Narrow Results

In this paper, we study the cubic–quintic nonlinear Helmholtz equation which enables a pulse propagating with Kerr-like and quintic properties further spatial dispersion. By noticing that the system is a nonintegrable one, we could get variety forms of solitary wave solutions by using a generalized $G^{\prime }∕G$-expansion method. In particular, we investigate four forms of the function solutions including soliton, bright soliton, singular soliton, periodic wave solutions. To perform this, the demonstrative pattern of the Helmholtz equation is made to show the probability and dependability of the protocol utilized in this research. The effect of the free variables on the behavior of the reached plots to a few achieved solutions for the nonlinear rational exact cases was also explored depending upon the nature of nonlinearities. The dynamical properties of the obtained solutions are analyzed and shown by plotting some density, two and three-dimensional images. We believe that our results would pave a way for future research generating optical memories based on the nonparaxial solitons.

In this paper, the investigation is conducted on a (2 + 1)-dimensional extended Boiti–Leon–Manna–Pempinelli equation for an incompressible fluid. Via the Riemann theta function, periodic-wave solutions are derived, and breather-wave solutions are constructed with the aid of the extended homoclinic test approach. Based on the polynomial expansion method, several traveling-wave solutions are derived. Besides, we observe that the amplitude of the breather keeps unchanged during the propagation and the traveling wave which is kink shaped propagates stably. Furthermore, we analyze the transition between the periodic-wave and soliton solutions, which implies that the periodic-wave solutions tend to the soliton solutions via a limiting procedure.

In this paper, the nonparaxial nonlinear Schrödinger (NNLS) equation by considering its integrability which enables the propagation of ultra-broad nonparaxial beams in a planar optical waveguide is studied. The plenty numbers of solitary wave solutions by using Hirota’s bilinear scheme are found, in addition, the bilinear transformation and also the related theorem for getting to the bilinear form of nonlinear system are considered. Two new simple approaches are implemented to recover periodic wave, bright soliton, singular, and singular soliton for this model. Because of the significance of the NNLS in modeling the propagation of solitons through an optical fiber, the recovered solitons are vital for describing and understanding a variety of fundamental physical processes. The effect of the free parameters on the behavior of acquired figures to a few obtained solutions by providing the feasibility and reliability of the used procedure was also discussed. For more physical illustration and knowledge of the physical characteristics of this equation, some important solutions are discussed graphically in the form of 2D and 3D plots by selecting suitable parameters.

In this research, the modified extended tanh-function (METF) and the extended Jacobi elliptic function expansion (EJEFE) techniques are used to investigate the generation and detection of soliton structures in the Hirota–Maccari (HM) model. Consequently, we obtain soliton solutions with advanced structures, including singular bright soliton, dark soliton, periodic waves, breather waves, periodic breather waves, and multiple bright and dark breather waves. In addition, a lump-type breather wave is also included in the presented solutions. Stability analysis of the obtained solutions is addressed by employing the Hamiltonian technique. 3D surfaces and 2D visuals of the outcomes are represented with the help of a computer application. These findings contribute to understanding nonlinear wave phenomena with potential applications in optics, fluid dynamics, and plasma physics.

The existence of solitary wave, kink wave and periodic wave solutions of a class of singular reaction–diffusion equations is obtained using some effective methods from the dynamical systems theory. Specially, for a class of nonlinear wave equations, fundamental properties of profiles of traveling wave solutions determined by some bounded orbits of the traveling wave systems are rigorously proved. Parametric conditions that guarantee the existence of the aforementioned solutions are derived and given explicitly.

In this paper, we consider a model which is a generalization of the nonlinear Schrödinger equation where the dispersive term was substituted by a nonlocal integral term with given kernel. The study on this model derives a planar dynamical system with two singular straight lines. On the basis of the investigation of the dynamical behavior and bifurcations of solutions of the planar dynamical system, we obtain all possible explicit exact parametric representations of solutions (including kink wave solutions, unbounded wave solutions, compactons, etc.) under different parameter conditions. The existence of bounded solutions of the planar dynamical system implies that there exist infinitely many breather solutions of this generalized nonlinear Schrödinger system.

