Degasperis Procesi Research Articles

Local Isometric Immersions and Breakdown of Manifolds Determined by Cauchy Problems of the Degasperis–Procesi Equation

Local Isometric Immersions and Breakdown of Manifolds Determined by Cauchy Problems of the Degasperis–Procesi Equation

Higher‐order integrable models for oceanic internal wave–current interactions

AbstractIn 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.

The stability of smooth solitary waves for the [formula omitted]-family of Camassa–Holm equations

The b-family of Camassa–Holm (b-CH) equation is a one-parameter family of PDEs, which includes the completely integrable Camassa–Holm and Degasperis–Procesi equations but possesses different Hamiltonian structures. Motivated by this, we study the existence and the orbital stability of the smooth solitary wave solutions with nonzero constant background to the b-CH equation for the special case b=1, whose the Hamiltonian structure is different from that of b≠1. We establish a connection between the stability criterion for the solitary waves and the monotonicity of a singular integral along the corresponding homoclinic orbits of the spatial ODEs. We verify the latter analytically using the framework for the monotonicity of period function of planar Hamiltonian systems, which shows that the smooth solitary waves are orbitally stable. In addition, we find that the existence and orbital stability results for 0<b<1 are similar to that of b>1, particularly the stability criteria are the same. Finally, combining with the results for the case b>1, we conclude that the solitary waves to the b-CH equation is structurally stable under the variation of the parameter b>0.

$H^{2}$ solutions for a Degasperis–Procesi-type equation

The Degasperis–Procesi equation can be regarded as a model for shallow water dynamics and its asymptotic accuracy is the same as for the Camassa–Holm equation. Here we consider a generalization of the Degasperis–Procesi equation and prove the existence and uniqueness of H^{2} solutions for the Cauchy problem.

Spectral and linear stability of peakons in the Novikov equation

AbstractThe Novikov equation is a peakon equation with cubic nonlinearity, which, like the Camassa–Holm and the Degasperis–Procesi, is completely integrable. In this paper, we study the spectral and linear stability of peakon solutions of the Novikov equation. We prove spectral instability of the peakons in . To do so, we start with a linearized operator defined on and extend it to a linearized operator defined on weaker functions in . The spectrum of the linearized operator in is proven to cover a closed vertical strip of the complex plane. Furthermore, we prove that the peakons are spectrally unstable on and linearly and spectrally stable on . The result on is in agreement with previous work about linear instability and our result on is in line with past work on orbital stability.

Analyzing fuzzy fractional Degasperis–Procesi and Camassa–Holm equations with the Atangana–Baleanu operator

Abstract This article presents a new approach for solving the fuzzy fractional Degasperis–Procesi (FFDP) and Camassa–Holm equations using the iterative transform method (ITM). The fractional Degasperis–Procesi (DP) and Camassa–Holm equations are extended from the classical DP and Camassa–Holm equations by incorporating fuzzy sets and fractional derivatives. The ITM is a powerful technique widely used for solving nonlinear differential equations. This approach transforms the fuzzy fractional differential equations into a series of ordinary differential equations, which are then solved iteratively using a recursive algorithm. Numerical simulations demonstrate the proposed approach’s accuracy and effectiveness. The results show that the ITM provides an efficient and accurate method for solving the FFDP and Camassa–Holm equations. The proposed method can be extended to solve other fuzzy fractional differential equations.

Soliton solutions to a negative flow in the Degasperis–Procesi hierarchy and its short-wave limit

Soliton solutions to a negative flow in the Degasperis–Procesi hierarchy and its short-wave limit

Highest waves for fractional Korteweg–De Vries and Degasperis–Procesi equations

Highest waves for fractional Korteweg–De Vries and Degasperis–Procesi equations

A New Iterative Method for Investigating Modified Camassa–Holm and Modified Degasperis–Procesi Equations within Caputo Operator

