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In this paper, we construct two families of new explicit exact traveling wave solutions of the Camassa–Holm equation which solve an open problem in the previous paper [Zhang G, Qiao ZJ, Liu F. Cusp...

We solve the open problem of spectral stability of smooth periodic waves in the Camassa–Holm equation. The key to obtaining this result is that the periodic waves of the Camassa–Holm equation can be characterized by an alternative Hamiltonian structure, different from the standard formulation common to the Korteweg-de Vries equation. The standard formulation has the disadvantage that the period function is not monotone and the quadratic energy form may have two rather than one negative eigenvalues. We prove that the nonstandard formulation has the advantage that the period function is monotone and the quadratic energy form has only one simple negative eigenvalue. We deduce a precise condition for the spectral and orbital stability of the smooth periodic travelling waves and show numerically that this condition is satisfied in the open region where the smooth periodic waves exist.

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.

The dynamical stability of the periodic peaked solitons for a generalized Camassa–Holm equation is studied in this paper. Its generalization version is known to admit a single-peaked soliton soluti...

We consider a stochastic Camassa-Holm equation driven by a one-dimensional Wiener process with a first order differential operator as diffusion coefficient. We prove the existence and uniqueness of local strong solutions of this equation. In order to do so, we transform it into a random quasi-linear partial differential equation and apply Kato's operator theory methods. Some of the results have potential to find applications to other nonlinear stochastic partial differential equations.

The nonlinear waves constitute a significant concern in the design in order to understand wave behaviors and structures. Due to the complexity of the waves phenomenon, many mathematical models have been coming up and extensively being studied until now. In this paper, we introduce a new aspect to study a wave model by coupling of the classical Camassa-Holm equation and the BBM-KdV equation with the dual-power law nonlinearities. The new model still satisfies the fundamental energy conservative property as the original models. As we have known, the use of an efficient and accurate numerical method can be a beneficial tool for studying the wave behaviors; therefore, we aim to propose a finite difference scheme for solving the new model. A three-level finite difference technique is applied here in order to be able to design a linear scheme. The highlight of our scheme is to propose an alternative way to discretize the highly nonlinear terms. The class of analytical solutions of the model is derived and used to validate our numerical scheme. Like the design, the energy conservation law is preserved through the numerical scheme. In the context of theoretical analysis, the existence and uniqueness is analyzed based on a priori estimate. The stability and convergence of the numerical solution with second-order accuracy on both space and time are also provided under a mild restriction on the ratio of τ / h . Many comparisons of our scheme and a modification of the existing finite difference scheme are preformed, and the consequences confirm that the proposed scheme gives a significant improvement over the other. Moreover, in the numerical simulations, the faithfulness of the proposed method is validated by the pieces of evidence of depression solitary waves under the effect of the power of nonlinearities.

Consideration in this paper is the generalized fractional Camassa–Holm equation. The local well-posedness is established in Besov space $$B^{s_0}_{2,1}$$ with $$s_0=2\nu -\frac{1}{2}$$ for $$\nu >\frac{3}{2} $$ and $$s_0=\frac{5}{2}$$ for $$1

This paper is devoted to constructing a globally rough solution for the higher order modified Camassa-Holm equation with randomization on initial data and periodic boundary condition. Motivated by the works of Thomann and Tzvetkov (Nonlinearity, 23 (2010), 2771–2791), Tzvetkov (Probab. Theory Relat. Fields, 146 (2010), 4679–4714), Burq, Thomann and Tzvetkov (Ann. Fac. Sci. Toulouse Math., 27 (2018), 527–597), the authors first construct the Borel measure of Gibbs type in the Sobolev spaces with lower regularity, and then establish the existence of global solution to the equation with the helps of Prokhorov compactness theorem, Skorokhod convergence theorem and Gibbs measure.

