On certain shallow water models, scaling invariance and strict self-adjointness

Abstract

Resumo: In this work we establish conditions for a class of third order partial differential equations to be strictly self-adjoint and scale invariant. The obtained family of equations includes the Benjamin-Bona-Mahony, Camassa-Holm and Novikov equations. Using the strict selfadjointness and Ibragimov's conservation theorem, we establish some local conservation laws for some of the mentioned equations.Palavras-chave: Strict self-adjointness, Ibragimov's conservation theorem, conservation laws. Historical surveyDuring the last century, a sequence of papers, starting with [27], showed and enlightened many properties of the well known Korteweg -de Vries equation(1)Later, in [1], a new equation called Benjamin-Bona-Mahoney (BBM), given bywas derived as an "alternative" for the KdV. Although equation (2) was derived using the same formal justification for obtaining (1), the differences between both equations are greater than the fact that (1) is an evolution equation whereas (2) is not. In [1] the authors found three conserved quantities on the solutions of (2). Later in [30], those conservation laws obtained were proved to be the only three admitted by (2). This fact shows a dramatic difference between BBM and KdV since the last one admits an infinite number of conserved quantities [28].More recently, Camassa and Holm [5] using Hamiltonian methods derived the famous CamassaHolm (CH) equationThe last equation possesses remarkable properties such as peakon solutions and a bi-hamiltoninan structure, see [5], which implies in the existence of an infinite number of conserved quantities, just like the KdV equation [15,28,26].Since then, a considerable number of papers have been dedicated to derive third order nonevolutionary dispersive equations having similar properties as those known to KdV and CH equation. To cite a few number of examples, it was derived in [8] an integrable equation having peakon solutions with first order nonlinearities, while in [9] another integrable equation, combining linear dispersion such as the KdV equation and a nonlinear dispersion like the CH

A generalised multicomponent system of Camassa-Holm-Novikov equations

“…see [3,8,7,10,11,23]. This integral corresponds to the Sobolev norm in H 1 (R) of the solutions u of (2).…”

“…Notice that, for the scalar case, the integral ( 25) can be derived by using the multiplier u (in the sense of [1, 2, 3]) or the fact that ( 1) is strictly self-adjoint (in the sense of [24,25]). In the latter case, this first integral is derived from the scaling generator X b in (3), see [7,10,26]. According to Table 1, the first integral ( 26) is derived as in the scalar case for b = 2.…”

“…More recently, equations with parameter-dependent nonlinearities have been considered, see for instance [3,10,11,19,20], where families of equations unifying both Camassa-Holm and Novikov equations [23,35] are studied.…”

“…Therefore, one may consider system (1) as a two-component generalisation of both CH and Novikov equations. Equation ( 2) is just the equation deduced in [10], by using symmetry arguments and techniques introduced in [24,25]. In [11] equation (2) was also re-obtained by imposing invariance under scalings and the existence of a certain multiplier (see [1,2] for further details).…”