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We show that the Cauchy problem for the Camassa–Holm equation has a unique, global, weak, and dissipative solution for any initial data u 0 ∈ H 1 (R) , such that u 0 , x is bounded from above almost everywhere. In particular, we establish a one-to-one correspondence between the properties specific to the dissipative solutions and a solution operator associating to each initial data exactly one solution. [ABSTRACT FROM AUTHOR]

In this paper, we derive a higher‐order Korteweg–de Vries (HKdV) equation as a model to describe the unidirectional propagation of waves on an internal interface separating two fluid layers of varying densities. Our model incorporates underlying currents by permitting a sheared current in both fluid layers, and also accommodates the effect of the Earth's rotation by including Coriolis forces (restricted to the Equatorial region). The resulting governing equations describing the water wave problem in two fluid layers under a "flat‐surface" assumption are expressed in a general form as a system of two coupled equations through Dirichlet–Neumann (DN) operators. The DN operators also facilitate a convenient Hamiltonian formulation of the problem. We then derive the HKdV equation from this Hamiltonian formulation, in the long‐wave, and small‐amplitude, asymptotic regimes. Finally, it is demonstrated that there is an explicit transformation connecting the HKdV we derive with the following integrable equations of a similar type: KdV5, Kaup–Kuperschmidt equation, Sawada–Kotera equation, and Camassa–Holm and Degasperis–Procesi equations. [ABSTRACT FROM AUTHOR]

In this paper, we propose two momentum-preserving finite difference schemes for solving one-dimensional Camassa–Holm equation with periodic boundary conditions. A two-level nonlinear difference scheme and a three-level linearized difference scheme are constructed by using the method of order reduction. For nonlinear scheme, we combine mid-point rule and a specific difference operator, which ensures that our obtained scheme is of second-order convergence in both temporal and spatial directions. For linearized scheme, we apply a linear implicit Crank–Nicolson scheme in the temporal direction, then unique solvability and momentum conservation are analysed in detail. Numerical experiments are provided for Camassa–Holm equation admitting different types of solutions, which demonstrate the convergence order and accuracy of the proposed methods coincide with theoretical analysis. Moreover, numerical results show that the nonlinear scheme exhibits better accuracy for mass conservation, while the linearized scheme is more time-saving in computation. [ABSTRACT FROM AUTHOR]

In this paper, we focus on zero-filter limit problem for the Camassa-Holm equation in the more general Besov spaces. We prove that the solution of the Camassa-Holm equation converges strongly in L ∞ (0 , T ; B 2 , r s (R)) to the inviscid Burgers equation as the filter parameter α tends to zero with the given initial data u 0 ∈ B 2 , r s (R) . Moreover, we also show that the zero-filter limit for the Camassa-Holm equation does not converges uniformly with respect to the initial data in B 2 , r s (R) . [ABSTRACT FROM AUTHOR]

In this short note, we prove that given initial data u 0 ∈ H s (R) with s > 3 2 and for some T > 0 , the solution of the Camassa-Holm equation does not converges uniformly with respect to the initial data in L ∞ (0 , T ; H s (R)) to the inviscid Burgers equation as the filter parameter α tends to zero. This is a complement of our recent result on the zero-filter limit. [ABSTRACT FROM AUTHOR]

We consider the propagation of smooth solitary waves in a two-dimensional generalization of the Camassa–Holm equation. We show that transverse perturbations to one-dimensional solitary waves behave similarly to the KP-II theory. This conclusion follows from our two main results: (i) the double eigenvalue of the linearized equations related to the translational symmetry breaks under a transverse perturbation into a pair of the asymptotically stable resonances and (ii) small-amplitude solitary waves are linearly stable with respect to transverse perturbations. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for the Camassa–Holm equation on the real line. By presenting a new construction of initial data, we show that the solution map in the smaller space B p , 1 1 ∩ C 0 , 1 with p ∈ (2 , ∞ ] is discontinuous at origin. More precisely, the initial data in B p , 1 1 ∩ C 0 , 1 can guarantee that the Camassa–Holm equation has a unique local solution in W 1 , p ∩ C 0 , 1 , however, this solution is instable and can have an inflation in B p , 1 1 ∩ C 0 , 1 . [ABSTRACT FROM AUTHOR]

