{"product_id":"9781470468774-quasi-periodic-traveling-waves-on-a","title":"Quasi-Periodic Traveling Waves on an Infinitely Deep Perfect Fluid Under Gravity","description":"\u003cmeta content=\"text\/html; charset=utf-8\" http-equiv=\"Content-Type\"\u003e\u003cp\u003e\u003cspan\u003eWe consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and establish the existence and linear stability of small amplitude, quasi-periodic in time, traveling waves, providing the first existence result of quasi-periodic water waves solutions bifurcating from a completely resonant elliptic fixed point.\u003cbr\u003eWe consider the gravity water waves system with a periodic one-dimensional interface in infinite depth and we establish the existence and the linear stability of small amplitude, quasi-periodic in time, traveling waves. This provides the first existence result of quasi-periodic water waves solutions bifurcating from a completely resonant elliptic fixed point. The proof is based on a Nash–Moser scheme, Birkhoff normal form methods and pseudo differential calculus techniques. We deal with the combined problems of small divisors and the fully-nonlinear nature of the equations.\u003cbr\u003e\u003cbr\u003eThe lack of parameters, like the capillarity or the depth of the ocean, demands a refined nonlinear bifurcation analysis involving several nontrivial resonant wave interactions, as the well-known \"Benjamin-Feir resonances\". We develop a novel normal form approach to deal with that. Moreover, by making full use of the Hamiltonian structure, we are able to provide the existence of a wide class of solutions which are free from restrictions of parity in the time and space variables.\u003cbr\u003e\u003cbr\u003e\u003c\/span\u003e\u003c\/p\u003e","brand":"Rarewaves","offers":[{"title":"Default Title","offer_id":55110064800118,"sku":"9781470468774","price":68.38,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0092\/7504\/8033\/files\/orig_36016474_be9d93cc-e9f5-4b52-8398-fe03bac7cf43.jpg?v=1757557684","url":"https:\/\/www.rarewaves.com\/products\/9781470468774-quasi-periodic-traveling-waves-on-a","provider":"Rarewaves.com","version":"1.0","type":"link"}