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In this paper, we are concerned with a one-parameter family of peakon equations with cubic nonlinearity parametrized by a parameter usually denoted by the letter b. This family is called the " b -Novikov" since it reduces to the integrable Novikov equation in the case b = 3. By extending the corresponding linearized operator defined on functions in H 1 (R) to one defined on weaker functions on L 2 (R) , we prove spectral and linear instability on L 2 (R) of peakons in the b -Novikov equations for any b. We also consider the stability on H 1 (R) and show that the peakons are spectrally or linearly stable only in the case b = 3. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For the cantilever beam vibration model without damping and forced terms, the corresponding differential system is a planar dynamical system with some singular straight lines. In this paper, by using the techniques from dynamical systems and singular traveling wave theory developed by [Li & Chen, 2007] to analyze its corresponding differential system, the bifurcations and the dynamical behaviors of the corresponding phase portraits are identified and analyzed. Under different parameter conditions, exact homoclinic and heteroclinic solutions, periodic solutions, compacton solutions, as well as peakons and periodic peakons, are all found explicitly. [ABSTRACT FROM AUTHOR]

A nonlinear shallow water wave equation containing the Fornberg–Whitham model is considered. The phase portrait analytical technique is employed to establish the existence of the smooth, peaked and cusped solitary wave solutions of the equation under inhomogeneous boundary conditions. Asymptotic and numerical analysis illustrates the dynamical features for the smooth, peaked and cusped solitary wave solutions. Our results are helpful to further understand the dynamical tendency of the solutions when the space variable tends to positive or negative infinite.

In this paper, we investigate the generalized Radhakrishnan–Kundu–Lakshmanan equation with polynomial law using the method of dynamical systems. By using traveling-wave transformation, the model can be converted into a singular integrable traveling-wave system. Then, we discuss the dynamical behavior of the associated regular system and we obtain bifurcations of the phase portraits of the traveling-wave system under different parameter conditions. Finally, under different parameter conditions, we obtain the exact periodic solutions, and the peakon, homoclinic and heteroclinic solutions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The homotopy analysis method is applied to a b-Novikov equation in order to obtain analytic approximations of its peakon solution for various values of b. The results demonstrate that there is an excellent agreement between the approximate solution and the known exact peakon solution of the equation. Moreover, the amplitude of the peakon is approximated and a conservation property of the obtained solution is validated.

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the generalized Radhakrishnan–Kundu–Lakshmanan equation with polynomial law using the method of dynamical systems. By using traveling-wave transformation, the model can be converted into a singular integrable traveling-wave system. Then, we discuss the dynamical behavior of the associated regular system and we obtain bifurcations of the phase portraits of the traveling-wave system under different parameter conditions. Finally, under different parameter conditions, we obtain the exact periodic solutions, and the peakon, homoclinic and heteroclinic solutions.

The homotopy analysis method is applied to a b-Novikov equation in order to obtain analytic approximations of its peakon solution for various values of b. The results demonstrate that there is an excellent agreement between the approximate solution and the known exact peakon solution of the equation. Moreover, the amplitude of the peakon is approximated and a conservation property of the obtained solution is validated. [ABSTRACT FROM AUTHOR]

Peakons and periodic peakons are two kinds of special symmetric traveling wave solutions, which have important applications in physics, optical fiber communication, and other fields. In this paper, we study the orbital stability of peakons and periodic peakons for a generalized Camassa–Holm equation with quadratic and cubic nonlinearities (mCH–Novikov–CH equation). It is a generalization of some classical equations, such as the Camassa–Holm (CH) equation, the modified Camassa–Holm (mCH) equation, and the Novikov equation. By constructing an inequality related to the maximum and minimum of solutions with the conservation laws, we prove that the peakons and periodic peakons are orbitally stable under small perturbations in the energy space. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Peakons and periodic peakons are two kinds of special symmetric traveling wave solutions, which have important applications in physics, optical fiber communication, and other fields. In this paper, we study the orbital stability of peakons and periodic peakons for a generalized Camassa–Holm equation with quadratic and cubic nonlinearities (mCH–Novikov–CH equation). It is a generalization of some classical equations, such as the Camassa–Holm (CH) equation, the modified Camassa–Holm (mCH) equation, and the Novikov equation. By constructing an inequality related to the maximum and minimum of solutions with the conservation laws, we prove that the peakons and periodic peakons are orbitally stable under small perturbations in the energy space.