This investigation employs advanced computational techniques to ascertain novel and precise solitary wave solutions of the Camassa–Holm ( ℋ) equation, a partial differential equation governing wave phenomena in one-dimensional media. Originally designed for the representation of shallow water waves, the ℋ equation has exhibited versatility across various disciplines, including nonlinear optics and elasticity theory. It intricately delineates the interplay between nonlinear and dispersive effects in wave systems, with nonlinearity arising from component interactions and dispersion rooted in the temporal spreading of waves. Furthermore, the ℋ equation governs the spatiotemporal evolution of wave profiles, encompassing both nonlinear and dispersive influences. Notably, the equation allows for soliton solutions — localized wave packets sustaining their form over extended distances. The identification of precise solitary wave solutions holds paramount significance for comprehending the ℋ equation's behavior in diverse physical contexts, such as fluid dynamics and nonlinear optics. Moreover, this study establishes a correlation between the investigated model and plasma physics, demonstrating the efficacy and efficiency of the employed computational techniques through benchmarking against alternative computational methods. This augmentation underscores the broader relevance of the ℋ equation, extending its applicability to provide insights into wave phenomena analogous to those encountered in plasma physics. [ABSTRACT FROM AUTHOR]

In this article, we mainly prove low regularity conservation laws for the Fokas-Lenells equation in Besov spaces with small initial data both on the line and on the circle. We develop a new technique in Fourier analysis and complex analysis to obtain the a priori estimates. It is based on the perturbation determinant associated with the Lax pair introduced by Killip, Vişan, and Zhang for completely integrable dispersive partial differential equations. Additionally, we also utilize the perturbation determinant to derive the global a priori estimates for the Schwartz solutions to the Camassa-Holm (CH) equation in H1{H}^{1}. Even though the energy conservation law of the CH equation is a fact known to all, the perturbation determinant method indicates that we cannot get any conserved quantities for the CH equation in Hk{H}^{k} except k=1k=1.

The objective of this paper is to examine the analytical properties of the nonlinear (1+1)-dimensional Camassa–Holm equation (ℂℍ), a fundamental model within the domain of nonlinear evolution equations. The aforementioned equation serves as a valuable tool in elucidating the unidirectional propagation of shallow water waves over a level terrain, as well as in representing certain nonlinear wave phenomena seen in cylindrical hyperelastic rods. We use the Khater III (핂hat.III) and improved Kudryashov (핀핂ud) technique to provide accurate solutions, drawing inspiration from the intricate mathematical framework of the ℂℍ issue. He’s variational iteration (ℍ핍핀) technique is used as a numerical methodology to assess the correctness of the generated answers. This strategy reveals a notable concurrence between the analytical and numerical outcomes. This alignment guarantees the suitability of the acquired solutions within the framework of the studied model.The importance of this investigation lies in its ability to improve our comprehension of the intricate dynamics regulated by the ℂℍ equation and its connections with other nonlinear evolution equations that describe shallow water wave behaviors and nonlinear wave propagation in cylindrical hyperelastic rods. The results of the study demonstrate novel analytical approaches, expanding the range of potential solutions and offering valuable insights into the physical characteristics of the interconnected wave phenomena. This research offers valuable insights and methodologies for addressing intricate mathematical models in shallow water wave theory and studying nonlinear waves in hyperelastic materials, therefore making substantial advances to the subject of nonlinear partial differential equations. [ABSTRACT FROM AUTHOR]

