Compatibility conditions to allow some large amplitude WKB analysis for Burger’s type systems

In this article, we discuss the problem of finding large amplitude asymptotic expansions for monophase oscillating solutions of the following multidimensional ( d > 1 ) Burger’s type system: ( ♦ ) ∂ t u + ( V ∘ u ⋅ ∇ x ) u = 0 , u ∈ R d , ( t , x ) ∈ R × R d , V ∈ C 1 ( R d ; R d ) . More precisely, we are concerned with families { u ε } ε ∈ ] 0 , 1 ] made of solutions to ( ♦ ) and having a development of the form u ε ( t , x ) = H ( t , x , Φ ( t , x ) ε ) + O ( ε ) where the function θ ⟼ H ( t , x , θ ) is periodic. In general, due to the formation of shocks, such a construction is not possible on a domain Ω which does not depend on ε ∈ ] 0 , 1 ] . In this article, we perform a detailed analysis of the restrictions to impose on the profile H and on the phase Φ in order to remedy this. Among these compatibility conditions, we can isolate some new interesting system of nonlinear partial differential equations. We explain how to solve them and we describe how the underlying structure is propagated through the evolution equation.