Oscillatory Integrals and Phenomena Beyond all Algebraic Orders: with Applications to Homoclinic Orbits in Reversible Systems

By Eric Lombardi. 2000

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Camassa–Holm, Korteweg–de Vries-5 and other asymptotically equivalent equations for shallow water waves (2003, background): “…with high wave number linear waves which give rise to exponentially small ripples at the tails of the solitary wave in accordance with the study of water wave solution of the full Euler equation, see for example Beale (1991), Grimshaw and Joshi (1995), Dias and Kharif (1999) and Lombardi (2000).”
On asymptotically equivalent shallow water wave equations (2003, background): “See, for example, [4, 19, 10] and [26] for more discussion and analysis of these fascinating resonances.”
Exponentially small splitting of separatrices beyond Melnikov analysis: rigorous results (2012, background): “Another work dealing with a singular case is [46], where the author proves the splitting of separatrices for a certain class of reversible systems in R4.”
Homoclinic and heteroclinic bifurcations in vector fields (2010, methodology): “We outline the results contained in the monograph [259] by Lombardi who studied homoclinic loops near 0iω resonances in both four- and infinite-dimensional state space.”