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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. • We propose a new 2+1-dimensional modified Camassa–Holm equation. • We propose a 2+1 Hunter-Saxton equation. • We connect the 2+1 mCH equation with a negative modified KP equation via a reciprocal transformation. • We obtain multi-soliton solutions of the 2+1 mCH equation. [ABSTRACT FROM AUTHOR]

Highlights • General Degasperis-Procesi model. Abstract We consider the general Degasperis-Procesi model of shallow water out-flows. This six parametric family of conservation laws contains, in particular, KdV, Benjamin-Bona-Mahony, Camassa-Holm, and Degasperis-Procesi equations. The main result consists of criterions which guarantee the existence of solitary wave solutions: solitons and peakons ("peaked solitons"). [ABSTRACT FROM AUTHOR]

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. • Periodic peakons are calculated and expressed in an explicit function formula.. [ABSTRACT FROM AUTHOR]

In this article, an energy-preserving finite difference scheme for solving the generalized Camassa–Holm (gCH) equation with the dual-power law nonlinearities is proposed. We first show that the solution of the initial–boundary-value gCH equation is unique and continuously dependent on the initial condition, then we construct a linear energy-preserving difference scheme for the gCH equation. The proposed difference scheme is three-level implicit, and the numerical convergence order is O (τ 2 + h 4). The energy conservation, unique solvability, convergence and stability of the finite difference scheme are rigorously proved by using the discrete energy method. Finally, some numerical examples show that the proposed numerical scheme is efficient and reliable. • A high-order finite difference scheme for solving the gCH equation is constructed. • The gCH equation has the dual-power law nonlinearities. • The presented scheme is unconditionally energy conservative. • The convergent accuracy of the presented scheme is O (τ 2 + h 4). • Numerical examples are given to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper a three-step solution scheme is employed to numerically explore the long-time solution behavior of the Camassa–Holm equation. In the present u − P − α formation, we conduct modified equation analysis to eliminate several leading discretization error terms and perform Fourier analysis for minimizing the wave-like type of error. A three-point seventh-order spatially accurate combined compact upwind scheme is developed for the approximation of first-order derivative term. For the purpose of retaining Hamiltonian and multi-symplectic geometric structures in the non-dissipative Camassa–Holm equation, the adopted time integrator conserves symplecticity. Another main emphasis of this study is to numerically shed light on the scenario of the soliton transport. [ABSTRACT FROM AUTHOR]

The exact solitary wave solutions of simplified modified Camassa–Holm equation with any power are investigated by using the method of undetermined coefficient and qualitative theory of planar dynamical system. The existence and numbers of bell solitary wave solutions, kink solitary wave solutions and periodic wave solutions are analyzed with the help of Maple software and phase portraits. The four new exact expressions of bell solitary wave solutions and kink solitary wave solutions are obtained. By applying the theory of orbital stability proposed by Grillakis, Shatah and Strauss and the explicit expressions of discrimination d ′ ′ (c) , the wave speed interval of orbital stable and unstable for bell solitary wave solutions with any power are given. Furthermore, we discuss the orbital stability of kink solitary wave solutions with first power and fractional power and deduce the wave speed interval of orbital unstable. Moreover, we simulate numerically the conclusion about orbital stability of the four solitary wave solutions obtained in this paper and show the orbital stable results visually. • The behaviours of bell, kink wave solutions and periodic wave solutions are analyzed. • The four new exact expressions of solitary wave solutions are obtained. • The wave speed interval of orbit stability for bell wave solutions with any power is given. • The wave speed interval of orbit stability of kink wave solutions with first power and fractional power is given. • Simulating numerically the conclusion of orbital stability of the four solitary wave solutions. [ABSTRACT FROM AUTHOR]