The aim of this paper is to study singular dynamics of solutions of Camassa-Holm equation. Based on the semigroup theory of linear operators and Banach contraction mapping principle, we prove the asymptotic stability of the explicit singular solution of Camassa-Holm equation.

Peaked periodic waves in the Camassa--Holm equation are revisited. Linearized evolution equations are derived for perturbations to the peaked periodic waves, and linearized instability is proven in...

In this article, we consider the Lipschitz metric of conservative weak solutions for the rotation-Camassa-Holm equation. Based on defining a Finsler-type norm on the tangent space for solutions, we first establish the Lipschitz metric for smooth solutions, then by proving the generic regularity result, we extend this metric to general weak solutions.

In this paper, we study global existence and blow-up for a periodic modified Camassa–Holm equation in nonhomogeneous Sobolev spaces. Also, we provide a key blow-up criteria to investigate norm infl...

In this work, a competent and efficient technique namely Exp-function method is used to find the solitary wave solutions of time fractional simplified modified Camassa-Holm equation (CH-equation). A generalized fractional complex transformation is accurately used for the conversion of this simplified modified CH-equation into an ordinary differential equation. The application of the method represent that this method is consistent, expedient and provides more general exact solutions.

Recent generalizations of the Camassa-Holm equation are studied from the point of view of existence of global solutions, criteria for wave breaking phenomena and integrability. We provide conditions, based on lower bounds for the first spatial derivative of local solutions, for global well-posedness for the family under consideration in Sobolev spaces. Moreover, we prove that wave breaking phenomena occurs under certain mild hypothesis. Regarding integrability, we apply the machinery developed by Dubrovin [Commun. Math. Phys. 267, 117--139 (2006)] to prove that there exists a unique bi-hamiltonian structure for the equation only when it is reduced to the Dullin-Gotwald-Holm equation. Our results suggest that a recent shallow water model incorporating Coriollis efects is integrable only in specific situations. Finally, to finish the scheme of geometric integrability of the family of equations initiated in a previous work, we prove that the Dullin-Gotwald-Holm equation describes pseudo-spherical surfaces., Some proofs corrected

In this paper, we first study the local well-posedness for the Cauchy problem of a modified Camassa–Holm equation in nonhomogeneous Besov spaces. Then we obtain a blow-up criteria and present a blow-up result for the equation. Finally, with proving the norm inflation we show the ill-posedness occurs to the equation in critical Besov spaces.

It is well-known that peakons in the Camassa–Holm equation are H 1 -orbitally stable thanks to conserved quantities and properties of peakons as constrained energy minimizers. By using the method of characteristics, we prove that piecewise C 1 perturbations to peakons grow in time in spite of their stability in the H 1 -norm. We also show that the linearized stability analysis near peakons contradicts the H 1 -orbital stability result, hence passage from linear to nonlinear theory is false in H 1 .

In this paper, we mainly study the Cauchy problem of a generalized Camassa–Holm equation. We prove the existence of global weak solutions for this generalized Camassa–Holm equation by the viscous a...

We consider the modernized Camassa—Holm equation with periodic boundary conditions. The quadratic nonlinearities in this equation differ substantially from the nonlinearities in the classical Camassa—Holm equation but have all its main properties in a certain sense. We study the so-called nonregular solutions, i.e., those that are rapidly oscillating in the spatial variable. We investigate the problem of constructing solutions asymptotically periodic in time and more complicated solutions whose leading terms of the asymptotic expansion are multifrequency. We study the problem of the possibility of a compact form of these asymptotic expansions and the problem of reducing the construction of the leading terms of the asymptotic expansions to the analysis of the solutions of special nonlinear boundary-value problems. We show that this is possible only for the classical Camassa—Holm equation.

This paper studies bifurcation solutions of the Camassa – Holm equation by using the local Lyapunov – Schmidt method. The Camassa–Holm equation is studied by reduction to an ODE. We find the key function that corresponds to the functional related to this equation and defined on a new domain. The bifurcation analysis of the key function is investigated by the angular singularities. We find the parametric equation of the bifurcation set (caustic) with its geometric description. Also, the bifurcation spreading of the critical points is found.

