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In this paper, we consider the bifurcations and traveling wave solutions for the generalized A ± equation, which we construct from the Alexeyev’s A ± equation. Based on the theory of dynamical system, we study the corresponding traveling wave system and bifurcations of its phase orbit portraits. Furthermore, according to the energy level curves of the phase portraits, we analyze all kinds of bounded solutions, and derive the exact representations for the solutions including periodic peakons, peakons, smooth periodic wave solutions, solitary wave solutions, kink wave solutions as well as compacton solution family. [ABSTRACT FROM AUTHOR]

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]

In this paper, we investigate the nonexistence of the periodic peakon and peakon for a highly nonlinear shallow-water model, which has been recently derived from the full governing equations for two dimensional flow with the Coriolis effect or with constant vorticity, under a larger scaling than the Camassa-Holm (CH) one. Note that the so obtained model not only has CH-type terms, but also exhibits cubic order nonlinearities. Thus it is interesting to study how the higher power nonlinear terms affect the existence of the (periodic) peakon. [ABSTRACT FROM AUTHOR]

In this paper, we explore how to generate solitary, peakon, periodic, cuspon and kink wave solution of the well-known partial differential equation Korteweg–de Vries (KdV) by using exp-function and modified exp-function methods. The presented methods construct more efficiently almost all types of soliton solution of KdV equation that can be rarely seen in the history. These methods appear to be straightforward and symbolic calculations are used to solve the problem. All resulting answers are verified for accuracy using the symbolic computation program with M a p l e. To show the physical appearance of the model, 3D plots of all the generated solutions are then displayed. The obtained solutions revealed the compatibility of the proposed techniques which provide the general solution with some free parameters. This is the key benefit of these methods over the other methods. [ABSTRACT FROM AUTHOR]

In this paper, we investigate several important properties of the generalized nonlinear Schrödinger equation with P T -symmetric δ -sech optical potentials, such as the phase breaking of P T -symmetric δ -sech potentials, and existence, dynamics and excitations of new soliton solutions. Specifically, we identify the fully real spectral region of the non-Hermitian Hamiltonian and observe the phase-breaking phenomenon. Additionally, we discover a new soliton solution that represents a peakon solution, a smooth soliton solution, and a double-hump soliton solution under different potential parameters. We analyze the stability of these three types of solutions and determine their stability domains. Furthermore, we study the numerical peakon solution and its stability. In particular, we investigate the interaction of peakon solutions and observe semi-elastic interactions. Finally, we explore the stable adiabatic excitations of peakons. These results contribute to a deeper understanding of P T -symmetric optical fields and provide insights for related experimental works. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analytical study of traveling wave solutions to a generalized Boussinesq equation with nonlinear dispersion. Utilizing the bifurcation method of dynamical systems, the existence of compacton and peakon solutions are established. Under certain parameter conditions, the compacton and peakon solutions can be bifurcated by smooth periodic wave solutions, smooth solitary wave solutions, and singular cusp solutions. Numerical simulations are supplied to corroborate the analytical results. Some previous results are extended. [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]

In this work, Painlevé analysis is used to examine the integrability of the (2+1)-dimensional 3-component coupled nonlinear Maccari's system. Subsequently, the truncated Painlevé approach is employed to solve the system, providing solutions in terms of arbitrary functions in space and time. By selecting appropriate arbitrary functions, various field excitations, such as dromion triplet solutions, rogue waves, lumps, peakons and compactons, are constructed. The results are represented graphically to illustrate the behavioral dynamics of the system using Mathematica software. [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]

This work investigates the exact solutions of the Raman soliton model with anti-cubic nonlinear in optical metamaterials. By travelling wave transformation, the model is transformed into a singular planar dynamical system with three singular straight lines. Using the bifurcation theory of dynamical systems, under different parameter conditions, bifurcations of phase portraits are studied. More than 30 exact explicit solutions of planar dynamical system are derived, such as exact periodic wave solutions, solitary wave solutions, kink and anti-kink wave solutions, periodic peakons and peakons as well as compacton solutions. In more general parametric conditions, all possible solutions are “caught in one dragnet”. [ABSTRACT FROM AUTHOR]

In this paper, the theory of dynamical systems is employed to investigate loop waves and cusp waves of the Fujimoto-Watanabe equation. These waves contain solitary loop waves, periodic loop waves, peakons and periodic cusp waves. Under fixed parameter conditions, their exact explicit parametric expressions are given. [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]

The -family is a one-parameter family of Hamiltonian partial differential equations of nonevolutionary type, which arises in shallow water wave theory. It admits a variety of solutions, including the celebrated peakons, which are weak solutions in the form of peaked solitons with a discontinuous first derivative at the peaks, as well as other interesting solutions that have been obtained in exact form and/or numerically. In each of the special cases and (the Camassa–Holm and Degasperis–Procesi equations, respectively), the equation is completely integrable, in the sense that it admits a Lax pair and an infinite hierarchy of commuting local symmetries, but for other values of the parameter it is nonintegrable. After a discussion of traveling waves via the use of a reciprocal transformation, which reduces to a hodograph transformation at the level of the ordinary differential equation satisfied by these solutions, we apply the same technique to the scaling similarity solutions of the -family and show that when or , this similarity reduction is related by a hodograph transformation to particular cases of the Painlevé III equation, while for all other choices of the resulting ordinary differential equation is not of Painlevé type. [ABSTRACT FROM AUTHOR]