• We propose a unified approach to establish Darboux-Bäcklund transformations of the Camassa-Holm (CH) type equations. The method simplifies the approach presented by Xia, Zhou and Qiao [J. Math. Phys. 57, 103502 (2016)] to obtain soliton solutions of the CH equation by the Darboux transformation, because it does not need to solve spectral problem of the CH equation at λ = 0 which is a singular point in the auxiliary problem, and avoids computing asymptotic properties of the wave functions. • The new method simplifies the procedure to construct Backlund transformations of the modified CH, the Degasperis-Procesi and the Novikov equations in references. • We present three Backlund transformations of the 2-CHsystem. We propose a unified approach to establish Darboux-Bäcklund transformations of the Camassa-Holm (CH) type equations. The method simplifies the approach presented by Xia, Zhou and Qiao [J. Math. Phys. 57, 103502 (2016)] to obtain soliton solutions of the CH equation by the Darboux transformation, because it does not need to solve spectral problem of the CH equation at λ = 0 which is a singular point in the auxiliary problem, and avoids computing asymptotic properties of the wave functions. We also study many other CH type equations such as the 2-CH, the modified CH, the Degasperis-Procesi and the Novikov via the method. Especially, we present three Bäcklund transformations of the 2-CH system. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we develop a novel multi-symplectic wavelet collocation method for solving multi-symplectic Hamiltonian system with periodic boundary conditions. Based on the autocorrelation function of Daubechies scaling functions, collocation method is conducted for the spatial discretization. The obtained semi-discrete system is proved to have semi-discrete multi-symplectic conservation laws and semi-discrete energy conservation laws. Then, appropriate symplectic scheme is applied for time integration, which leads to full-discrete multi-symplectic conservation laws. Numerical experiments for the nonlinear Schrödinger equation and Camassa–Holm equation show the high accuracy, effectiveness and good conservation properties of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: In this contribution we describe the role of several two-component integrable systems in the classical problem of shallow water waves. The starting point in our derivation is the Euler equation for an incompressible fluid, the equation of mass conservation, the simplest bottom and surface conditions and the constant vorticity condition. The approximate model equations are generated by introduction of suitable scalings and by truncating asymptotic expansions of the quantities to appropriate order. The so obtained equations can be related to three different integrable systems: a two component generalization of the Camassa–Holm equation, the Zakharov–Ito system and the Kaup–Boussinesq system. The significance of the results is the inclusion of vorticity, an important feature of water waves that has been given increasing attention during the last decade. The presented investigation shows how – up to a certain order – the model equations relate to the shear flow upon which the wave resides. In particular, it shows exactly how the constant vorticity affects the equations. [Copyright &y& Elsevier]

Abstract: We present some geometric investigations on some recently derived shallow water equations. While the Camassa–Holm equation can be expressed as an Euler equation on the diffeomorphism group of the circle, this is not the case for the Degasperis–Procesi equation. [Copyright &y& Elsevier]

Abstract: A new procedure for finding exact travelling wave solutions to the modified Camassa–Holm and Degasperis–Procesi equations is proposed. It turns out that many new solutions are obtained. Furthermore, these solutions are in general forms, and many known solutions to these two equations are only special cases of them. [Copyright &y& Elsevier]

Abstract: In this paper we take up the question of a small dispersion limit for the Camassa–Holm equation. The particular limit we study involves a modification of the Camassa–Holm equation, seen in the recent theoretical developments by Himonas and Misiołek, as well as the first author, where well-posedness is proved in weak Sobolev spaces. This work led naturally to the question of how solutions actually behave in these modified equations as time evolves. While the dispersive limit studied here is inspired by the work of Lax and Levermore on the zero dispersion limit of the Korteweg–de Vries equation to the inviscid Burgers’ equation, here there is no known Inverse Scattering theory. Consequently, we resort to a sophisticated numerical simulation to study two representative (one smooth and one peakon), but by no means exhaustive, initial conditions in the modified Camassa–Holm equation. In both cases there appears to be a strong limit of the modified Camassa–Holm equation to the Camassa–Holm equation as the dispersive parameter tends to zero, provided that solutions have not evolved for too long (time sufficiently small). For the smooth initial condition considered, this time must be chosen before the solution approaches steepening; beyond this time the computed solution becomes increasingly complicated as the dispersive term tends towards zero, and there does not appear to be a limit. By contrast, for the peakon initial condition this limit does appear to exist for all times considered. While in many cases the computations required few discretization points, there were some very challenging cases (particularly for the small dispersion computations) where an enormous number of unknowns were required to properly resolve the solution. [Copyright &y& Elsevier]