A periodic boundary value problem is considered for a modified Camassa–Holm equation, which differs from the well-known classical equation by several additional quadratic terms. Three important conditions on the coefficients of the equation are formulated under which the original equation has the Camassa–Holm type. The dynamic properties of regular solutions in neighborhoods of all equilibrium states are investigated. Special nonlinear boundary value problems are constructed to determine the “leading” components of solutions. Asymptotic formulas for the set of periodic solutions and finite-dimensional tori are obtained. The problem of infinite-dimensional tori is studied. It is shown that the normalized equation in this problem can be compactly written in the form of a partial differential equation only for the classical Camassa–Holm equation. An asymptotic analysis is presented in the cases when one of the coefficients in the linear part of the equation is sufficiently small, while the period in the boundary conditions is sufficiently large.

In present paper, we study the Cauchy problem for a generalized Camassa-Holm equation, which was discovered by Novikov. Our purpose here is to establish persistence properties and some unique continuation properties of the solutions of this equation in weighted spaces.

A generalized Camassa–Holm equation proposed by Novikov is considered. The existence and uniqueness of a positive weak solution for the equation is established by using a classical method.

The two-component Novikov equation is an integrable generalization of the Novikov equation, which has the peaked solitons in the sense of distribution as the Novikov and Camassa-Holm equations. In this paper, we prove the existence of the \begin{document}$ H^1 $\end{document} -weak solution for the two-component Novikov equation by the regular approximation method due to the existence of three conserved densities. The key elements in our approach are some a priori estimates on the approximation solutions.

In this paper we study a periodic two-component Camassa-Holm equation with generalized weakly dissipation. The local well-posedness of Cauchy problem is investigated by utilizing Kato’s theorem. The blow-up criteria and the blow-up rate are established by applying monotonicity. Finally, the global existence results for solutions to the Cauchy problem of equation are proved by structuring functions.

In this article, we consider the exact solutions to the simplified modified Camassa-Holm (SMCH) equation which has many potential applications in mathematical physics and engineering sciences. We examine the exact travelling wave solutions by means of the modified simple equation (MSE) method by making use of travelling transformation. The attained solutions are in the form of trigonometric and hyperbolic functions. We demonstrate that the method is more general, straightforward and powerful and can be used to examine more general travelling wave solutions of various kinds of fractional nonlinear differential equations arising in mathematical physics and better than other method. Finally, we show the graphical representation and discuss the physical significance of the obtained solutions for its definite values of the involved parameters through depicting 3D and 2D figures in order to know the physical phenomena.

The Cauchy problem for the Dullin-Gottwald-Holm (DGH) equation \[u_t-\alpha^2 u_{xxt}+2\omega u_x +3uu_x+\gamma u_{xxx}=\alpha^2 (2u_x u_{xx} + uu_{xxx})\] with zero boundary conditions (as \(|x|\to\infty\)) is treated by the Riemann-Hilbert approach to the inverse scattering transform method. The approach allows us to give a representation of the solution to the Cauchy problem, which can be efficiently used for further studying the properties of the solution, particularly, in studying its long-time behavior. Using the proposed formalism, smooth solitons as well as non-smooth cuspon solutions are presented.

In this study, an integrable dispersive modified Camassa–Holm equation is considered with the essence of fractional beta derivative. The aforesaid equation is a shallow water equation and a bi-Hamiltonian having an associated isospectral problem of second order. Three diverse techniques namely the extended Jacobi’s elliptic function expansion, the new version of Kudryashov and the E x p a function methods are enforced. A variety of complex solitary wave solutions including, Jacobi’s elliptic function solutions, bright and dark solitons and many other analytical solutions are developed. The obtained results are explicated graphically depending upon the physical and fractional parameters. These results may also be used to illuminate the significance of applied methods to many other related non-linear physical phenomena.

