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We prove that both local and non-local formulations of the Degasperis–Procesi equation possess a pseudospherical nature. As a result, solutions determined by Cauchy problems with non-trivial initial data and a minimal specific regularity define a coframe for a pseudospherical metric within a designated strip. While the region is entirely described by the initial condition, the one-forms are determined by the corresponding solutions. Moreover, we establish that any non-trivial initial condition yields a second fundamental form, which is locally determined within the strip. The surface may collapse within a finite-height strip when the initial condition leads to wave breaking. We determine conditions for the coframe to be defined on the upper plane. We investigate possible integrability structures related to the triad of fundamental one-forms. [ABSTRACT FROM AUTHOR]

We study nonlocal symmetries of the Degasperis–Procesi equation, which are shown to be closely related to its integrable structure. First, applying the Hamiltonian operator to the gradients of the spectral parameter, we construct nonlocal symmetries of the Kaup–Kupershmidt equation. Next, we show that the nonlocal symmetries can be prolonged to local symmetries for a prolonged system by introducing new dependent variables. Finally, applying the Liouville transformation relating the Degasperis–Procesi and Kaup–Kupershmidt hierarchies, we obtain the corresponding nonlocal symmetries of the Degasperis–Procesi equation. [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 study, we establish the existence and nonexistence of smooth and peaked solitary wave solutions (or periodic) to the Degasperis-Procesi-Kadomtsev-Petviashvili (DP-KP) equation with a weak transverse effect. We have also shown that DP-KP equation possesses periodic shock waves similar to that of the Degasperis-Procesi equation.

In this paper, a reciprocal transformation is applied to connect a negative flow of the Degasperis–Procesi (DP) equation with the Kaup–Kupershmidt (KK) equation. Via Darboux transformation, we obtained multisoliton solutions of a negative flow in the DP hierarchy and its short-wave limit. [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]

This article is concerned with the convergence properties of the Strang splitting method for the Degasperis-Procesi equation, which models shallow water dynamics. The challenges of analyzing splitting methods for this equation lie in the fact that the involved suboperators are both nonlinear. In this paper, instead of building the second order convergence in L 2 for the proposed method directly, we first show that the Strang splitting method has first order convergence in H 2 . In the analysis, the Lie derivative bounds for the local errors are crucial. The obtained first order convergence result provides the H 2 boundedness of the approximate solutions, thereby enabling us to subsequently establish the second order convergence in L 2 for the Strang splitting method. [ABSTRACT FROM AUTHOR]

Main aim of the current study is to examine the outcomes of nonlinear partial modified Degasperis-Procesi equation of arbitrary order by using two analytical methods. Both methods are based on homotopy and a novel adjustment with generalized Laplace transform operator. Nonlinear terms are handled by using He's polynomials. The fractional order modified Degasperis-Procesi (FMDP) equation, is capable to describe the nonlinear aspects of dispersive waves. The Katugampola derivative of fractional order in the caputo type is employed to model this problem. The numerical results and graphical representation demonstrate the efficiency and accuracy of applied techniques.

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This article studies the Stochastic Degasperis-Procesi equation on R with an additive noise. Applying the kinetic theory, and considering the initial conditions in L 2 (R) ∩ L 2 + δ (R) , for arbitrary small δ > 0 , we establish the existence of a global pathwise solution. Restricting to the particular case of zero noise, our result improves the deterministic solvability results that exist in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the local-in-time existence and uniqueness of solution to the Degasperis-Procesi equation in B ∞ , 1 1 (R) . Moreover, we prove that the data-to-solution of this equation is continuous but not uniformly continuous in B ∞ , 1 1 (R) . [ABSTRACT FROM AUTHOR]

The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the DP equation on the real line, extending our previous work on their spectral stability [J. Math. Pures Appl. (9) 142 (2020), pp. 298–314]. The main difficulty stems from the fact that the natural energy space is a subspace of L^3, but the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the L^2-norm, resulting in L^3 higher-order nonlinear terms in the augmented Hamiltonian. But the usual coercivity estimate is in terms of L^2 norm for DP equation, which cannot be used to control the L^3 higher order term directly. The remedy is to observe that, given a sufficiently smooth initial condition satisfying some mild constraint, the L^\infty orbital norm of the perturbation is bounded above by a function of its L^2 orbital norm, yielding the higher order control and the orbital stability in the L^2\cap L^\infty space. [ABSTRACT FROM AUTHOR]