In the paper, we revisit the uniform continuity properties of the data-to-solution map of the Camassa–Holm equation on the real-line case. We show that the data-to-solution map of the Camassa–Holm equation is not uniformly continuous on the initial data in Besov spaces B p , r s (R) with s > 1 2 and 1 ≤ p , r < ∞ , which improves the previous works Himonas et al. (2007) [23] , Li et al. (2020) [32] and Li et al. (2021) [31]. Furthermore, we present a strengthening of our previous work in Li et al. (2020) [32] and prove that the data-to-solution map for the Camassa–Holm equation is nowhere uniformly continuous in B p , r s (R) with s > max ⁡ { 1 + 1 / p , 3 / 2 } and (p , r) ∈ [ 1 , ∞ ] × [ 1 , ∞). The method applies also to the b-family of equations which contain the Camassa–Holm and Degasperis–Procesi equations. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain the sharp estimate for the lowest positive eigenvalue for the Camassa-Holm equation y ″ = 1 4 y + λ m (t) y , with the Neumann-Dirichlet boundary conditions, where potential m admits to change sign. We first study the optimal lower bound for the smallest positive eigenvalue in the measure differential equations. Then based on the relationship between the minimization problem of the smallest positive eigenvalue for the ODE and the one for the MDE, we find the explicit optimal lower bound of the smallest positive eigenvalue for this indefinite Camassa-Holm equation. [ABSTRACT FROM AUTHOR]

In this paper we discuss pseudo-peakons, a new class of weak solutions found in the study of higher order equations of Camassa-Holm (CH) type. A pseudo-peakon is a weak bounded solution with differentiable first derivative and continuous and bounded second derivative, but such that any higher order derivative blows up. We recall that pseudo-peakons appear if we change the momentum m appearing in the Camassa-Holm equation from m = (1 − ∂ x 2) u to m = (1 − α ∂ x 2) (1 − β ∂ x 2) u (α and β are two real parameters). Here we note that they also appear if we look for "geometrically integrable" fifth order equations, that is, for equations describing one-parametric families of pseudo-spherical surfaces in a sense explained in Section 1. [ABSTRACT FROM AUTHOR]

We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa–Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for dissipative weak martingale solutions to this SPDE, with general finite-energy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi-Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second-order Stratonovich–Itô correction term. [ABSTRACT FROM AUTHOR]

This paper investigates a generalized Camassa–Holm equation with cubic nonlinearities (alias the mCH-Novikov equation), which is a generalization of some special equations. The mCH-Novikov equation possesses well-known peaked solitary waves that are called peakons. The peakons were proved orbital stable by Chen et al. in [Stability of peaked solitary waves for a class of cubic quasilinear shallow-water equations. Int Math Res Not. 2022;1–33]. We mainly prove the orbital stability of the multi-peakons in the mCH-Novikov equation. In this paper, it is proved that the sum of N fully decoupled peaks is orbitally stable in the energy space by using energy argument, combining the orbital stability of single peakons and local monotonicity of the method. [ABSTRACT FROM AUTHOR]

In this paper, we prove the orbital stability of periodic peaked traveling waves (peakons) for a nonlinear quartic μ -Camassa–Holm equation. The equation is a μ -version of the nonlinear quartic Camassa–Holm equation which was proposed by Anco and Recio (J Phys A Math Theor 52:125–203, 2019). The equation admits the periodic peakons. It is shown that the periodic peakons are orbitally stable under small perturbations in the energy space by finding inequalities related to the three conservation laws with global maximum and minimum of the solution. [ABSTRACT FROM AUTHOR]

The Boussinesq and Camassa-Holm equations are, respectively, used to describe the bidirectional and unidirectional motions of small-amplitude waves on shallow water surfaces, but their wave dynamics remain a challenge for almost all conventional numerical methods. In this paper, we use physicsinformed neural networks (PINNs), a mesh-free deep learning method, to accurately predict the soliton (peakon) interaction or rogue wave behaviours of both equations with only a few initial and boundary data, providing a method for solving certain numerically unstable fluid systems or extreme wave solutions of regular fluid systems. It is revealed that by decomposing both of the equations into lower-order coupled systems, one can obtain higher-precision wave behaviours, especially for the Camassa-Holm equation. To retrieve certain unknown hydraulic parameters based on the observed wave data, e.g. the water depth, we use a multiple PINN method and stably identify the dynamic parameters in the Boussinesq equation through the association of localized soliton and rogue wave solutions. Furthermore, we compare the PINNs with conventional high-precision timesplitting Fourier spectral (TSFS) method and find that to achieve the same split feature of the Y-shaped soliton of the Boussinesq equation, PINNs require only one-third of the initial and boundary data of the TSFS method. [ABSTRACT FROM AUTHOR]

