Hadronic temperature and black solitons

Hawking has shown that event-horizon-containing classical soliton (black-hole) solutions of Einstein's equation radiate all species of particles with a thermal spectrum, the temperature being defined in terms of surface gracity. For a spinless soliton, the temperature is inversely proportional to the radius of its event-horizon. Assuming that there exists a fundamental strongly-interacting (massive) spin-2 field satisfying an Einstein-like equation with a strong coupling parameter, we propose to identify temperature in hadronic physics with strong surface gravity effects. The existence of black-body solitonic solutions for such an equation may then explain the thermal spectrum in E_T observed in high-energy collisions.     

Optical soliton perturbation for Gerdjikov–Ivanov equation via two analytical techniques

This paper employs two integration procedures to obtain soliton solutions to the perturbed Gerdjikov–Ivanov equation. They are G′/G²–expansion method and the sine–cosine method. Bright, dark and singular solitons are revealed along with a few of the combo–soliton solutions. The existence criteria of these solitons are also given.     

Cusp and loop soliton solutions of short-wave models for the Camassa–Holm and Degasperis–Procesi equations

A simple method is developed for constructing the solutions of the short-wave model equations associated with the Camassa–Holm (CH) and Degasperis–Procesi (DP) shallow-water wave equations. Taking an appropriate scaling limit of the N-soliton solution of the CH equation, we obtain the N-cusp soliton solution for the CH short-wave model. The similar procedure also leads to the N-loop soliton solution for the DP short-wave model. We describe the property of the solutions. In particular, we derive the large-time asymptotics of the solutions as well as the formulas for the phase shift.     

Microscopic theory of substrate-induced gap effect on real AFM susceptibility in graphene

In this article, the novel (G ^′/G)-expansion method is successfully applied to construct the abundant travelling wave solutions to the KdV–mKdV equation with the aid of symbolic computation. This equation is one of the most popular equation in soliton physics and appear in many practical scenarios like thermal pulse, wave propagation of bound particle, etc. The method is reliable and useful, and gives more general exact travelling wave solutions than the existing methods. The solutions obtained are in the form of hyperbolic, trigonometric and rational functions including solitary, singular and periodic solutions which have many potential applications in physical science and engineering. Many of these solutions are new and some have already been constructed. Additionally, the constraint conditions, for the existence of the solutions are also listed.     

Dispersionless envelope lattice solitons on a D-dimensional nonlinear lattice

We show by using the real exponential approach that the d-dimensional discrete nonlinear Schrödinger equation has more general dispersionless envelope lattice soliton solutions than the known bright soliton and kink solutions. Depending on the values of the parameters, the new solutions can describe both bright and dark lattice solitons. Especially, we find novel “W”-like envelope lattice solitons.     

Vandermonde-like determinants’ representations of Darboux transformations and explicit solutions for the modified Kadomtsev-Petviashvili equation

Recently, the (2+1)-dimensional modified Kadomtsev-Petviashvili (mKP) equation was decomposed into two known (1+1)-dimensional soliton equations by Dai and Geng [H.H. Dai, X.G. Geng, J. Math. Phys. 41 (2000) 7501]. In the present paper, a systematic and simple method is proposed for constructing three kinds of explicit N-fold Darboux transformations and their Vandermonde-like determinants’ representations of the two known (1+1)-dimensional soliton equations based on their Lax pairs. As an application of the Darboux transformations, three explicit multi-soliton solutions of the two (1+1)-dimensional soliton equations are obtained; in particular six new explicit soliton solutions of the (2+1)-dimensional mKP equation are presented by using the decomposition. The explicit formulas of all the soliton solutions are also expressed by Vandermonde-like determinants which are remarkably compact and transparent.     

Codimension One Threshold Manifold for the Critical gKdV Equation

We construct the “threshold manifold” near the soliton for the mass critical gKdV equation, completing results obtained in Martel et al. (Acta Math 212:59–140, 2014, J Math Eur Soc 2015). In a neighborhood of the soliton, this C¹ manifold of codimension one separates solutions blowing up in finite time and solutions in the “exit regime”. On the manifold, solutions are global in time and converge locally to a soliton. In particular, the soliton behavior is strongly unstable by blowup.     

Conte Truncated Expansion and Applications

In the special Conte truncated expansion approach one obtains different solutions of the Prigogine–Lefever equation by use of various solutions of a type of Riccati equation, including the periodic soliton solutions and singular soliton solutions. In order to acquire conveniently the soliton solutions of the Boussinesq equation, a proper transformation is applied. Using the special Conte truncated expansion approach yields the known bell-shape solutions and some new soliton solutions like cot² × sec², tan² × c sec², tanh² × sech², etc. We also study the soliton solutions of the modified Burgers equation (MBE). Using leading term analysis, we find the exponent is a fraction, i.e., – . Therefore, the special Conte truncated expansion approach cannot be used directly. A transformation is first made to them another form of the MBE. Various soliton solutions of MBE are then presented, including the periodic solutions and singular soliton solutions.     

Some extensions on the soliton solutions for the Novikov equation with cubic nonlinearity

In this paper, by using the bifurcation method of dynamical systems, we derive the traveling wave solutions of the nonlinear equation UU_τyy ? U_yU_τy + U²U_τ + 3U_y = 0. Based on the relationship of the solutions between the Novikov equation and the nonlinear equation, we present the parametric representations of the smooth and nonsmooth soliton solutions for the Novikov equation with cubic nonlinearity. These solutions contain peaked soliton, smooth soliton, W-shaped soliton and periodic solutions. Our work extends some previous results.     