Important analytical methods such as the methods of exp-function, rational hyperbolic method (RHM) and sec–sech method are applied in this paper to solve fractional nonlinear partial differential equations (FNLPDEs) with a truncated [Formula: see text]-fractional derivative (TMFD), which consist of exponential terms. A general modified fractional Degasperis–Procesi–Camassa–Holm equation (GM-FDP-CHE) is investigated with TMFD. The exp-function method is also applied to derive a variety of traveling wave solutions (TWSs) with distinct physical structures for this nonlinear evolution equation. The RHM is used to obtain single-soliton solutions for this equation. The sec–sech method is used to derive multiple-soliton solutions of the GM-FDP-CHE. These techniques can be implemented to find various differential equations exact solutions arising from problems in engineering. The analytical solution of the [Formula: see text]-fractional heat equation is found. Graphical representations are also given.

In this paper, a Lie symmetry method is used for the nonlinear generalized Camassa–Holm equation and as a result reduction of the order and computing the conservation laws are presented. Furthermore,μ-symmetry andμ-conservation laws of the generalized Camassa–Holm equation are obtained.

In this paper, we mainly devote to investigate a weakly dissipative shallow water wave model. By overcoming the difficulties caused by the complicated higher-order nonlinear terms and dissipative t...

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.

In this paper, we first establish the local well-posedness and the rotational speed limit for the rotation Camassa–Holm equation modelling the equatorial water waves with the weak Coriolis effect i...

In this paper, we deal with the existence of global mild solutions and asymptotic behavior to the viscous Camassa–Holm equation in the locally uniform spaces. First we establish the global well-posedness for the Cauchy problem of viscous Camassa–Holm equation in $${\mathbb {R}}^1$$ for any initial data $$u_0\in {\dot{H}}^1_U({\mathbb {R}}^1).$$ Then we study the long time dynamical behavior of non-autonomous viscous Camassa–Holm equation on $${\mathbb {R}}^1$$ with a new class of external forces and show the existence of $$(H^1_U({\mathbb {R}}^1),H^1_\phi ({\mathbb {R}}^1))$$ -uniform(w.r.t. $$g\in \mathcal {H}_{L^2_U({\mathbb {R}}^1)}(g_0)$$ ) attractor $$\mathcal {A}_{\mathcal {H}_{L^2_U({\mathbb {R}}^1)}(g_0)}$$ with locally uniform external forces being translation uniform bounded but not translation compact in $$L_b^2({\mathbb {R}};L^2_U({\mathbb {R}}^1))$$ .

In this paper, we study the periodic Cauchy problem for a mathematical model of the equatorial water waves propagating mainly in one direction with the weak Coriolis effect due to the Earth's rotation, which reduces to the Camassa-Holm equation as the Coriolis effect vanishes. We first prove that this model is well-posed in the sense of Hadamard when initial data belongs to the Sobolev space H s ( T ) with s > 3 / 2 . Then we show that the well-posedness is sharp in the sense that the continuity of the data-to-solution map is not better than continuous, whose proof rests upon the method of approximate solutions and the well-posedness estimate. However, if H s ( T ) is equipped with a weaker H r -topology for 0 ≤ r s , we demonstrate that the data-to-solution map is Holder continuous with the exponent depending on r and s. Finally, we establish a Cauchy-Kowalevski type theorem for this model.

In this paper, we mainly devote to study an integrable generalized Camassa–Holm equation with both quadratic and cubic nonlinearity proposed by Xia, Qiao and Li [An integrable system with peakon, c...

This paper studies the decay properties of solutions around the traveling wave solutions for Cauchy problem of the second-order Camassa–Holm equation. Applying the extended pseudo-conformal transformation methods, appearing the relevant works on the generalized Korteweg–de Vries equation (KdV) and nonlinear Schrodinger equation (NLS) from Martel and Merle, the solution is controlled by the decaying function with exponential speed, corresponding to the initial data and its second derivative with exponential decay.

