamics by analyzing the attained soliton solutions in nonlinear optical fibers systems, which can be used for further studies.

Kinks in higher-order polynomial models

We consider a family of field-theoretic models with a real scalar field in (1+1)-dimensional space–time. The field dynamics in each model is determined by a polynomial potential with two degenerate minima. We obtain exact general formulas for kink solutions with power-law asymptotic behavior. We also write out formulas for the asymptotics of all found kinks. In addition, we analyze some other properties of the obtained kinks: stability potentials, zero modes, positions of the centers of mass.

Existence of traveling wave solutions for a generalized Burgers–Fisher equation with weak convection

The existence of traveling wave solutions for a generalized Burgers–Fisher equation with weak convection term is investigated in current paper. The corresponding traveling wave equation is converted to a regularly perturbed Hamiltonian system by rescaling of the wave speed. Then by using Melnikov’s method, we show that the generalized Burgers–Fisher equation contains kink and anti-kink wave solutions with small wave speeds, and the wave speed selection principle is presented as well. By analyzing the ratio of Abelian integrals, we also prove that the generalized Burgers–Fisher equation admits periodic wave solutions whose rescaled limit wave speed is monotonically increasing. Moreover, the upper and lower bounds of the limit wave speed are given, and the relationship between wave speed and wavelength is established. Numerical simulations are performed to illustrate our theoretical analysis.

Variety interaction between [formula omitted]-lump and [formula omitted]-kink solutions for the (3+1)-D Burger system by bilinear analysis

In this paper, we investigate the (3+1)-dimensional Burger system which is employed in soliton theory and generated by considering the Hirota bilinear equation. We conclude some novel analytical solutions, including 2-lump-type, interaction between 2-lump and one kink, two lump and two kink of type I, two lump and two kink of type II, two lump and one periodic, two lump and kink-periodic, and two lump and periodic–periodic wave solutions for the considered system by symbolic estimations. The main ingredients for this scheme are to recover the Hirota trilinear forms and their generalized equivalences. Then we apply explicit numerical methods, most of which are recently introduced by many scholars, to reproduce the analytical solutions. The test results show that the best algorithms, especially the Hirota bilinear, are very efficient and severely outperform the other methods.

Kink solutions with power law tails

We present a comprehensive review about the various facets of kink solutions with a power law tail, which have received considerable attention during the last few years. This area of research is in its early stages; although several aspects have become clear by now, there are a number of issues which have only been partially understood or not understood at all. We first discuss the aspects which are reasonably well known and then address in some detail the issues which are only partially or not understood at all. We present a wide class of higher (than sixth) order field theory models admitting implicit kink as well as mirror kink solutions where the two tails facing each other have a power law or a power-tower type fall off, whereas the other two ends not facing each other could have either an exponential or a power law tail. The models admitting implicit kink solutions where the two ends facing each other have an exponential tail while the other two ends have a power law tail are also discussed. Moreover, we present several field theory models which admit explicit kink solutions with a power law fall off; we note that in all these polynomial models while the potential V(ϕ) is continuous, its derivative is discontinuous. We also discuss one of the most important and only partially understood issues of the kink–kink and the kink–antikink forces in case the tails facing each other have a power law fall off. Finally, we briefly discuss the kink–antikink collisions at finite velocity and present some open questions.

Solitary, kink and periodic wave solutions of the (3+1)-dimensional Hirota–Satsuma–Ito-like equation

In this article, we consider new solutions to the (3+1)-dimensional Hirota–Satsuma–Ito-like (HSIl) equation by using the exp-function and its expansion methods. This model describes the unidirectional propagation of shallow water waves. The is rewritten as an ordinary differential equation by the traveling wave transformation. Some exact traveling wave solutions of this equation are obtained based on the homogeneous balance method, which includes solitons, kinks, periodic wave solutions, etc. The derived results are innovative and have important applications in the current field of mathematical physics research. To further investigate the properties of these solutions, the dynamical behaviors of these solutions are analyzed. The results show that the scheme is completely powerful and effective.

