Modified Camassa Holm Equation Research Articles

Integrable Semi-Discretization for a Modified Camassa-Holm Equation with Cubic Nonlinearity

Integrable Semi-Discretization for a Modified Camassa-Holm Equation with Cubic Nonlinearity

Diverse soliton wave profile analysis in ion-acoustic wave through an improved analytical approach

In engineering and applied sciences, several physical phenomena can be more precisely characterized by employing nonlinear fractional partial differential equations. The primary goal of this research is to examine the traveling wave solution in closed form for the nonlinear acoustic wave propagation model known as the space-time fractional simplified modified Camassa–Holm equation, which is used to explained the unidirectional propagation of shallow-water waves with non-hydrostatic pressure and explains the dispersion properties of numerous phenomena like fluid flow, control theory, liquid drop patterning in plasma, acoustics, fusion and fission processes, etc. The utmost potential approach, namely the new auxiliary equation technique, is applied for analyzing the space-time nonlinear fractional simplified modified Camassa-Holm equation in the logic of the newest established truncated M-fractional derivative. The fractional partial differential equations have been transformed to the ordinary differential equation using the complex wave transformation in the sense of truncated M-fractional derivative. A variety of soliton solutions, including anti-kink, single soliton, anti-bell, bell, kink, multiple soliton, double soliton, peakon shape, anti-peakon shape, singular-kink, compacton shape, periodic shape, and so many, which are displayed in the diagram of 3D and contour plots by taking into account a number of various parameters. It is essential to point out that all derived outcomes are directly compared to the original solutions to certify their exactness. Results show that the used scheme is capable, simple, and straightforward and can be useful to a variety of complex phenomena. The acquired results are unique for the model equation and could be applied to the analysis of several nonlinear study fields.

Orbital stability of smooth solitons for the modified Camassa-Holm equation

The modified Camassa-Holm equation with cubic nonlinearity is completely integrable and is considered a model for the unidirectional propagation of shallow-water waves. The localized smooth-wave solution exists uniquely, up to translation, within a certain range of the linear dispersive parameter. By constructing conserved H1 and L1 quantities in terms of the momentum variable m, this study demonstrates that the smooth soliton, when regarded as a solution of the initial-value problem for the modified Camassa-Holm equation, is orbitally stable to perturbations in the Sobolev space H3. Furthermore, the global well-posedness of the solution is established for certain initial data in Hs with s≥3.

Transient asymptotics of the modified Camassa–Holm equation

Transient asymptotics of the modified Camassa–Holm equation

Blow-up, global existence and propagation speed for a modified Camassa–Holm equation both dissipation and dispersion in $$H^{s,p}(\mathbb {R})$$

Blow-up, global existence and propagation speed for a modified Camassa–Holm equation both dissipation and dispersion in $$H^{s,p}(\mathbb {R})$$

Uniqueness of conservative solutions to the modified Camassa-Holm equation via characteristics

In this paper, for a given conservative solution, we introduce a set of auxiliary variables tailored to this particular solution, and prove that these variables satisfy a particular semilinear system having unique solutions. In turn, we get the uniqueness of the conservative solution in the original variables.

New Traveling Wave Solutions to the Simplified Modified Camassa–Holm Equation and the Landau-Ginsburg-Higgs Equation

Researchers are interested in the (1+1)-dimensional Camassa-Holm and Landau-Ginzburg-Higgs equations as they allow for the study of unidirectional wave propagation in shallow waters with a flat seabed, as well as nonlinear media exhibiting dispersion systems and superconductivity. This work has effectively developed exact wave solutions to the stated models, which may have significant consequences for characterising the nonlinear dynamical behaviour related to the phenomena. The extended -expansion technique is employed to procure a diverse array of progressive wave solutions characterized by hyperbolic, trigonometric, and rational functions. The solutions are shown as 3D profiles with a variety of shapes, including kink, singular kink, periodic, singular periodic, etc. The physical significance of the solutions is discussed by these plots, and the approach used in this study is considered efficient and capable of finding analytical solutions for the nonlinear models. Dhaka Univ. J. Sci. 72(1): 7-13, 2024 (January)

Rotation number and eigenvalues of two-component modified Camassa–Holm equations

In this paper, we study the spectral problem y1y2x=−1212λm−12λn12y1y2for the two-component modified Camassa–Holm equation, where m,n are two potentials. By introducing the rotation number ρ(λ) and studying its properties, we prove that for any integer k, the periodic or anti-periodic eigenvalues are the endpoints of the interval λ∈R:ρ(λ)=−k2. Moreover, we prove that as nonlinear functionals of potentials, such eigenvalues are continuous in potentials with respect to the weak topologies in the Lebesgue space L1[0,T]. Finally, we apply the trace formula to give some estimates of the periodic eigenvalues when the L1 norms of potentials are given.

Novel solitary wave solutions in dual-mode simplified modified Camassa-Holm equation in shallow water waves

Novel solitary wave solutions in dual-mode simplified modified Camassa-Holm equation in shallow water waves

Some new decay properties of solutions for a modified Camassa–Holm equation with cubic nonlinearity

In this article, we derive some new asymptotic properties of solution for a modified Camassa–Holm equation with cubic nonlinearity in weighted Sobolev spaces. Specifically, we prove that the strong solutions of Eq. (1.2) will maintain the corresponding decay properties provided the initial data ω0(x), ω0,x(x) decay logarithmically, algebraically and exponentially, respectively, which improves and extends the persistence properties of strong solutions to Eq. (1.2) obtained by Wu and Guo in paper [J. Math. Phys. 55, 081504 (2014)]. This method can be used to study the asymptotic decay properties for a class of Camassa–Holm equations.