In this article, high temporal and spatial resolution schemes are combined to solve the Camassa-Holm and Degasperis-Procesi equations. The differential quadrature method is strengthened by using modified uniform algebraic trigonometric tension B-splines of order four to transform the partial differential equation (PDE) into a system of ordinary differential equations. Later, this system is solved considering an optimized hybrid block method. The good performance of the proposed strategy is shown through some numerical examples. The stability analysis of the presented method is discussed. This strategy produces a saving of CPU-time as it involves a reduced number of grid points. [ABSTRACT FROM AUTHOR]

Based on the ∂ ‾ -generalization of the Deift-Zhou steepest descent method, we extend the long-time and Painlevé asymptotics for the Camassa-Holm (CH) equation to the solutions with initial data in a weighted Sobolev space H 4 , 2 (R). With a new scale (y , t) and a Riemann-Hilbert problem associated with the initial value problem, we derive different long time asymptotic expansions for the solutions of the CH equation in different space-time solitonic regions. The half-plane { (y , t) : − ∞ < y < ∞ , t > 0 } is divided into four asymptotic regions: 1. Fast decay region, y / t ∈ (− ∞ , − 1 / 4) with an error O (t − 1 / 2) ; 2. Modulation-soliton region, y / t ∈ (2 , + ∞) , the result can be characterized with an modulation-solitons with residual error O (t − 1 / 2) ; 3. Zakhrov-Manakov region, y / t ∈ (0 , 2) and y / t ∈ (− 1 / 4 , 0). The asymptotic approximation is characterized by the dispersion term with residual error O (t − 3 / 4) ; 4. Two transition regions, | y / t | ≈ 2 and | y / t | ≈ − 1 / 4 , the asymptotic results are described by the solution of Painlevé II equation with error order O (t − 1 / 2). [ABSTRACT FROM AUTHOR]

The Camassa-Holm equation is a shallow water model possessing solutions breaking in finite time, whereas from a geometric viewpoint, its smooth solutions determine pseudospherical surfaces. Despite having been known for a long time, these two different aspects have never been considered conjunctively. As a result, it is not know if a solution developing breaking wave can also determine a pseudospherical surface nor, that being so, whether or how the singularities of the solutions can be, or even are always, inherited by the corresponding first fundamental form. We prove that certain solutions, with enough regularity and developing breaking wave, determine a pseudospherical surface defined on a strip of finite height whose corresponding metric tensor blows up near the strip's boundary. To this end, we introduce the notion of pseudospherical surface modelled by a function space and reformulate the notion of generic solution of an equation describing pseudospherical surfaces. These two new viewpoints enable us to proceed with a geometric analysis study of the surfaces determined by solutions of Cauchy problems involving the Camassa-Holm equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for the Camassa–Holm equation in Sobolev spaces. Firstly, we establish the energy conservation for weak solutions of the Camassa–Holm equation in H 1 (R) ∩ B 3 , 2 1 (R) and prove that every weak solution in H 7 6 (R) is unique by the embedding H 7 6 (R) ↪ B 3 , 2 1 (R). Then, we obtain the local well-posedness for the Camassa–Holm equation in W 2 , 1 (R). It is worth noting that B 1 , 1 2 (R) ↪ W 2 , 1 (R) and the Camassa–Holm equation is well-posed in B 1 , 1 2 (R) and is ill-posed in W 1 + 1 p , p (R) (1 < p ≤ ∞) by the pervious papers. Our result implies that W 2 , 1 (R) is the critical Sobolev spaces for the well-posedness of the Camassa–Holm equation. [ABSTRACT FROM AUTHOR]