Under investigation in this work is a nonlinear shallow water equation, Camassa–Holm equation, which is very important in fluid dynamics, nonlinear dynamics and physical application. The new equivariant moving frames method is implemented to obtain a finite generating set of differential invariants, recurrence relations, and syzygies among the generating differential invariants, for Lie symmetry group of Camassa–Holm equation. This method is very efficient, only using the infinitesimal determining equations and choice of cross-section normalization. The results are useful support for describing the invariant properties and trend of physical, oceanic or atmospheric motion.

In the present paper, we derive generalized symmetries of order three of the Camassa–Holm equation by infinite prolongation of a generalized vector field and applying infinitesimal symmetry criterion. In addition, one-dimensional optimal system of Lie subalgebras investigated by applying the adjoint representation. Furthermore, determining equation for multipliers and the 2- dimensional homotopy formula employed to construct higher–order conservation laws for the Camassa–Holm equation.

In this paper, we study orbital stability of peakons for the generalized modified Camassa–Holm (gmCH) equation, which is a natural higher-order generalization of the modified Camassa–Holm (mCH) equation, and admits Hamiltonian form and single peakons. We first show that the single peakon is the usual weak solution of the PDEs. Some sign invariant properties and conserved densities are presented. Next, by constructing the corresponding auxiliary function h ( t , x ) and establishing a delicate polynomial inequality relating to the two conserved densities with the maximal value of approximate solutions, the orbital stability of single peakon of the gmCH equation is verified. We introduce a new approach to prove the key inequality, which is different from that used for the mCH equation. This extends the result on the stability of peakons for the mCH equation (Qu et al. 2013) [36] successfully to the higher-order case, and is helpful to understand how higher-order nonlinearities affect the dispersion dynamics.

We consider the Camassa–Holm (CH) equation, a nonlinear dispersive wave equation that models one-way propagation of long waves of moderately small amplitude. We discretize in space the periodic initial-value problem for CH (written in its original and in system form), using the standard Galerkin finite element method with smooth splines on a uniform mesh, and prove optimal-order $$L^{2}$$ -error estimates for the semidiscrete approximation. Using the fourth-order accurate, explicit, “classical” Runge–Kutta scheme for time-stepping, we construct a highly accurate, stable, fully discrete scheme that we employ in numerical experiments to approximate solutions of CH, mainly smooth travelling waves and nonsmooth solitons of the ‘peakon’ type.

In this paper, we consider the Cauchy problem for the fractional Camassa–Holm equation which models the propagation of small-but-finite amplitude long unidirectional waves in a nonlocally and nonlinearly elastic medium. Using Kato’s semigroup approach for quasilinear evolution equations, we prove that the Cauchy problem is locally well-posed for data in $$H^{s}({\mathbb {R}})$$ , $$s>{\frac{5}{2}}$$ .

In this paper, we develop two linear conservative Fourier pseudo-spectral schemes for the Camassa–Holm equation. We first apply the Fourier pseudo-spectral method in space for the Camassa–Holm equation to arrive at a spatial semi-discretized system in which a corresponding discrete momentum conservation law is preserved. Then we employe the linear-implicit Crank–Nicolson scheme and the leap-frog scheme for the semi-discrete system, respectively. The two new fully discrete methods are proved to conserve the discrete momentum conservation law of the original system, which implies the numerical solutions are bounded in the discrete L∞ norm. Furthermore, the proposed methods are unconditionally stable, second order in time and high order in space, and uniquely solvable. Numerical experiments are presented to show the convergence property as well as the efficiency and accuracy of the new schemes. The proposed methods in this paper could be readily utilized to design linear momentum-preserving numerical approximations for many other Hamiltonian PDEs.