1 department of mathematics and statistics brock university st. catharines, on l2s3a1, canada 2 department of mathematics faculty of sciences, university of cádiz puerto real, cádiz, spain, 11510 Abstract. A general family of peakon equations is introduced, involving two arbitrary functions of the wave amplitude and the wave gradient. This family contains all of the known breaking wave equations, including the integrable ones: Camassa-Holm equation, Degasperis-Procesi equation, Novikov equation, and FORQ/modified Camassa-Holm equation. One main result is to show that all of the equations in the general family possess weak solutions given by multi-peakons which are a linear superposition of peakons with timedependent amplitudes and positions. In particular, neither an integrability structure nor a Hamiltonian structure is needed to derive N -peakon weak solutions for arbitrary N > 1. As a further result, single peakon travelling-wave solutions are shown to exist under a simple condition on one of the two arbitrary functions in the general family of equations, and when this condition fails, generalized single peakon solutions that have a time-dependent amplitude and a time-dependent speed are shown to exist. An interesting generalization of the Camassa-Holm and FORQ/modified Camassa-Holm equations is obtained by deriving the most general subfamily of peakon equations that possess the Hamiltonian structure shared by the Camassa-Holm and FORQ/modified Camassa-Holm equations. Peakon travellingwave solutions and their features, including a variational formulation (minimizer problem), are derived for these generalized equations. A final main result is that 2-peakon weak solutions are investigated and shown to exhibit several novel kinds of behaviour, including the formation of a bound pair consisting of a peakon and an anti-peakon that have a maximum finite separation. emails: sanco@brocku.ca, elena.recio@uca.es

Nonlinear generalizations of integrable equations in one dimension, such as the KdV and Boussinesq equations with p-power nonlinearities, arise in many physical applications and are interesting in analysis due to critical behaviour. This paper studies analogous nonlinear p-power generalizations of the integrable KP equation and the Boussinesq equation in two dimensions. Several results are obtained. First, for all p = 0, a Hamiltonian formulation of both generalized equations is given. Second, all Lie symmetries are derived, including any that exist for special powers p = 0. Third, Noether's theorem is applied to obtain the conservation laws arising from the Lie symmetries that are variational. Finally, explicit line soliton solutions are derived for all powers p > 0, and some of their properties are discussed.

1 department of mathematics and statistics brock university st. catharines, on l2s3a1, canada 2 department of mathematics faculty of sciences, university of cádiz puerto real, cádiz, spain, 11510Abstract. A class of generalized nonlinear p-Laplacian evolution equations is studied. These equations model radial diffusion-reaction processes in n ≥ 1 dimensions, where the diffusivity depends on the gradient of the flow. For this class, all local conservation laws of low-order and all Lie symmetries are derived. The physical meaning of the conservation laws is discussed, and one of the conservation laws is used to show that the nonlinear equation can be mapped invertibly into a linear equation by a hodograph transformation in certain cases. The symmetries are used to derive exact group-invariant solutions from solvable threedimensional subgroups of the full symmetry group, which yields a direct reduction of the nonlinear equation to a quadrature. The physical and analytical properties of these exact solutions are explored, some of which describe moving interfaces and Green's functions.

1 department of mathematics and statistics brock university st. catharines, on l2s3a1, canada 2 department of mathematics faculty of sciences, university of cádiz puerto real, cádiz, spain, 11510Abstract. All nonlinear dispersive wave equations in the general classare known to possess multi-peakon weak solutions. A classification is presented for families of multi-peakon equations in this class that possess conserved momentum; conserved H 1 norm of u; conserved H 2 norm of u; conserved L 2 norm of m; related conservation laws. The results yield, among others, two interesting wide families of equations:for which the L 2 norm of m is conserved. The overlap of these two families yields a singular equation which is nevertheless found to possess both smooth solitary wave solutions and peakon travelling wave solutions.

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The generalized Kawahara equation $u_t=a(t) u_{xxxxx} +b(t)u_{xxx} +c(t)f(u) u_x$ appears in many physical applications. A complete classification of low-order conservation laws and point symmetries is obtained for this equation, which includes as a special case the usual Kawahara equation $u_t = \alpha u u_x+\beta u^2u_x +\gamma u_{xxx}+\mu u_{xxxxx}$. A general connection between conservation laws and symmetries for the generalized Kawahara equation is derived through the Hamiltonian structure of this equation and its relationship to Noether's theorem using a potential formulation.Comment: 6 page

In this paper, for the Cooper‐Shepard‐Sodano equation, some conservation laws are obtained by applying the multiplier method. Furthermore, we study this equation from the point of view of Lie symmetries. We perform an analysis of the symmetry reductions taking into account the similarity variables and the similarity solutions, which allow us to transform our equation into ordinary differential equations.

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