In this paper, we consider two singular nonlinear planar dynamical systems created from the studies of one-dimensional bright and dark spatial solitons for one-dimensional beams in a nonlocal Kerr-like media. On the basis of the investigation of the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we obtain all possible explicit exact parametric representations of solutions (including solitary wave solutions, periodic wave solutions, peakon and periodic peakons, compacton solutions, etc.) under different parameter conditions.

The dynamical model of a nonlinear wave is governed by a partial differential equation which is a special case of the $b$-family equation. Its traveling system is a singular system with a singular straight line. On this line, there exist two degenerate nodes of the associated regular system. By using the method of dynamical systems and the theory of singular traveling wave systems, in this paper we show that, corresponding to global level curves, this wave equation has global periodic wave solutions and anti-solitary wave solutions. We obtain their exact representations. Specially, we discover some new phenomena. (i) Infinitely many periodic orbits of the traveling wave system pass through the singular straight line. (ii) Inside some homoclinic orbits of the traveling wave system there is not any singular point. (iii) There exist periodic wave bifurcation and double anti-solitary waves bifurcation.

Using the dynamical systems analysis and singular traveling wave theory developed by Li and Chen [2007] to the classical and modified Serre shallow water wave equations, it is shown that, in different regions of the parameter space, all possible bounded solutions (solitary wave solutions, kink wave solutions, peakons, pseudo-peakons and periodic peakons as well as compactons) can be obtained. More than 28 explicit and exact parametric representations are precisely derived. It is demonstrated that, more interestingly, the modified Serre equation has uncountably infinitely many smooth solitary wave solutions and uncountably infinitely many pseudo-peakon solutions. Moreover, it is found that, differing from the well-known peakon solution of the Camassa–Holm equation, the modified Serre equation has four new forms of peakon solutions.

For the generalized Serre–Green–Naghdi equations with surface tension, using the methodologies of dynamical systems and singular traveling wave theory developed by Li and Chen [2007] for their traveling wave systems, in different parameter conditions of the parameter space, all possible bounded solutions (solitary wave solutions, kink wave solutions, peakons, pseudo-peakons and periodic peakons as well as compactons) are obtained. More than 26 explicit exact parametric representations are given. It is interesting to find that this fully nonlinear water waves equation coexists with uncountably infinitely many smooth solitary wave solutions or infinitely many pseudo-peakon solutions with periodic solutions or compacton solutions. Differing from the well-known peakon solution of the Camassa–Holm equation, the generalized Serre–Green–Naghdi equations have four new forms of peakon solutions.

For the modified Camassa–Holm equation, by using the methodologies of dynamical systems and singular traveling wave theory developed by Li and Chen [2007] to its corresponding traveling wave system, under different parameter conditions, all possible exact explicit bounded solutions (solitary wave solutions, periodic wave solutions, peakon, periodic peakons, as well as compactons) are obtained. More than 23 explicit exact parametric representations of the above-mentioned traveling wave system are presented.

This paper studies traveling wave solutions of a generalized Sasa–Satsuma equation introduced in [Adem et al., 2020]. For the cases of $n=0,1,2$, under given parameter conditions, the bifurcations of traveling wave solutions in the parameter space are investigated for the corresponding traveling systems. All possible explicit exact parametric representations of various solutions are obtained.

In this paper, the methodologies of dynamical systems and singular traveling wave theory developed by [Li & Chen, 2007] are applied to find the solutions in the form of $u(x,t)=\varphi (x-ct)e^{i(\kappa x-\omega t)}$ for the Hirota-type peakon equation posed by [Anco & Mobasheramini, 2017; Anco et al., 2021]. For the function $\varphi (\xi )$ therein, under the parameter conditions of $\kappa =\frac{5}{3},$$c=\frac{807\omega }{100}$ or $c\ne \frac{807\omega }{100}$, the existence of some possible bounded solutions (solitary wave solutions, periodic wave solutions and periodic peakons, as well as compactons) is proved, with exact explicit parametric representations obtained, except for the peakon solution.