The generalized modified Camassa-Holm equation possesses well-known peaked solitary waves that are called peakons. Their stability has been established by Z. Guo et al. (J Differ Equ 266:7749–7779, 2019) by using the Constantin-Strauss approach. In this paper, using energy argument and combining the method of the orbital stability of a single peakon with monotonicity of the local energy norm, we prove that the sum of N sufficiently decoupled peakons is orbitally stable in the energy space. [ABSTRACT FROM AUTHOR]

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The Camassa–Holm equation governs the dynamics of shallow water waves or in its reduced form models nonlinear dispersive waves in hyperelastic rods. By using the straightforward Bogning-Djeumen Tchaho-Kofané method, explicit expressions of many solitary wave solutions with different profiles not previously derived in the literature are constructed and classified. Geometric characterizations of the solutions in terms of three new mappings are presented. Intensive numerical simulations carried confirm the stability of the solutions even with relatively high critical velocities and reveal that solitary waves with large widths are more stable than the ones with small widths. [ABSTRACT FROM AUTHOR]

We study the Lyapunov stability of the periodic Camassa–Holm equation in terms of the periodic/anti-periodic eigenvalues and the associated spectral intervals. We consider the case with definite potentials as well as the case with indefinite potentials. In particular, we prove a Lyapunov-type stability criterion. [ABSTRACT FROM AUTHOR]

We prove that the two-component peakon solutions are orbitally stable in the energy space. The system concerned here is a two-component Novikov system, which is an integrable multi-component extension of the integrable Novikov equation. We improve the method for the scalar peakons to the two-component case with genuine nonlinear interactions by establishing optimal inequalities for the conserved quantities involving the coupled structures. Moreover, we also establish the orbital stability for the train-profiles of these two-component peakons by using the refined analysis based on monotonicity of the local energy and an induction method. [ABSTRACT FROM AUTHOR]

The paper aims at developing the Riemann–Hilbert (RH) approach for the modified Camassa–Holm (mCH) equation on the line with non-zero boundary conditions, in the case when the solution is assumed to approach two different constants at different sides of the line. We present detailed properties of spectral functions associated with the initial data for the Cauchy problem for the mCH equation and obtain a representation for the solution of this problem in terms of the solution of an associated RH problem. [ABSTRACT FROM AUTHOR]

In this paper, solutions of the Camassa-Holm equation near the soliton Q is decomposed by pseudo-conformal transformation as follows: λ1/2(t)u(t, λ(t)y + x(t)) = Q(y) +ε(t, y), and the estimation formula with respect to ε(t, y) is obtained: ∣ε(t, y)∣ ≤ Ca3Te−θ∣y∣ + ∣λ1/2(t)ε0∣. For the CH equation, we prove that the solution of the Cauchy problem and the soliton Q is sufficiently close as y → ∞, and the approximation degree of the solution and Q is the same as that of initial data and Q, besides the energy distribution of ε is consistent with the distribution of the soliton Q in H2. [ABSTRACT FROM AUTHOR]

We study special properties of solutions to the IVP associated to theCamassa-Holm equation on the line related to the regularity and the decay ofsolutions. The first aim is to show how the regularity on the initial data istransferred to the corresponding solution in a class containing the "peakonsolutions". In particular, we shall show that the local regularity is similarto that exhibited by the solution of the inviscid Burger's equation with thesame initial datum. The second goal is to prove that the decay results obtainedin a paper of Himonas, Misio{\l}ek, Ponce, and Zhou extend to the class ofsolutions considered here.