The Cauchy problem for a unified family of integrable U (1) -invariant peakon equations from the NLS hierarchy is studied. As main results, local well-posedness is proved in Besov spaces, and blow-up is established through use of an L 1 conservation law. [ABSTRACT FROM AUTHOR]

In this paper, we study a generalized scale-invariant analogue of the well-known Korteweg–de Vries (KdV) equation. This generalized equation can be thought of as a bridge between the KdV equation and the SIdV equation that was discovered recently, and shares the same one-soliton solution as the KdV equation. By employing the auxiliary equation method, we are able to obtain a wide variety of traveling wave solutions, both bounded and singular, which are kink and bell types, periodic waves, exponential waves, and peaked (peakon) waves. As far as we know, these solutions are new and their explicit closed-form expressions have not been reported elsewhere in the literature.

In this paper, the method of dynamical systems developed in [Li & Chen, 2007] is applied to the rotation-two-component Camassa–Holm system. Through qualitative analysis, under given parameter conditions, exact explicit solitary wave solution, pseudo-peakon solution, peakon and periodic peakon, as well as compacton solution, are obtained. Some parameter conditions constraints are derived for ensuring the existence of these solutions. [ABSTRACT FROM AUTHOR]

This paper studies the bifurcations of phase portraits for the regularized Saint-Venant equation (a two-component system), which appears in shallow water theory, by using the theory of dynamical systems and singular traveling wave techniques developed in [Li & Chen, 2007] under different parameter conditions in the two-parameter space. Some explicit exact parametric representations of the solitary wave solutions, smooth periodic wave solutions, periodic peakons, as well as peakon solutions, are obtained. More interestingly, it is found that the so-called u -traveling wave system has a family of pseudo-peakon wave solutions, and their limiting solution is a peakon solution. In addition, it is found that the u -traveling wave system has two families of uncountably infinitely many solitary wave solutions and compacton solutions. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we study the symmetric travelling wave solutions of the b-family of the Novikov equation. We show that the b-family of the Novikov equation can provide symmetric travelling wave solutions, such as peakon, kink and smooth soliton solutions. In particular, the single peakon, two-peakon, stationary kink, anti-kink, two-kink, two-anti-kink, bell-shape soliton and hat-shape soliton solutions are presented in an explicit formula.

The Camassa–Holm equation can be used in fluids and other fields. Under investigation in this paper, the bilinear form, implicit soliton solution and multi-peakon solution of the generalized nonautonomous Camassa–Holm equation under constraints are derived. Based on these, time varying influence factors of solution amplitude, velocity and background are discussed, which are caused by inhomogeneity of boundaries and media. Furthermore, the phenomena of nonlinear tunnelling, soliton collision and split are constructed to show the characteristic of nonautonomous solitons and peakons in the propagation. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

A two component generalization Camassa–Holm equation with peakon is proposed, which is associated with a 2 × 2 matrix spectral problem with two potentials. As its reduction, a new integrable peakon equation is derived. Then positive flows associated with the 2 × 2 matrix spectral problem are discussed, a new hierarchy of nonlinear evolution equations is obtained. Subsequently, N -peakon of the peakon equation are constructed and the dynamical system with which the N -peakon evolve is deduced. As an example, exact single peakon and 2-peakon are given and their figures are plotted. Moreover, infinitely many conserved quantities of the peakon equation are obtained. [ABSTRACT FROM AUTHOR]