Dynamical Analysis of Nonlocalized Wave Solutions of 2 + 1 -Dimensional CBS and RLW Equation with the Impact of Fractionality and Free Parameters

This study retrieves some new exact solutions to the 2 + 1 -dimensional Calogero-Bogoyavlenskii-Schilf (CSB) equation and regularized long wave (RLW) equation in the context of nonlinear traveling wave phenomena. In this regard, the advanced exp − φ ξ -expansion method is imposed to the 2 + 1 -dimensional CBS and RLW equation, and consequently, rogue, kink, singular kink, periodic, singular, and multiple soliton solutions are exhibited in terms of trigonometric, hyperbolic, and rational function solutions. To enucleate the underlying nonlocalized traveling wave features, accomplished exact solutions are established by making their dynamic behavior of the exact solutions exhibited in three-dimensional (3D) and two-dimensional (2D) combined chart with the help of computational software Maple 18. All of our accomplished solutions are claimed to be new in the sense of conformable derivative, chosen a unique fractional type wave transformation, dynamical behavior of fractionally and free variable, and the imposed method on our preferred equations.

RETRACTED: Lie symmetry analysis, optimal system and exact solutions for variable-coefficients (2 + 1)-dimensional dissipative long-wave system

In this article, the symmetry, optimal system, exact solutions and conservation laws of (2+1)-dimensional dissipative long-wave system with variable-coefficients (vcDLW) are investigated. These are important for mathematicians and physicists to solve problems about shallow water waves. The infinitesimal generators of independent variables are solved using Lie symmetry analysis method firstly. Based on the linear combination of vector fields, the representative elements of one-dimensional subalgebras optimal system are introduced by Olver's method. Some new exact solutions of the (2+1)-dimensional vcDLW are obtained by using Riccati equation method and tanh expansion method, such as kink solutions and soliton solutions. It is shown that this system has different solutions for different reductions. These solutions are important for describing a number of physical phenomena. The difference in this article is that the coefficients of some terms are functions. The unique feature of this article is that we can obtain many different types of equations when the coefficient functions change. This makes the (2+1)-dimensional vcDLW describe more general physical phenomena. In addition, the dynamical behaviour of the exact solutions of the (2+1)-dimensional vcDLW is analysed. And the evolution of several types of solutions is studied depending on the change in time. This process can be described better with the help of figures. Finally, The conservation laws of the (2+1)-dimensional vcDLW are obtained using the new conservation theorem with two modificatory rules.

A variety of new soliton structures and various dynamical behaviors of a discrete electrical lattice with nonlinear dispersion via variety of analytical architectures

Nonlinear electrical lattices are powerful experimental tool for studying nonlinear dispersive media and creating opportunities for the realistic modeling of electrical solitons. In this work, we adopt the generalized auxiliary equation methods, the first integral method, and the Möbius transformation approach to extract a variety of traveling and solitary wave solutions of the nonlinear Salerno equation describing the nonlinear discrete electrical lattice with nonlinear dispersion. The kink, antikink, dark, bright, peakons, antipeakons, and periodic wave solutions are all derived. The existence criteria of solitons are described. It has been shown that for the selective values of arbitrary constants in an auxiliary equation, the generalized auxiliary equation method II is reduced to the new ‐model expansion method as well as the new extended auxiliary equation method. The impact of free parameters on the obtained solutions is investigated and graphically depicted using physical descriptions. The acquired results are important for the validity of numerical and experimental results and further understanding of the wave propagation in the nonlinear discrete electrical lattice. In addition, the stability analysis of the obtained solitary wave solutions is studied. Furthermore, using the maple software, all derived solutions were checked by re‐entering them into the considered equation.