In this paper, we develop cell-average-based neural network (CANN) method to approximate solutions of nonlinear Cahn–Hilliard equation and Camassa–Holm equation. The CANN method is motivated by the finite volume scheme and evolved from the integral or weak formulation of partial differential equations. The major idea of cell-average-based neural network method is to explore a neural network to approximate the solution average difference or evolution between two neighboring time steps. Unlike traditional numerical methods, CANN method is not limited by the CFL restriction and can adapt large time steps for solution evolution, which is a significant advantage that classical numerical methods do not have. Once well trained, this method can be implemented as an fixed explicit finite volume scheme and applied to certain groups of initial value conditions for Cahn–Hilliard equation and Camassa–Holm equation without retraining the neural network. Furthermore, the CANN method also performs very well in handling data with corruption or low-quality data generated by Gaussian white noise. Numerical examples are presented to demonstrate effectiveness, accuracy and capability of the proposed method. [ABSTRACT FROM AUTHOR]

The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the DP equation on the real line, extending our previous work on their spectral stability [J. Math. Pures Appl. (9) 142 (2020), pp. 298–314]. The main difficulty stems from the fact that the natural energy space is a subspace of L^3, but the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the L^2-norm, resulting in L^3 higher-order nonlinear terms in the augmented Hamiltonian. But the usual coercivity estimate is in terms of L^2 norm for DP equation, which cannot be used to control the L^3 higher order term directly. The remedy is to observe that, given a sufficiently smooth initial condition satisfying some mild constraint, the L^\infty orbital norm of the perturbation is bounded above by a function of its L^2 orbital norm, yielding the higher order control and the orbital stability in the L^2\cap L^\infty space. [ABSTRACT FROM AUTHOR]

In this paper, we design a novel class of arbitrarily high-order, linearly implicit and energy-preserving numerical schemes for solving the nonlinear dispersive equations. Based on the idea of the energy quadratization technique, the original system is firstly rewritten as an equivalent system with a quadratization energy. The prediction-correction strategy, together with the Partitioned Runge-Kutta method, is then employed to discretize the reformulated system in time. The resulting semi-discrete system is high-order, linearly implicit and can preserve the quadratic energy of the reformulated system exactly. Finally, we take the Camassa-Holm equation as a benchmark to show the efficiency and accuracy of the proposed schemes. [ABSTRACT FROM AUTHOR]

The Camassa–Holm hierarchy can be regarded as isospectral flows of the inhomogeneous string. This paper is devoted to the exploration of the second flow in the Camassa–Holm hierarchy (2ndCH) together with its peakon dynamical system as well as their nonisospectral generalizations. It is shown that a reduction of the peakon dynamical system of the 2ndCH equation results in the two-component modified Camassa–Holm (2mCH) interlacing peakon dynamical system. This reduction result is then extended to the nonisospectral case. More precisely, a nonisospectral extension of the 2ndCH is proposed together with its multipeakons based on classical determinant technique. It is also demonstrated that the corresponding peakon dynamical system can be reduced to the generalized nonisospectral 2mCH interlacing peakon dynamical system. Moreover, a special case of the proposed equation is investigated and a new phenomenon of 2-peakon is observed. [ABSTRACT FROM AUTHOR]

We initiate the process by introducing a nonisospectral Lax pair, from which we derive an integrable nonisospectral hierarchy associate with Camassa-Holm equation. Through the inverse scattering transform method, we obtain parameter expressions for the N-soliton solution of the integrable nonisospectral hierarchy associate with Camassa-Holm equation. To derive the precise expression of the solution without the parameters, a coordinate transformation is performed. In order to work out accurately the soliton solution through the Gel'fand-Levitan-Marchenko equation. Finally, we present the graphical representation of the 1-soliton solution and analyze its dynamic behavior. [ABSTRACT FROM AUTHOR]

In this study, we use cutting-edge analytical and numerical approaches to the Gilson–Pickering (GP) problem in order to get precise soliton solutions. This model explains wave propagation in plasma physics and crystal lattice theory. A variety of evolution equations have been developed from the GP model, including the Fornberg–Whitham (FW) equation, the Rosenau–Hyman (RH) equation, and the Fuchssteiner-Fokas-Camassa–Holm (FFCH) equation, to name a few. The GP model has been studied using these evolution equations. To investigate the characterizations of new waves, crystal lattice theory and plasma physics use the Khater II, and He’s variational iteration approaches. Many alternative responses may be achieved by utilizing various formulae; each of these solutions is shown by a distinct graph. The validity of such methods and solutions may be demonstrated by assessing how well the relevant techniques and solutions match up. The results of this study suggest that the technique is preferred for successfully resolving nonlinear equations that emerge in mathematical physics.