We prove spectral instability of peakons in the b-family of Camassa-Holm equations that includes the integrable cases of b = 2 and b = 3. We start with a linearized operator defined on functions in H1(R) ∩W1,∞(R) and extend it to a linearized operator defined on weaker functions in L2(R). For b ̸= 5 2, the spectrum of the linearized operator in L2(R) is proved to cover a closed vertical strip of the complex plane. For b = 5 2, the strip shrinks to the imaginary axis, but an additional pair of real eigenvalues exists due to projections to the peakon and its spatial translation. The spectral instability results agree with the linear instability results in the case of the Camassa-Holm equation for b = 2. [ABSTRACT FROM AUTHOR]

In this paper, we study the orbital stability of peakons and periodic peakons for a nonlinear quartic Camassa-Holm equation. We first verify that the QCHE has global peakon and periodic peakon solutions. Then by the invariants of the equation and controlling the extrema of the solution, we prove that the shapes of the peakons and periodic peakons are 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]

In this paper, by using the dynamic system method and the known conservation laws of the gCH equation, and underlying features of the peakons, we study the peakon solutions and the orbital stability of the peakons for a nonlinear generalization of the Camassa-Holm equation (gCH). The gCH equation is first transformed into a planar system. Then, by the first integral and algebraic curves of this system, we obtain one heteroclinic cycle, which corresponds to a peakon solution. Moreover, we give a proof of the orbital stability of the peakons for the gCH equation. [ABSTRACT FROM AUTHOR]

Considered herein is a multi-component Novikov equation, which admits bi-Hamiltonian structure, infinitely many conserved quantities and peaked solutions. In this paper, we deduce two blow-up criteria for this system and global existence for some two-component case in H s . Finally, we verify that the system possesses peakons and periodic peakons. [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.

This paper investigates the rotation-Camassa–Holm equation, which appears in long-crested shallow-water waves propagating in the equatorial ocean regions with the Coriolis effect due to the earth's rotation. The rotation-Camassa–Holm equation contains the famous Camassa–Holm equation and is a special case of the generalized Camassa–Holm equation. By using the approach of dynamical systems and singular traveling wave theory to its traveling wave system, in different parameter conditions of the five-parameter space, the bifurcations of phase portraits are studied. Some exact explicit parametric representations of the smooth solitary wave solutions, periodic wave solutions, peakons and anti-peakons, periodic peakons as well as compacton solutions are obtained. [ABSTRACT FROM AUTHOR]

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]

The orbital stability of peakons and hyperbolic periodic peakons for the Camassa-Holm equation has been established by Constantin and Strauss in [A. Constantin, W. Strauss, Comm. Pure. Appl. Math. 53 (2000) 603-610] and Lenells in [J. Lenells, Int. Math. Res. Not. 10 (2004) 485-499], respectively. In this paper, we prove the orbital stability of the elliptic periodic peakons for the modified Camassa-Holm equation. By using the invariants of the equation and controlling the extrema of the solution, it is demonstrated that the shapes of these elliptic periodic peakons are stable under small perturbations in the energy space. Throughout the paper, the theory of elliptic functions and elliptic integrals is used in the calculation. [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.

We prove that the peakons are asymptotically stable, under the flow of the Degasperis–Procesi equation, in the class of functions with a momentum density that belongs to M + (R). The key argument is a rigidity result for uniformly in time exponentially decaying global solutions that is shared by the Holm–Staley b -family of equations for b ≥ 1. This extends previous results obtained for the Camassa–Holm equation (b = 2). [ABSTRACT FROM AUTHOR]

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 wide class of nonlinear dispersive wave equations are shown to possess a novel type of peakon solution in which the amplitude and speed of the peakon are time-dependent. These novel dynamical peakons exhibit a wide variety of different behaviours for their amplitude, speed, and acceleration, including an oscillatory amplitude and constant speed which describes a peakon breather. Examples are presented of families of nonlinear dispersive wave equations that illustrate various interesting behaviours, such as asymptotic travelling-wave peakons, dissipating/anti-dissipating peakons, direction-reversing peakons, runaway and blow up peakons, among others. [ABSTRACT FROM AUTHOR]

Stable rotating polygonal hollow patterns were predicted theoretically and measured for Leidenfrost drops confined by circular boundary. Liquid free surface modes occur with different number of edges. This two-dimensional incompressible and inviscid flow is controlled by surface tension, evaporation reaction and gravitation. The model is based on shallow water theory and predicts the existence of sharp rotational polygonal waves as solitary waves solutions. These patterns may represent a signature of universality in shallow water rotating models since similar rotating hollow polygons are noticed in tropical cyclones, fast rotated liquids, Saturn's hexagon, and some plasma systems. [ABSTRACT FROM AUTHOR]

The numerical solution of initial value problems is used to obtain compacton and kovaton solutions of K(f m, g n) equations generalizing the Korteweg–de Vries K(u2, u1) and Rosenau–Hyman K(u m, u n) equations to more general dependences of the nonlinear and dispersion terms on the solution u. The functions f(u) and g(u) determining their form can be linear or can have the form of a smoothed step. It is shown that peakocompacton and peakosoliton solutions exist depending on the form of the nonlinearity and dispersion. They represent transient forms combining the properties of solitons, compactons, and peakons. It is shown that these solutions can exist against an inhomogeneous and nonstationary background. [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]