The NLS Approximation Makes Wrong Predictions for the Water Wave Problem in Case of Small Surface Tension and Spatially Periodic Boundary Conditions

From the Klein–Gordon–Zakharov system to the Klein–Gordon equation

“…There are not so many examples of amplitude equations which, although derived in a formally correct way, make wrong predictions about the dynamics of the original system (cf. [9][10][11]). The KG equation without modification for the KGZ system on the torus can be added to this list of examples.…”

“…Remark There are not so many examples of amplitude equations which, although derived in a formally correct way, make wrong predictions about the dynamics of the original system (cf. ). The KG equation without modification for the KGZ system on the torus can be added to this list of examples.Remark The error can be estimated with the help of the previously mentioned energy estimates by using Gronwall's inequality if the so‐called residual is sufficiently small.…”

Justification of the Nonlinear Schrödinger Equation for the Evolution of Gravity Driven 2D Surface Water Waves in a Canal of Finite Depth

“…Assume that the A j , A jl , A n jl satisfy (28)- (30), where the A j , A jl , A n jl solve (18)- (26). Then for all s ≥ 0 there exist C Res , C , ε 0 > 0 depending on C A such that for all ε ∈ (0, ε 0 ) the corresponding approximation ε satisfies…”

“…We note that this fact should not be taken for granted. There are modulation equations (for examples see [11,26,30]) which although derived by reasonable formal arguments do not reflect the true dynamics of the original equations. However, our theorem is not optimal, since in general we cannot prove T 1 = T 0 .…”

On the dynamics of the mean-field polaron in the high-frequency limit

“…Such approximation results should not be taken for granted. There are various counterexamples showing that formally derived limit equations make wrong predictions about the original system [Sch95,SSZ15].…”