In this paper, we employ the new iterative method to investigate two prominent nonlinear partial differential equations, namely the modified Camassa–Holm (mCH) equation and the modified Degasperis–Procesi (mDP) equation, both within the framework of the Caputo operator. The mCH and mDP equations are fundamental in studying wave propagation and soliton dynamics, exhibiting complex behavior and intriguing mathematical structures. The new iterative method (NIM), a powerful numerical technique, is utilized to obtain analytical and numerical solutions for these equations, offering insights into their dynamic properties and behavior. Through systematic analysis and computation, we unveil the unique features of the mCH and the mDP equations, shedding light on their applicability in various scientific and engineering domains. This research contributes to the ongoing exploration of nonlinear wave equations and their solutions, emphasizing the versatility of the new iterative method in tackling complex mathematical problems. Numerical results and comparative analyses are presented to validate the effectiveness of the new iterative method in solving these equations, highlighting its potential for broader applications in mathematical modeling and analysis.

Probabilistic global well-posedness to the nonlocal Degasperis–Procesi equation

We study the almost sure existence of a global solution of the nonlocal Degasperis–Procesi equation. After a suitable randomization, we obtain the local existence and uniqueness of the mild solution for a large set of the supercritical initial data. Then, we establish an energy estimate of the solution and get the global existence in probability. The methods can be used to get the well-posedness of fractional Burgers equation with the supercritical initial data.

Bäcklund transformations of multi-component Boussinesq and Degasperis–Procesi equations

The finding of new integrable coupling systems has become an important area of research in mathematical physics and their study will aid in the classification of multi-component integrable systems. A basic method for generating integrable coupling systems is algebraic expansion, for example, the Frobenius algebra, the Lie algebra, the superalgebra, and so on. In this paper, we introduce a Frobenius Boussinesq equation based on the Frobenius algebra, and then we present a Lax pair of it. It follows that we give a Bäcklund transformation of the Frobenius Boussinesq equation. Furthermore, the lattice equation of the Frobenius Boussinesq equation is presented by using three Bäcklund transformations, and then obtain the exact solutions. Additionally, we obtain the conservation laws of the Frobenius Boussinesq equation via the Bäcklund transformation. Strongly coupled and weakly coupled systems physically represent strong and weak interactions, respectively. In this paper, we introduce a weakly coupled Degasperis–Procesi (DP) equation, and construct a Lax pair of it. In addition, the Bäcklund transformation and superposition principle are applied to investigate the weakly coupled DP equation. We also obtain the conservation laws of the weakly coupled DP equation. Then, we introduce a strongly coupled DP equation, and use the same method to study the strongly coupled DP equation. The exact solutions of these two equations are obtained. Moreover, we introduce a [Formula: see text]-DP equation. Considering the superposition principle, we obtain the solution of an associated [Formula: see text]-DP equation by using Bäcklund transformations. These new multi-component integrable systems can enrich the existing integrable models and possibly describe new nonlinear phenomena.

Peakons and Persistence Properties of Solution for the Interacting System of Popowicz

This paper focuses on a two-component interacting system introduced by Popowicz, which has the coupling form of the Camassa–Holm and Degasperis–Procesi equations. Using distribution theory, single peakon solutions and several double peakon solutions of the system are described in an explicit expression. Moreover, dynamic behaviors of several types of double peakon solutions are illustrated through figures. In addition, the persistence properties of the solutions to the Popowicz system in weighted Lp spaces is considered via a large class of moderate weights.

Remarks on strong global solutions of the [formula omitted]-equation

A bounded first-order derivative with respect to x of a solution to the b-equation, which encompasses the Camassa–Holm and Degasperis–Procesi equations, serves as a crucial condition that triggers a process leading to the establishment of bounds for derivatives of the solution up to an order determined by the regularity of the initial data. Consequently, this condition ensures the global existence of strong solutions in Sobolev spaces.

Higher dimensional Degasperis–Procesi–Camassa–Holm systems

The (1+1)-dimensional models can be applied to derive corresponding higher dimensional models by a deformation algorithm with the help of their conservation laws, their Lax integrability can be proved in the same way. In this paper, we modify the original deformation algorithm such that the conserved density should not be derivative dependent. The modified deformation algorithm is mainly applied to the (1+1)-dimensional Degasperis-Procesi equation and Camassa–Holm equation to establish some brand-new (N+1)-dimensional coupled Degasperis–Procesi–Camassa–Holm systems and their (1+1)-dimensional reductions, whose Lax pairs have been given. On the other hand, we can find a new type of peakon solution of the (2+1)-dimensional deformed Degasperis-Procesi equation.

