相关论文: A dispersive regularization for the modified Camas…

We present a comprehensive discretization scheme for linear and nonlinear stochastic differential equations (SDEs) driven by either Brownian motions or $\alpha$-stable processes. Our approach utilizes compound Poisson particle…

A new two-component system with cubic nonlinearity and linear dispersion: \begin{eqnarray*} \left\{\begin{array}{l} m_t=bu_{x}+\frac{1}{2}[m(uv-u_xv_x)]_x-\frac{1}{2}m(uv_x-u_xv), \\ n_t=bv_{x}+\frac{1}{2}[ n(uv-u_xv_x)]_x+\frac{1}{2}…

We construct a global continuous semigroup of weak periodic conservative solutions to the two-component Camassa-Holm system, $u_t-u_{txx}+\kappa u_x+3uu_x-2u_xu_{xx}-uu_{xxx}+\eta\rho\rho_x=0$ and $\rho_t+(u\rho)_x=0$, for initial data…

We develop KAM theory close to an elliptic fixed point for quasi-linear Hamiltonian perturbations of the dispersive Degasperis-Procesi equation on the circle. The overall strategy in KAM theory for quasi-linear PDEs is based on Nash-Moser…

We present a compact parametric representation of the smooth bright multisolution solutions for the modified Camassa-Holm (mCH) equation with cubic nonlinearity. We first transform the mCH equation to an associated mCH equation through a…

In this paper we mainly study the Cauchy problem for a generalized Camassa-Holm equation. First, by using the Littlewood-Paley decomposition and transport equations theory, we establish the local well-posedness for the Cauchy problem of the…

In this paper, we rigorously investigate the truncation method for the Cauchy problem of Helmholtz equations which is widely used to model propagation phenomena in physical applications. The method is a well-known approach to the…

Developments in numerical methods for problems governed by nonlinear partial differential equations underpin simulations with sound arguments in diverse areas of science and engineering. In this paper, we explore the regularization method…

An explicit reciprocal transformation between a 2-component generalization of the Camassa-Holm equation, called the 2-CH system, and the first negative flow of the AKNS hierarchy is established, this transformation enables one to obtain…

The Cauchy problem for the inhomogeneous Helmholtz equation with non-uniform refraction index is considered. The ill-posedness of this problem is tackled by means of the variational form of mollification. This approach is proved to be…

We provide a closed Poisson algebra involving the Ragnisco--Bruschi generalization of peakon dynamics in the Camassa--Holm shallow-water equation. This algebra is generated by three independent matrices. From this presentation, we propose a…

We analyze a Fourier spectral Galerkin method for the fractional Camassa-Holm (fCH) equation involving a fractional Laplacian of exponent $\alpha \in [1,2]$ with periodic boundary conditions. The semi-discrete scheme preserves both mass and…

This work is devoted to the studies of a Hamilton--Jacobi equation with a quadratic and degenerate Hamiltonian, which comes from the dynamics of a multipeakon in the Camassa--Holm equation. It is given by a quadratic form with a singular…

This paper develops a hybridizable discontinuous Galerkin method for the two-dimensional Camassa--Holm--Kadomtsev--Petviashvili equation. The method employs Cartesian meshes with tensor-product polynomial spaces, enabling separate treatment…

The Novikov equation is a Camassa-Holm type equation with cubic nonlinearity. This paper aims to prove the asymptotic stability of peakons solutions under $H^1(\mathbb{R})$-perturbations satisfying that their associated momentum density…

The paper is devoted to investigating a Cauchy problem for nonlinear elliptic PDEs in the abstract Hilbert space. The problem is hardly solved by computation since it is severely ill-posed in the sense of Hadamard. We shall use a modified…

A generalized two-component model with peakon solutions is proposed in this paper. It allows an arbitrary function to be involved in as well as including some existing integrable peakon equations as special reductions. The generalized…

We consider a two-component Hamiltonian system of partial differential equations with quadratic nonlinearities introduced by Popowicz, which has the form of a coupling between the Camassa-Holm and Degasperis-Procesi equations. Despite…

In this paper we derive generalized forms of the Camassa-Holm (CH) equation from a Boussinesq-type equation using a two-parameter asymptotic expansion based on two small parameters characterizing nonlinear and dispersive effects and…

In this paper we introduce a two-component system, depending on a parameter $b$, which generalises the Camassa-Holm ($b=1$) and Novikov equations ($b=2$). By investigating its Lie algebra of classical and higher symmetries up to order $3$,…