To realize and comprehend the physical phenomena of nonlinear system, exploration of traveling wave solutions plays an important role. Among the class of dispersive PDEs of traveling wave solutions the Degasperis-Procesi (DP) equation comprises high order nonlinear derivatives and is considered as a well known model for shallow water dynamics having similar asymptotic accuracy as for the Camassa-Holm (CH) equation. In this study we investigate solutions of some high order nonlinear dispersive PDEs namely generalized Degasperis-Procesi (DP), Camassa-Holm (CH) and Korteweg-de Vries (KdV) equations by the use of Radial Basis Function (RBF) combined with Finite Differences (RBF-FD) and Pseudo-Spectral (RBF-PS) methods. For the time derivative approximation, the fourth-order Runge-Kutta (RK) technique is accomplished. The efficiency and accuracy of our suggested approaches are demonstrated using examples and results. [ABSTRACT FROM AUTHOR]

In this paper, the extended simple equation method (ESEM) and the generalized Riccati equation mapping (GREM) method are applied to the nonlinear third-order Gilson–Pickering (GP) model to obtain a variety of new exact wave solutions. With the suitable selection of parameters involved in the model, some familiar physical governing models such as the Camassa–Holm (CH) equation, the Fornberg–Whitham (FW) equation, and the Rosenau–Hyman (RH) equation are obtained. The graphical representation of solutions under different constraints shows the dark, bright, combined dark–bright, periodic, singular, and kink soliton. For the graphical representation, 3D plots, contour plots, and 2D plots of some acquired solutions are illustrated. The obtained wave solutions motivate researchers to enhance their theories to the best of their capacities and to utilize the outcomes in other nonlinear cases. The executed methods are shown to be practical and straightforward for approaching the considered equation and may be utilized to study abundant types of NLEEs arising in physics, engineering, and applied sciences. [ABSTRACT FROM AUTHOR]

In the present paper, we study the integration of the periodic Camassa–Holm equation with an integral-type source. Physically, sources arise in solitary waves with a variable speed and lead to a variety of dynamics of physical models. With regard to their applications, these kinds of systems are usually used to describe interactions between different solitary waves. We show that the periodic Camassa–Holm equation with an integral-type source is also an important theoretical model as it is a completely integrable system. We obtain a representation for the solution of periodic Camassa–Holm equation with an integral-type source in the framework of the inverse spectral problem for a weighted Shturm–Liouville operator. [ABSTRACT FROM AUTHOR]

A solitary wave is the dispersion-less solution of nonlinear evolutionary equations that travels at a constant speed without dissipating its energy. The purpose of this article is to provide insight into the discovery and history of solitons. The different types of the solitons are discussed in brief that is helpful for the researchers. For the discussion of the nature of solitons, the solution behavior of the Korteweg de Vries equation (KdV), the sine-Gordon (SG), the Camassa-Holm (CH) equation, and the nonlinear Schrodinger (NLS) equation are considered. This article deals with the various applications of solitons in different fields such as biophysics, nonlinear optics, Bose-Einstein condensation, plasma physics, Josephson junction, etc. focusing on the properties of solitons based on their classification.

In Zhang et al. (2007) and Zhang (2021) we constructed all single-peak traveling wave solutions of the Camassa–Holm equation including some explicit solutions. In general it is a challenge to construct exact multi-peak traveling wave solutions. As an example a periodic traveling wave (or wavetrain), a special type of spatiotemporal oscillation that is a periodic function of both space and time, plays a fundamental role in many mathematical equations such as shallow water wave equations. In this paper we will construct some new exact periodic traveling wave solutions of the Camassa–Holm equation.

