Review
Mathematics, Applied
XiaoHua Liu
Summary: The exact solitary wave solutions of the simplified modified Camassa-Holm equation with any power are investigated using the method of undetermined coefficient and the qualitative theory of planar dynamical system. The existence and number of bell solitary wave solutions, kink solitary wave solutions, and periodic wave solutions are analyzed using Maple software and phase portraits. New exact expressions for bell solitary wave solutions and kink solitary wave solutions are obtained. The orbital stability of the wave solutions is discussed and the wave speed intervals for orbital stability and instability are determined. Numerical simulations are performed to verify the results and visualize the orbital stability.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2022)
Article
Mathematics, Applied
Shuguan Ji, Yonghui Zhou
Summary: This paper investigates the global solutions for the modified Camassa-Holm equation. The global existence and uniqueness of strong solutions for the initial datum u(0)(x) belonging to H-s(R) with s>3/2 are obtained. Additionally, norm estimates for strong solutions and its momentum density are provided. Moreover, by utilizing the compensated compactness method and regularization technique, the global existence and uniqueness of weak solutions for the initial datum u(0)(x) belonging to H-1(R) are established.
ZAMM-ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK
(2022)
Article
Mathematics, Applied
Jiaqi Yang
Summary: This paper improves the results of Gan et al. by establishing regularity criteria for the fractional Camassa-Holm equations and obtaining some corollaries. Additionally, for n=3, the authors prove the boundedness of suitable weak solutions.
ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK
(2022)
Article
Mathematics, Applied
Lina Chen, Chunxia Guan
Summary: This paper investigates the Cauchy problem of the generalized Camassa-Holm equation, proving the existence of global strong solutions and the existence and uniqueness of global weak solutions under a certain sign condition of the initial data.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2021)
Article
Mathematics, Applied
Zheng Dou, Kexin Luo
Summary: This paper investigates the nonlinear rotation-Camassa-Holm equation with the Coriolis effect, which physically depicts the motion of equatorial water waves. With the help of the viscous approximation tool, upper bound estimates for the space derivative of the viscous solution and high order integrable estimates for the time-space variables are obtained. Using these two estimates, the existence of H1(R) global weak solutions to the rotation-Camassa-Holm model is proven.
Article
Mathematics, Applied
Byungsoo Moon
Summary: The paper discusses the nonexistence of periodic peaked traveling wave solution for rotation-Camassa-Holm equation. This equation differs from others such as Camassa-Holm equation, modified Carnassa-Holm equation, and Novikov equation in terms of having no nontrivial periodic Camassa-Holm peaked solution.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2021)
Article
Mathematics
Giuseppe Maria Coclite, Lorenzo di Ruvo
Summary: Camassa-Holm type equations are used as models for the unidirectional propagation of shallow water waves over a flat bottom. They also describe finite length, small amplitude radial deformation waves in cylindrical compressible hyperelastic rods. Under appropriate assumptions on the initial data, time T, and equation coefficients, we prove the well-posedness of the classical solutions for the Cauchy problem.
Article
Mathematics, Applied
Giuseppe Maria Coclite, Lorenzo di Ruvo
Summary: This paper investigates the evolution of shallow water waves described by the high order Camassa-Holm equation and the manifold of smooth orientation-preserving diffeomorphisms of the unit circle in the plane S1. The well-posedness of classical solutions for the Cauchy problem associated with these equations is proven.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Physics, Mathematical
Zhaonan Luo, Zhijun Qiao, Zhaoyang Yin
Summary: This paper presents globally conservative solutions to the Cauchy problem for the modified Camassa-Holm equation by transforming the equation into an equivalent semi-linear system, giving global solutions through the standard ordinary differential equation theory and conservation law, and obtaining the solutions back in the original variables.
JOURNAL OF MATHEMATICAL PHYSICS
(2021)
Article
Mathematics
Shuguan Ji, Yonghui Zhou
Summary: In this paper, we focus on weakly dissipative periodic Camassa-Holm type equations with quadratic and cubic non-linearities. We establish local well-posedness of solutions in the Sobolev space, analyze the relation between the behavior of u(x) and wave breaking phenomena of solutions, and provide a sufficient condition on initial data for wave breaking to occur. Additionally, we prove the global existence of solutions.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2022)
Article
Mathematics, Interdisciplinary Applications
Shilong Huang, Hongmin Li
Summary: We propose a unified approach to establish Darboux-Backlund transformations of the Camassa-Holm (CH) type equations. The method simplifies the approach to obtain soliton solutions of the CH equation, and has been successfully applied to other CH type equations.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Materials Science, Multidisciplinary
Asim Zafar, M. Raheel, Kamyar Hosseini, Mohammad Mirzazadeh, Soheil Salahshour, Choonkil Park, Dong Yun Shin
Summary: This study investigates an integrable dispersive modified Camassa-Holm equation with fractional beta derivative, utilizing techniques such as extended Jacobi's elliptic function expansion, the new version of Kudryashov, and the Exp(a) function methods. Various complex solitary wave solutions are developed, including Jacobi's elliptic function solutions, bright and dark solitons, and others, with graphical explanations based on physical and fractional parameters. These results can illuminate the significance of applied methods in studying related nonlinear physical phenomena.
RESULTS IN PHYSICS
(2021)
Article
Mathematics
Xinyu Tu, Chunlai Mu, Shuyan Qiu
Summary: This article investigates the Lipschitz metric of conservative weak solutions for the rotation-Camassa-Holm equation by defining a Finsler-type norm on the tangent space and proving a generic regularity result. The metric is first established for smooth solutions and then extended to general weak solutions.
ACTA MATHEMATICA SCIENTIA
(2021)
Article
Mathematics, Applied
Jinlu Li, Yanghai Yu, Weipeng Zhu
Summary: The aim of this paper is to answer the question left in Gui and Liu (2015). We prove that given initial data u0 ∈ Hs(R) with s > 32 and for some T > 0, the solution of the Camassa-Holm equation converges strongly in L∞(0,T; Hs(R)) to the inviscid Burgers equation as the filter parameter α tends to zero.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics, Applied
Tomasz Cieslak, Wojciech Krynski
Summary: Geometric tools were applied to study the dynamics of two- and three-peakon solutions of the Camassa-Holm equation. New proofs of the solutions' asymptotic behavior were provided, recovering well-known collision conditions. Additionally, the Gauss curvature (in the two-peakon case) and sectional curvature (in the three-peakon case) of corresponding manifolds were computed.
