Search

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]

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]