Main aim of the current study is to examine the outcomes of nonlinear partial modified Degasperis-Procesi equation of arbitrary order by using two analytical methods. Both methods are based on homotopy and a novel adjustment with generalized Laplace transform operator. Nonlinear terms are handled by using He's polynomials. The fractional order modified Degasperis-Procesi (FMDP) equation, is capable to describe the nonlinear aspects of dispersive waves. The Katugampola derivative of fractional order in the caputo type is employed to model this problem. The numerical results and graphical representation demonstrate the eficiency and accuracy of applied techniques. [ABSTRACT FROM AUTHOR]

To realize and comprehend the physical phenomena of nonlinear system, exploration of traveling wave solutions plays an important role. Among the class of dispersive PDEs of traveling wave solutions the Degasperis-Procesi (DP) equation comprises high order nonlinear derivatives and is considered as a well known model for shallow water dynamics having similar asymptotic accuracy as for the Camassa-Holm (CH) equation. In this study we investigate solutions of some high order nonlinear dispersive PDEs namely generalized Degasperis-Procesi (DP), Camassa-Holm (CH) and Korteweg-de Vries (KdV) equations by the use of Radial Basis Function (RBF) combined with Finite Differences (RBF-FD) and Pseudo-Spectral (RBF-PS) methods. For the time derivative approximation, the fourth-order Runge-Kutta (RK) technique is accomplished. The efficiency and accuracy of our suggested approaches are demonstrated using examples and results. [ABSTRACT FROM AUTHOR]

In the present study, we construct a new blow-up of strong solution to show wave breaking for the well-known Degasperis–Procesi equation. Unlike the previous related results for the shallow water wave models, no conservation law is used here. [ABSTRACT FROM AUTHOR]

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.

The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model as an asymptotic approximation for the unidirectional propagation of shallow water waves. This work is to establish the L 2 ∩ L ∞ orbital stability of a wave train containing N smooth solitons which are well separated. The main difficulties stem from the subtle nonlocal structure of the DP equation. One consequence is that the energy space of the DE equation based on the conserved quantity induced by the translation symmetry is only equivalent to the L 2 -norm, which by itself can not bound the higher-order nonlinear terms in the Lagrangian. Our remedy is to introduce a priori estimates based on certain smooth initial conditions. Moreover, another consequence is that the nonlocal structure of the DP equation significantly complicates the verification of the monotonicity of local momentum and the positive definiteness of a refined quadratic form of the orthogonalized perturbation. [ABSTRACT FROM AUTHOR]

An efficient and robust hybrid scheme based on radial basis functions (RBFs) and finite difference is implemented to solve nonlinear partial differential equations (PDEs). In the proposed method, first-order finite difference and the θ -weighted scheme are coupled for temporal discretization, while RBFs are used for spatial discretization. The key feature of the scheme is to use quasilinearization with a collocation approach to reduce nonlinear PDEs to linear algebraic system of equations which are easy to solve. Stability analysis is carried out to examine the spectral radius of the amplification matrix versus the shape parameter. Furthermore, the scheme is applied to solve some nonlinear PDEs including Fornberg–Whitham, Degasperis–Procesi equations, and their modified forms. Efficiency of the proposed technique is demonstrated via different error norms and conservative quantities. Moreover, the computed results are compared with the existing results in the literature. Simulations reveal better accuracy for the considered problems. [ABSTRACT FROM AUTHOR]

The Degasperis–Procesi (DP) equation is a significant model of shallow-water waves, which has been well investigated in the current study. Notably, the existence of solitary wave solutions without perturbation has been first proved. However, the persistence of solitary wave solutions of the perturbed DP equation by employing the geometric singular perturbation theory and the Melnikov method has been analyzed. Therefore, the perturbed DP equation possesses a solitary wave solution from the demonstration of having a homoclinic orbit. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem of the Degasperis–Procesi equation with weak dissipation. We establish a new local-in-space blow-up criterion of the dissipative Degasperis–Procesi equation on line R and on circle S, respectively. [ABSTRACT FROM AUTHOR]

In this paper we consider the Degasperis-Procesi equation, which is an approximation to the incompressible Euler equation in shallow water regime. First we provide the existence of solitary wave solutions for the original DP equation and the general theory of geometric singular perturbation. Then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the geometric singular perturbation theory and invariant manifold theory. According to the relationship between solitary wave and homoclinic orbit, the Degasperis-Procesi equation is transformed into the slow-fast system by using the traveling wave transformation. It is proved that the perturbed equation also has a homoclinic orbit, which corresponds to a solitary wave solution of the delayed Degasperis-Procesi equation. [ABSTRACT FROM AUTHOR]