Orbital stability of the periodic peakons for the higher-order modified Camassa-Holm equation

Orbital stability of the periodic peakons for the higher-order modified Camassa-Holm equation

The new complex travelling wave solutions of the simplified modified camassa holm equation

The new complex travelling wave solutions of the simplified modified camassa holm equation

Integrable generalization of the modified Camassa–Holm equation in [formula omitted] dimensions

We propose a new (2+1)-dimensional modified Camassa–Holm equation by applying a reciprocal transformation to the negative modified Kadomtsev–Patvishvili II equation. We construct multi-soliton solutions of the new equation by combining the reciprocal transformation and the Darboux transformation of the negative modified Kadomtsev–Patvishvili II equation.

Analytical soliton solutions of the beta time-fractional simplified modified Camassa-Holm equation in shallow water wave propagation

In this study, the analytical soliton solutions of the beta time-fractional simplified modified Camassa-Holm equation, a mathematical model used to describe the propagation of shallow water waves characterized by weak dispersion and nonlinearities, is determined. The (G′/G,1/G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({G}^{\\prime}/G,1/G)$$\\end{document}-expansion method, a powerful and reliable technique, is exploited to formulate the soliton solutions for the equation. The process yields a wide range of solutions, including trigonometric, rational, and hyperbolic functions with free parameters. Various original soliton solutions are generated for different parameter values, including bell-shaped, anti-bell-shaped, periodic, compacton, singular bell-shaped, singular periodic and flat kink solitons. To visually comprehend the physical characteristics of the obtained solutions, two- and three-dimensional graphs, as well as contour plots, are plotted. Thorough comparisons with previous results are conducted to ensure the originality of the derived solutions. The insights gained from understanding soliton behavior in shallow water can be helpful in improving tsunami warnings, coastal protection systems, underwater data transmission, submarine cables, and marine sensing networks.

Orbital stability of periodic peakons for the higher-order modified Camassa–Holm equation

Orbital stability of periodic peakons for the higher-order modified Camassa–Holm equation

Solving the modified Camassa–Holm equation via the inverse scattering transform

Solving the modified Camassa–Holm equation via the inverse scattering transform

Solving the modified Camassa-Holm equation via the inverse scattering transform

С помощью преобразования взаимности и ассоциативного уравнения изучается обратная задача рассеяния на основе матричной задачи Римана-Гильберта для модифицированного уравнения Камассы-Холма с ненулевыми граничными условиями на бесконечности. С помощью униформизующей переменной представлены прямая и обратная задачи рассеяния для ассоциированного модифицированного уравнения Камассы-Холма. С помощью преобразования взаимности и формулы восстановления потенциала этого уравнения построено $N$-солитонное решение для модифицированного уравнения Камассы-Холма с ненулевыми граничными условиями. В качестве приложения представлены различные решения как светлого, так и темного типа для модифицированного уравнения Камассы-Холма: гладкие солитонные решения, сингулярные солитонные решения, многозначные сингулярные солитонные решения, а также их взаимодействия.

Numerical Solutions of the Nonlinear Dispersive Shallow Water Wave Equations Based on the Space–Time Coupled Generalized Finite Difference Scheme

This study applies the space–time generalized finite difference scheme to solve nonlinear dispersive shallow water waves described by the modified Camassa–Holm equation, the modified Degasperis–Procesi equation, the Fornberg–Whitham equation, and its modified form. The proposed meshless numerical scheme combines the space–time generalized finite difference method, the two-step Newton’s method, and the time-marching method. The space–time approach treats the temporal derivative as a spatial derivative. This enables the discretization of all partial derivatives using a spatial discretization method and efficiently handles mixed derivatives with the proposed mesh-less numerical scheme. The space–time generalized finite difference method is derived from Taylor series expansion and the moving least-squares method. The numerical discretization process only involves functional data and weighting coefficients on the central and neighboring nodes. This results in a sparse matrix system of nonlinear algebraic equations that can be efficiently solved using the two-step Newton’s method. Additionally, the time-marching method is employed to advance the space–time domain along the time axis. Several numerical examples are presented to validate the effectiveness of the proposed space–time generalized finite difference scheme.

On conservative sticky peakons to the modified Camassa-Holm equation

We use a sticky particle method to show global existence of (energy) conservative sticky N-peakon solutions to the modified Camassa-Holm equation. A dispersion regularization is provided as a selection principle for the uniqueness of conservative N-peakon solutions. The dispersion limit avoids the collision between peakons, and numerical results show that the dispersion limit is exactly the sticky peakons. At last, when the splitting of peakons is allowed, we give an example to show the non-uniqueness of conservative solutions.

Curvature blow-up and peakons for the Camassa–Holm type equation

In this paper we are concerned with a periodic Camassa–Holm type equation, which includes several integrable systems as particular cases such as the Camassa–Holm equation, the Novikov equation and the modified Camassa–Holm equation. By applying the method of characteristics, the blow-up criterion and the precise blow-up scenario are established. Then the curvature blow-up result is presented. Finally, periodic peakons are expressed in an explicit function formula.