For the cubic Camassa–Holm type equation, by using the techniques from dynamical systems and singular traveling wave theory developed by Li and Chen [2007] to analyze its corresponding traveling wave system, it was found that under different parameter conditions, its bifurcation portraits exhibit all possible exact explicit bounded solutions (solitary wave solutions, periodic wave solutions, peakon as well as periodic peakons). A total of 19 explicit exact parametric representations of the traveling wave system of the Camassa–Holm type equation are presented.

For the generalized Serre–Green–Naghdi system with weak Coriolis effect and surface tension, by using the dynamical system methods and singular traveling wave theory developed by Li and Chen [2007] to its associate traveling wave system, under different parameter conditions, all possible bounded solutions (solitary wave solutions, periodic wave solutions, peakons, periodic peakons as well as compacton solution families) are obtained. Exact explicit parametric representations are given.

For the generalized Camassa–Holm–Degasperis–Procosi (CH–DP) type equation, by using the techniques from dynamical systems and singular traveling wave theory developed by Li and Chen [2007] to analyze its corresponding traveling wave systems, which depend on four parameters, it is found that under different parameter conditions its bifurcation portraits exhibit all possible exact explicit bounded solutions, such as solitary wave solutions, periodic wave solutions, peakon as well as periodic peakons. A total of 30 explicit exact parametric representations of the traveling wave system of the CH–DP type equation are presented.

This paper focuses on the persistence of solitary waves and periodic waves of a singularly perturbed generalized Drinfel’d–Sokolov system. Geometric singular perturbation theory is first employed to reduce the higher-dimensional system to the perturbed planar system. By perturbation analysis and Abelian integrals theory, we are then able to find some sufficient conditions about the wave speed to guarantee the existence of homoclinic orbits and periodic orbits, which indicates the existence of solitary waves and periodic waves. Furthermore, we find the lower and upper bounds of the limit wave speed.

For the optical soliton model in fifth-order weakly nonlocal nonlinear media, to find its exact explicit solutions, the corresponding traveling wave system is formulated as a planar dynamical system with a singular straight line. Then, by using techniques from dynamical systems and singular traveling wave theory developed by [Li & Chen, 2007] to analyze the planar system and find the corresponding phase portraits, the dynamical behavior of the amplitude component can be assessed. Under different parameter conditions, exact explicit solitary wave solutions, periodic wave solutions, kink, and anti-kink wave solutions, compacton solutions, as well as peakons and periodic peakons are found with precise formulations.

To find the exact explicit solution of the concatenation model, the corresponding differential system of the amplitude component is used, which is a planar dynamical system with a singular straight line. In this paper, by using the techniques from dynamical systems and singular traveling wave theory developed by Li and Chen [2007], its corresponding traveling wave system is solved and analyzed, obtaining the corresponding phase portraits and showing the dynamical behavior of the amplitude component. Under different parameter conditions, exact explicit solitary wave solutions, periodic wave solutions, kink and anti-kink wave solutions, compacton solutions, as well as peakons and periodic peakons, are found explicitly.

In this study, we present a fractal generalized fourth-order Boussinesq equation which can describe the shallow water waves with the non-smooth boundary (such as the fractal boundary). Aided by the semi-inverse method, we establish its variational principle, which is proved to have a strong minimum condition via the He–Weierstrass theorem. Then, two powerful approaches namely the variational method (VM) and energy balance theory (EBT) are utilized to search for the periodic wave solutions. As expected, the results obtained by the two methods are almost the same. Furthermore, the impact of the fractal orders on the periodic wave structure is illustrated via the 3D plot and 2D curve. The results of this paper are expected to provide a reference for the study of periodic wave theory in fractal space.