For s < 3 / 2 , it is known (see [24]) that the Cauchy problem for the b-family of equations is ill-posed in Sobolev spaces H s when b > 1. The proof of ill-posedness depends naturally on the value of b , and is based on the construction of peakon–antipeakon solutions with interesting properties which allows to make conclusion on ill-posedness. In this context the construction of such type of the solution for b < 1 is very attractive problem which help shed light on the ill-posedness problem in this case. Thus, in the present paper we consider the ill-posedness of the b-family of equations with additional term for insufficiently investigated case b < 1 on the line. [ABSTRACT FROM AUTHOR]

The Degasperis-Procesi equation is an approximating model of shallow-water wave propagating mainly in one direction to the Euler equations. Such a model equation is analogous to the Camassa-Holm approximation of the two-dimensional incompressible and irrotational Euler equations with the same asymptotic accuracy, and is completely integrable with the bi-Hamiltonian structure. In the present study, we establish existence and spectral stability results of localized smooth solitons to the Degasperis-Procesi equation on the real line. The stability proof relies essentially on refined spectral analysis of the linear operator corresponding to the second-order variational derivative of the local Hamiltonian of the Degasperis-Procesi equation. [ABSTRACT FROM AUTHOR]

We consider the Bäcklund transformation for the Degasperis-Procesi (DP) equation. Using the reciprocal transformation and the associated DP equation, we construct the Bäcklund transformation for the DP equation involving both dependent and independent variables. We also obtain the corresponding nonlinear superposition, which we use together with the Bäcklund transformation to derive some soliton solutions of the DP equation. [ABSTRACT FROM AUTHOR]

We propose a new approach for calculating multisoliton solutions of the Degasperis–Procesi equation and its shortwave limit by combining a reciprocal transformation with the Darboux transformation of the negative flow of the Kaup–Kupershmidt hierarchy. In particular, different specifications of the soliton parameters lead to two different types of soliton solutions of the Degasperis–Procesi equation. [ABSTRACT FROM AUTHOR]

A class of nonlocal symmetries of the Camassa-Holm type equations with bi-Hamiltonian structures, including the Camassa-Holm equation, the modified Camassa-Holm equation, Novikov equation and Degasperis-Procesi equation, is studied. The nonlocal symmetries are derived by looking for the kernels of the recursion operators and their inverse operators of these equations. To find the kernels of the recursion operators, the authors adapt the known factorization results for the recursion operators of the KdV, modified KdV, Sawada-Kotera and Kaup-Kupershmidt hierarchies, and the explicit Liouville correspondences between the KdV and Camassa-Holm hierarchies, the modified KdV and modified Camassa-Holm hierarchies, the Novikov and Sawada-Kotera hierarchies, as well as the Degasperis-Procesi and Kaup-Kupershmidt hierarchies. [ABSTRACT FROM AUTHOR]

Abstract The nonlinear Degasperis-Procesi (DP) equation allows shock wave solutions and possesses bi-Hamiltonian properties. Considering both of the two properties, we introduce a numerical scheme which involves three key features. First, a stable moving knots strategy is introduced to simulate the shock waves with less knots. Second, the moving knots equation together with the DP equation are simulated using the meshless multi-quadric (MQ) quasi-interpolation method, which allows scattered data approximation. Third, the proposed moving knots meshless scheme preserves the conservation laws. Both theoretical analysis and numerical examples demonstrate that the algorithm saves the computational time, improves the accuracy, and possesses a long-time tracking capability. [ABSTRACT FROM AUTHOR]

In this work we present new results on the convergence of diagonal sequences of certain mixed type Hermite-Padé approximants of a Nikishin system. The study is motivated by a mixed Hermite-Padé approximation scheme used in the construction of solutions of a Degasperis-Procesi peakon problem and germane to the analysis of the inverse spectral problem for the discrete cubic string. [ABSTRACT FROM AUTHOR]