The analysis of solitonic, supernonlinear, periodic, quasiperiodic, bifurcation and chaotic patterns of perturbed Gerdjikov–Ivanov model with full nonlinearity

In this manuscript, the perturbed Gerdjikov–Ivanov equation is considered for the investigating model. It is studied to govern the dynamics of soliton propagation through optical fibers, metamaterials or PCF. The observed model is subjected to an extended simple equation technique, which disclosed an abundant of exact solutions including trigonometric function, singular, kink, peakon and compacton solutions. These solutions are made available with their essential conditions, which guarantee the persistence of such optical solitons and portraits using appropriate physical parameters in 3D, 2D and density plots. After that, the traveling wave transformation is used to turn the mth-order nonlinear perturbed Gerdjikov–Ivanov equation into a planar dynamical system. The dynamical and chaotic behaviors of the considered equation are also discussed. The qualitative analysis of the dynamical system and the chaotic behaviors of the perturbed system are investigated using the theory of plane dynamic systems. In addition, we utilize the RK4 method to discover patterns in the dynamical system that seem to be super nonlinear, periodic, and quasiperiodic. The reported results are new and have not been investigated. They can be used in the explanation of the physical phenomena modeled and will give information about the long-term dynamic behavior. Numerical simulations reveal that by changing the frequency and amplitude parameters have an impact on the dynamic behaviors of the system. It is established that the extended simple equation method and dynamical observations offer further influential mathematical tools for constructing exact solutions and their qualitative analysis in NLEEs in mathematical physics.

Bifurcation of some new traveling wave solutions for the time–space M- fractional MEW equation via three altered methods

In this work,1/G′, modified G′/G2 and new extended direct algebraic methods are proposed to construct the novel exact traveling wave solutions in the form of trigonometric, hyperbolic and exponential functions of the time-fractional modified equal-width (MEW) equation in the sense of M- truncated fractional derivative. These methods contribute a variety of exact solutions in terms of the hyperbolic, trigonometric and rational functions to the scientific literature. The obtained solutions are verified for aforesaid equation through symbolic soft computations. To promote the essential propagated features, some investigated solutions are exhibited in the form of 2D and 3D graphics by passing on the precise values to the parameters under the constrain conditions. Further, the dynamical behavior is investigated. Based on the bifurcation constrains on the system’s parameters, we constructed also some new wave solution which are assorted into solitary, kink, periodic, and super periodic wave solutions. The influence of the included parameters on the solution is clarified. Moreover, we guarantee that all the solutions are new and an excellent contribution in the existing literature of solitary wave theory.

The Weierstrass and Jacobi elliptic solutions along with multiwave, homoclinic breather, kink-periodic-cross rational and other solitary wave solutions to Fornberg Whitham equation

We investigate multiwave, homoclinic breather, M-shaped, Kink cross-rational, periodic cross kink solutions for the nonlinear generalized Fornberg–Whitham (gFW)-equation by using appropriate polynomial function schemes. We also obtain some interactions for governing model including; interaction of M-shaped soliton with single exp-function, interactional solution with double exp-functions. In addition, we obtain dark, periodic soliton, Weierstrass and Jacobi elliptic, bell and kink type solutions for gFW-equation via newly developed sub-ODE method.

Orbital stability and instability of periodic wave solutions for the $$\phi ^4$$-model

In this work we find explicit periodic wave solutions for the classical $$\phi ^4$$ -model, and study their corresponding orbital stability/instability in the energy space. In particular, for this model we find at least four different branches of spatially-periodic wave solutions, which can be written in terms of Jacobi elliptic functions. Two of these branches correspond to superluminal waves (speeds larger than the speed of light), the third-one corresponds to sub-luminal waves and the remaining one corresponds to stationary complex-valued waves. In this work we prove the orbital instability of real-valued sub-luminal traveling waves. Furthermore, we prove that under some additional hypothesis, complex-valued stationary waves as well as the real-valued zero-speed sub-luminal wave are all stable. This latter case is related (in some sense) to the classical Kink solution of the $$\phi ^4$$ -model.

Breather, kink and rogue wave solutions of Sharma-Tasso-Olver-like equation

Breather, kink and rogue wave solutions of Sharma-Tasso-Olver-like equation

Superposed hyperbolic kink and pulse solutions of coupled ϕ4, NLS and mKdV equations

In this paper, we obtain novel solutions of a coupled [Formula: see text], a coupled nonlinear Schrödinger equation and a coupled modified Korteweg de Vries equation which can be re-expressed as a linear superposition of either the sum or the difference of two hyperbolic pulse solutions or the sum of either a two-kink or a kink and an antikink solution. These results demonstrate that the notion of superposed solutions extends to coupled nonlinear equations as well.