This paper is concerned with surface waves, solutions of hyperbolic nonlinear boundary value problems. We construct BKW solutions in the weakly nonlinear regime with infinite expansion in powers of ε. We rigorously justify this expansion, constructing exact solutions, which admit the asymptotic expansions. We also show that the solution is not necessarily localized at the order O(ε ∞ ) in the interior, even if the data are.

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This paper is concerned with the reflection of nonlinear discontinuous waves, for weakly well-posed hyperbolic boundary value problems, satisfying the (WR) condition, that is in a case where the IBVP is neither strongly stable, nor strongly unstable. We study how the singularities of a striated solution are reflected when the solution hits the boundary. We prove striated estimates and L ∞ estimates and observe the loss of one derivative: we show that a discontinuity of the gradient of the solution across a hyperplane can be reflected in a discontinuity across a hyperplane of the solution itself.The propagation field associated to the equationD = 0 isthe derivatives being calculated at (ω, k).From (6.3), we then obtain the following proposition. Proposition 6.3. H = H Lop and H Lop is tangent to the striation.We then deduce from the (WR) condition.Proof. From the (WR) condition, the derivative ∂τD of the Lopatinski determinant D does not vanish at (ω, k), that is w 2 = 0. Since ∂ỹ = P ∂ y and ∂t = a∂The coefficient of ∂ t in H is equal to −aw 2 and thus does not vanish.Proposition 2.17 follows from Propositions 6.1, 6.2 and 6.4. Additional L 2 regularityWe have the equation:The field X − being outgoing (σ − < 0), the problem X − v = h is well-posed without boundary condition, thus the following proposition.Proposition 6.5. For all f in L 2 x (H 0,m (Ω 0 T )) and g in H 0,m (Ω 0 T ) functions vanishing in the past, the solution u of the problem (5.1) vanishing in the past satisfieswith C a constant independent of T, γ, u, f and g.Remark 6.6. From now on, C will be a constant, independent of T , γ, u, f and g, which may vary from an expression to another.Proof of Proposition 6.5. The energy method on the equationγ v 0,γ,0,T ≤ C h 0,γ,0,T ≤ C( f 0,γ,0,T + u 2,γ,0,T ).Applying Y α to (6.4), for |α| ≤ m − 2, we obtain, since Y α exactly commutes with the equation: γ X + u m−2,γ,0,T ≤ C( f m−2,γ,0,T + u m,γ,0,T ).