Topological solitons of resonant nonlinear Schödinger'sequation with dual-power law nonlinearity by G′/G-expansion technique

This paper considers the resonant nonlinear Schrödinger's equation with dual-power law nonlinearity. The G′/G-expansion method is applied to integrate this equation. The soliton solutions are thus obtained. Both constant coefficients as well as time-dependent coefficients are considered. The results for parabolic law nonlinearity fall out as a special case.     

Bilinearization of discrete soliton equations and singularity confinement

The singularity confinement method is applied to the systematic derivation of the bilinear equations for discrete soliton equations. Using the bilinear forms, the N-soliton and algebraic solutions of the discrete potential mKdV equation are constructed.     

一类非线性演化方程新的显式行波解

借助Mathematica软件,采用三角函数法和吴文俊消元法,获得了一类非线性演化方程u_tt+au_xx+bu+cu³=0的三组行波解,其中包括新的行波解、扭状孤波解和钟状孤波解.从而作为该方程的特例,如Duffing方,Klein-Gordon方程、Landau-Ginburg-Higgs方程和⁴方程等也都获得了相应的若干行波解.这种方法也适用于其他非线性方程. 关键词：     

Optical solitons in (1 + 1) and (2 + 1) dimensions

This paper addresses the nonlinear Schrödinger's equation that serves as the model to study the propagation of optical solitons through nonlinear optical fibers. The main focus of this paper is the aspect of integrability. There are a couple of integration tools that are employed to obtain the exact solutions to the model. Fan's F-expansion approach is applied to extract several forms of solutions to the model. This integration mechanism displays cnoidal waves, snoidal waves and several other solutions; needless to mention that these solutions, in the limiting case, leads to bright, dark and singular soliton solutions. The study then rolls over to the (2 + 1)-dimensions where, in addition, the semi-inverse variational principle is applied to extract a bright soliton solution, along with the necessary constraint conditions. There is also a display of several numerical simulations.     

Bright and dark soliton solutions of the (3 + 1)-dimensional generalized Kadomtsev–Petviashvili equation and generalized Benjamin equation

In this paper, we obtain the 1-soliton solutions of the (3?+?1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation and the generalized Benjamin equation. By using two solitary wave ansatz in terms of sech ^p and $\tanh^{p}$ functions, we obtain exact analytical bright and dark soliton solutions for the considered model. These solutions may be useful and desirable for explaining some nonlinear physical phenomena in genuinely nonlinear dynamical systems.     

Multisoliton solutions of the (2+1)-dimensional Harry Dym equation

The exact N-soliton solutions of the (2+1)-dimensional Harry Dym equation are constructed analytically. Different types of two-soliton interactions are singled out in the general N-soliton solution. The existence of inelastic soliton interaction and two-soliton resonances are shown.     

Incoherently coupled anti-dark soliton pairs

The solutions of the intensity envelopes of a novel type of incoherently coupled spatial soliton pair, named anti-dark soliton pair, are deduced by careful theoretical analyses. Simultaneously the forming conditions of the anti-dark soliton pair are derived in the paper, according to which an example of anti-dark soliton family that exists in cubic-quintic nonlinear media is given.     

Abundant exact solutions for the (3+1)-dimensional generalized nonlinear Schrödinger equation with variable coefficients

The (3+1)-dimensional generalized nonlinear Schrodinger equation with variable coefficients (3D-VcgNLSE) and optical lattice is investigated. Bright and dark soliton solutions are presented by two direct ansätz. Two similar solutions are obtained in terms of the elliptic and the second type of Painlevé transcendent functions. Furthermore, hyperbolic and trigonometric solutions are studied via the G′/G-expansion method. The dynamical behaviors are demonstrated in some 3D- and contour plots.     

Stationary solitons of the fifth order KdV-type. Equations and their stabilization

Exact stationary soliton solutions of the fifth order KdV type equation, u_t + αu^pu_x + βu_3x + γu_5x = 0, are obtained for any p (> 0) in case αβ > 0, Dβ > 0, βγ < 0 (where D is the soliton velocity), and it is shown that these solutions are unstable with respect to small perturbations in case p ≥ 5. Various properties of these solutions are discussed. In particular, it is shown that for any p these solitons are lower and narrower than the corresponding γ = 0 solitons. Finally, for p = 2 we obtain an exact stationary soliton solution even when D, α, β, γ are all > 0 and discuss its various properties.     

Optical solitons with complex Ginzburg–Landau equation for two nonlinear forms using F-expansion

This paper implements F-expansion scheme to obtain Jacobi’s elliptic function to complex Ginzburg–Landau equation with two nonlinear forms. In the limiting case of the modulus of ellipticity, bright and dark soliton solutions emerge.     

Topological defects with long-range interactions

We investigate a modified sine-Gordon equation which possesses soliton solutions with long-range interaction. We introduce a generalized version of the Ginzburg-Landau equation which supports long-range topological defects in D = 1 and D > 1. The interaction force between the defects decays so slowly that it is possible to enter the non-extensivity regime. These results can be applied to non-equilibrium systems, pattern formation and growth models.