Abstract The (2+1)-dimensional generalized sine-Gordon equations can be used to describe the propagation of femtosecond laser pulse in a systems of two-level atoms. In this paper, bilinear forms of the equations have been constructed through variable transformation and Bell polynomial. Based on the bilinear method and truncated Painlevé expansion, multi-soliton and quasi-periodic peakon solutions for such a system are derived, respectively. Overtaking elastic interactions between the two solitons and among the three solitons have been found. It is presented that the soliton with the smaller amplitude can travel faster than the larger one. Apart from the soliton, other type of nonlinear wave propagating in the systems of two-level atoms has been displayed, namely the quasi-periodic peakon. The peakon oscillates in both space and time, while its left and right derivatives at peak point are relatively large, which means the electric-field intensity of the femtosecond laser pulse is rather strong. [ABSTRACT FROM AUTHOR]

Raman soliton model in nanoscale optical waveguides, with metamaterials, having polynomial law nonlinearity is investigated by the method of dynamical systems. The functions in the solutions satisfy a singular planar dynamical system having two singular straight lines. By using the bifurcation theory method of dynamical systems to the equations of , under 23 different parameter conditions, bifurcations of phase portraits and exact periodic solutions, homoclinic and heteroclinic solutions, periodic peakons and peakons as well as compacton solutions for this planar dynamical system are obtained. 92 exact explicit solutions of system (6) are derived. [ABSTRACT FROM AUTHOR]

In this paper, we study a model of generalized Dullin-Gottwald-Holm equation, depending on the power law nonlinearity, that derives a series of planar dynamical systems. The study of the traveling wave solutions for this model derives a planar Hamiltonian system. By investigating the dynamical behavior and bifurcation of solutions of the traveling wave system, we derive all possible exact explicit traveling wave solutions, under different parametric conditions. These results completely improve the study of traveling wave solutions to the mentioned model stated in [Biswas & Kara, 2010]. [ABSTRACT FROM AUTHOR]

In this paper, we propose a multi-component system of the Camassa–Holm equation, denoted by CH( N , H ), with 2 N components and an arbitrary smooth function H . This system is shown to admit Lax pair and infinitely many conservation laws. We particularly study the case N = 2 and derive the bi-Hamiltonian structures and peaked soliton (peakon) solutions for some examples. [ABSTRACT FROM AUTHOR]

Peakon traveling wave solutions, both on the line and on the circle, are derived for a novel ab-family of nonlocal evolution equations with cubic nonlinearities. At least two members of this ab-family, namely the Fokas-Olver-Rosenau-Qiao equation and the Novikov equation, are known to be integrable. Furthermore, a generalization of the ab-family with nonlinearities of order k ∈ N, ≥ 2, is considered and its multi-peakon on the line is obtained. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We investigate the blow-up mechanism of solutions to a class of quasilinear integrable equations which could possess peakons. The dynamics of the blow-up quantity along the characteristics is established by the Riccati-type differential inequality which involves the interaction among three parts: a local nonlinearity, a nonlocal term, and a term stemming from the weak linear dispersion. To analyse the interplay among these quantities, we provide two different approaches. The first one is designed for the case when the equations do not exhibit a weak linear dispersion and hence focuses on the interplay between the first two parts. The method is based on a refined analysis on either evolution of the solution u and its gradient u x , that is, C u ± u x or the growth rate of the relative ratio u x / u . The second one handles the general situation when all of three parts are present. The idea is to extract the “truly” blow-up component from the Riccati-type differential inequality and utilizes the Morawetz-type identity or higher order conservation laws to show that such a component blows up in finite time before the other component degenerates. [ABSTRACT FROM AUTHOR]

In this paper, we consider a modulated equation in a discrete nonlinear electrical transmission line. This model is an integrable planar dynamical system having three singular straight lines. By using the theory of singular systems to investigate the dynamical behavior for this system, we obtain bifurcations of phase portraits under different parameter conditions. Corresponding to some special level curves, we derive exact explicit parametric representations of solutions (including smooth solitary wave solutions, peakons, compactons, periodic cusp wave solutions) under different parameter conditions. [ABSTRACT FROM AUTHOR]