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.

Ill-posedness for the two component Degasperis–Procesi equation in critical Besov space

Ill-posedness for the two component Degasperis–Procesi equation in critical Besov space

On the applications of collocation method for numerically analyzing the nonlinear Degasperis–Procesi and Benjamin–Bona–Mahony equations

This research uses the Sinc Collocation method to numerically examine the Degasperis–Procesi and Benjamin–Bona–Mahony equations, achieving a high level of precision and accuracy on computational grounds with a variety of mesh points. The proposed technique involves global collocation using Sinc bases function (SBF) as an activation function. Initially, the time derivative discretization has been performed through finite difference scheme and the remaining spatial derivatives are approximated by [Formula: see text]-weighted scheme to minimize the complexity in numerical computations. The main contribution of research is in modifying the proposed scheme to overcome the complexity involved in the whole process of finding solutions with a cardinal expansion of Sinc functions in nonlinear and singular systems. The solutions of Degasperis–Procesi and Benjamin–Bona–Mahony stiff nonlinear systems are illustrated numerically and graphically and the validation and verification of the obtained results are examined through stability analysis by calculating the spectral radius by bounding the value of [Formula: see text]. A comparative study is presented with some of the well-known numerical techniques, which assured the accuracy and reliability of our technique.

Efficient computational approaches for fractional-order Degasperis-Procesi and Camassa–Holm equations

In this study, we present a comprehensive comparison of two powerful analytical techniques, Aboodh Adomian decomposition method (AADM) and homotopy perturbation transform method (HPTM), for obtaining series solutions of nonlinear partial differential equations, specifically focusing on Camassa–Holm (CH) and Degasperis–Procesi (DP) equations. These equations are widely used to describe various nonlinear wave phenomena in fluid mechanics, optical fibers, and other applications. By applying both AADM and HPTM to CH and DP equations, we demonstrate the effectiveness and efficiency of each method in terms of accuracy, convergence, and computational complexity. Furthermore, we provide a detailed analysis of the series solutions obtained by each method and discuss their respective advantages and limitations. The results reveal that both methods are capable of providing accurate and convergent series solutions for the considered equations. However, AADM shows a slightly better performance in terms of convergence rate and ease of implementation, making it a preferable choice for solving CH and DP equations. This comparative study serves as a useful reference for researchers and practitioners working in the field of nonlinear partial differential equations and their applications.

Finite-Difference Hermite WENO Scheme for Degasperis-Procesi Equation

We present a fifth-order finite-difference Hermite weighted essentially non-oscillatory (HWENO) method for solving the Degasperis–Procesi (DP) equation in this paper. First, the DP equation can be rewritten as a system of equations consisting of hyperbolic equations and elliptic equations by introducing an auxiliary variable, since the equations contain nonlinear higher order derivative terms. Then, the auxiliary variable equations are solved using the Hermite interpolation, while the HWENO scheme is performed for the hyperbolic equations. Compared with the popular WENO-type scheme, the most important feature of the HWENO scheme mentioned in this paper is the compactness of its spatial reconstruction stencil, which can achieve the fifth-order accuracy of the expected design with only three points, while the WENO method requires five points. Finally, we demonstrate the effectiveness of the HWENO method in various aspects by conducting some benchmark numerical tests.

A new 3-component Degasperis–Procesi hierarchy

We propose a new 3-component Degasperis–Procesi (3-DP) system. By investigating the bi-Hamiltonian structure of the corresponding hierarchy, we find that the Hamiltonian functionals of the 3-DP hierarchy are local in the negative direction and homogeneous in both directions. We show that the 3-DP system is related to the first negative flow in a modified Yajima–Oikawa hierarchy via a reciprocal transformation. With the help of this reciprocal transformation and a Darboux transformation of the associated 3-DP system, we construct a Bäcklund transformation for the 3-DP system. Furthermore, we present a short-wave model of the 3-DP system.