In this talk, we consider the existence of solitons (travelling waves of finite energy) for a class of higher order Boussinesq system of the form
 (0.1) (I − b∂2x + b2∂4x) ηt + ∂xu − a∂3xu + a2∂5xu + ∂x (G1(η, ηx, ηxx, u, ux, uxx)) =0,

(I − d∂2x + d2∂4x) ut + ηx − c∂3xη + c2∂5xη + ∂x (G2(η, ηx, ηxx, u, ux, uxx)) = 0,
where η = η(x, t) and u = u(x, t) are real-valued functions, the constants a, b, c, d, a2, b2, c2, d2 are related and the nonlinearity G = (G1, G2)t has the variational structure

G1(q, r, s, t, v, w) = Fs(q, r, s, t) − rFqt(q, r, s, t) − zFrt(q, r, s, t)

− tFst(q, r, s, t) − wFtt(q, r, s, t),

G2(q, r, s, t, v, w) = Fq(q, r, s, t) − rFqr(q, r, s, t) − zFrr(q, r, s, t)

− tFsr(q, r, s, t) − wFtr(q, r, s, t),

where F is a p + 2-homogeneous c2 function.

It is important to point out that these systems are formally second-order approximations of the full, two-dimensional Euler equations in the case of special nonlinearities. As happens
in the case of the abcd system and the KdV equation, the higher order Boussinesq system
(0.1) is somehow related with the fifth-order KdV equation
ut + uxxxxx + buxxx = (f(u, ux, uxx))x,

where the nonlinearity has the variational form

f(q, r, s) = Fq(q, r) − rFqr(q, r) − sFrr(q, r),

for some C2 function F that is homogeneous of degree p+2 for some  p> 1.

This particular modes can be consider as long-wave approximations to the water in the cases  F(u) =−u^3 ,F(u, ux) =- uux^2 and F(u, ux) = uux^2 − u^3

The existence of solitons for such a system is a direct consequence of the characterization of the travelling waves as critical points and the Lions concentration-compactness principle.