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This paper deals with the Cauchy problem for the interacting system of the Camassa-Holm and Degasperis-Procesi equations mt=-3m(2ux+vx)-mx(2u+v),nt=-2n(2ux+vx)-nx(2u+v),where m=u-uxx and n=v-vxx. By the transport equations theory and the classical Friedrichs regularization method, the local well-posedness of solutions for this system in nonhomogeneous Besov spaces Bp,rs×Bp,rs with 1≤p,r≤+∞ and s>max2+1p,52 is obtained, and the local well-posedness in critical Besov space B2,15/2×B2,15/2 is also established. Moreover, by the approach for approximate solutions and well-posedness estimates, we obtain two sequences of solution for this equation, which are bounded in the Sobolev space Hs(R)×Hs(R) with s>5/2, and the distance between the two sequences is lower-bounded by a positive constant for any time t, but converges to zero at the initial time. This implies that the solution map is not uniformly continuous. [ABSTRACT FROM AUTHOR]

In this paper, we propose a two-component b-family system with cubic nonlinearity and peaked solitons (peakons) solutions, which includes the celebrated Camassa-Holm equation, Degasperis-Procesi equation, Novikov equation and its two-component extension as special cases. Firstly, we study single peakon and multi-peakon solutions to the system. Then the local well-posedness for the Cauchy problem of the system is discussed. Furthermore, we derive the precise blow-up scenario and global existence for strong solutions to the two-component b-family system with cubic nonlinearity. Finally, we investigate the asymptotic behaviors of strong solutions at infinity within its lifespan provided the initial data decay exponentially and algebraically. [ABSTRACT FROM AUTHOR]

We summarize the results of our recent work on Bäcklund transformations (BTs), particularly focusing on the relation between BTs and infinitesimal symmetries. We present a BT for an associated Degasperis-Procesi (aDP) equation and its superposition principle and investigate the solutions generated by applying this BT. Following our general methodology, we use the superposition principle of the BT to generate the infinitesimal symmetries of the aDP equation. [ABSTRACT FROM AUTHOR]

In this paper, persistence properties of solutions are investigated for a 4-parameter family ( k − a b c equation) of evolution equations having ( k + 1 ) -degree nonlinearities and containing as its integrable members the Camassa–Holm, the Degasperis–Procesi, Novikov and Fokas–Olver–Rosenau–Qiao equations. These properties will imply that strong solutions of the k − a b c equation will decay at infinity in the spatial variable provided that the initial data does. Furthermore, it is shown that the equation exhibits unique continuation for appropriate values of the parameters k , a , b , and c . [ABSTRACT FROM AUTHOR]

In this paper, the formation of singularities for the nonlocal Whitham-type equations is studied. It is shown that if the lowest slope of flows can be controlled by its highest value with the bounded Whitham-type integral kernel initially, then the finite-time blow-up will occur in the form of wave-breaking. This refined wave-breaking result is established by analyzing the monotonicity and continuity properties of a new system of the Riccati-type differential inequalities involving the extremal slopes of flows. Our theory is illustrated via the Whitham equation, Camassa–Holm equation, Degasperis–Procesi equation, and their μ -versions as well as hyperelastic rod equation. [ABSTRACT FROM AUTHOR]

For $$s<3/2$$ , it is shown that the Cauchy problem for the b-family of equations is ill-posed in Sobolev spaces $$H^s$$ on both the torus and the line when $$b>1$$ . The proof of ill-posedness depends on the value of b, where the most interesting case arises for $$b=3$$ , the Degasperis-Procesi equation. Considering that the b-family of equations is locally well-posed in $$H^s$$ for $$s>3/2$$ , this work establishes 3 / 2 as the critical index of well-posedness in Sobolev spaces for $$b>1$$ . [ABSTRACT FROM AUTHOR]

Peakon traveling wave solutions, both on the line and on the circle, are derived for a novel ab-family of nonlocal evolution equations with cubic nonlinearities. At least two members of this ab-family, namely the Fokas-Olver-Rosenau-Qiao equation and the Novikov equation, are known to be integrable. Furthermore, a generalization of the ab-family with nonlinearities of order k ∈ N, ≥ 2, is considered and its multi-peakon on the line is obtained. [ABSTRACT FROM AUTHOR]

This paper is concerned with an optimal distributed control problem of the viscous Degasperis-Procesi equation. The Dubovitskii and Milyutin functional analytical approach is adopted in the investigation of the Pontryagin maximum principle of the system. The necessary condition for optimality is established for the problem in the fixed final horizon case and, subsequently, a remark on the applicability of the obtained results is made for illustration. [ABSTRACT FROM AUTHOR]