In the present paper we prove the validity of the Camassa-Holm equation as a long wave limit to the double dispersion equation which describes the propagation of bidirectional weakly nonlinear and dispersive waves in an infinite elastic medium. First we show formally that the right-going wave solutions of the double dispersion equation can be approximated by the solutions of the Camassa-Holm equation in the long wave limit. Then we rigorously prove that the solutions of the double dispersion and the Camassa-Holm equations remain close over a long time interval, determined by two small parameters measuring the effects of nonlinearity and dispersion. [ABSTRACT FROM AUTHOR]

For the famous Camassa-Holm equation, the well-posedness in B p , 1 1 + 1 p (R) with p ∈ [ 1 , ∞) and the ill-posedness in B p , r 1 + 1 p (R) with p ∈ [ 1 , ∞ ] , r ∈ (1 , ∞ ] had been studied in [13,14,16,23] , that is to say, it only left an open problem in the critical case B ∞ , 1 1 (R) proposed by Danchin in [13,14]. In this paper, we solve this problem by proving the norm inflation and hence the ill-posedness for the Camassa-Holm equation in B ∞ , 1 1 (R). Therefore, the well-posedness and ill-posedness for the Camassa-Holm equation in all critical Besov spaces B p , 1 1 + 1 p (R) with p ∈ [ 1 , ∞ ] have been completed. Finally, since the norm inflation occurs by choosing an special initial data u 0 ∈ B ∞ , 1 1 (R) but u 0 x 2 ∉ B ∞ , 1 0 (R) (an example implies B ∞ , 1 0 (R) is not a Banach algebra), we then prove that this condition is necessary. That is, if u 0 x 2 ∈ B ∞ , 1 0 (R) holds, then the Camassa-Holm equation has a unique solution u (t , x) ∈ C T (B ∞ , 1 1 (R)) ∩ C T 1 (B ∞ , 1 0 (R)) and the norm inflation will not occur. [ABSTRACT FROM AUTHOR]

The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model as an asymptotic approximation for the unidirectional propagation of shallow water waves. This work is to establish the L 2 ∩ L ∞ orbital stability of a wave train containing N smooth solitons which are well separated. The main difficulties stem from the subtle nonlocal structure of the DP equation. One consequence is that the energy space of the DE equation based on the conserved quantity induced by the translation symmetry is only equivalent to the L 2 -norm, which by itself can not bound the higher-order nonlinear terms in the Lagrangian. Our remedy is to introduce a priori estimates based on certain smooth initial conditions. Moreover, another consequence is that the nonlocal structure of the DP equation significantly complicates the verification of the monotonicity of local momentum and the positive definiteness of a refined quadratic form of the orthogonalized perturbation. [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 is devoted to the study of initial-boundary value problems for time-fractional analogues of Korteweg-de Vries, Benjamin-Bona-Mahony, Burgers, Rosenau, Camassa-Holm, Degasperis-Procesi, Ostrovsky and time-fractional modified Korteweg-de Vries-Burgers equations on a bounded domain. Sufficient conditions for the blowing-up of solutions in finite time of aforementioned equations are presented. We also discuss the maximum principle and influence of gradient non-linearity on the global solvability of initial-boundary value problems for the time-fractional Burgers equation. The main tool of our study is the Pohozhaev nonlinear capacity method. We also provide some illustrative examples.

This paper proposes a higher-order μ -Camassa–Holm equation, which is regarded as a higher-order extension of the μ -Camassa–Holm equation, and preserves some properties of the μ -Camassa–Holm equation. We first show that the equation admits the peaked traveling wave solution, which is given by a Green function of the momentum operator. Local well-posedness of the Cauchy problem in the suitable Sobolev space is established. Finally, the blow-up criterion and wave breaking mechanism for solutions with certain initial profiles are studied. It turns out that all the nonlinearities even the first-order nonlinearity may have the effect on the blow up. [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]