A plethora of three-dimensional periodic travelling gravity-capillary water waves with multipulse transverse profiles

Homoclinic and Heteroclinic Bifurcations in Vector Fields

“…Systems of the above type arise in coupled nonlinear Schrödinger equations [429] and in the study of three-dimensional water waves [158]. The key feature in (5.10) is the semisimple eigenvalue of multiplicity two that is unfolded by the parameter µ: in contrast to the situation studied in §5.3.3, the unfolded eigenvalues are always real.…”

“…We refer to [158,429] and [10] for conditions on g(u 1 , u 2 ) that guarantee the existence of transverse homoclinic orbits. The next theorem shows that (5.10) admits N -homoclinic orbits near µ = 0.…”

Three‐dimensional travelling gravity‐capillary water waves

“…the domain D s of the vector field on the right-hand side of (16) is the subset of X s+1 defined by the boundary conditions (14), (15). Furthermore, equation (16) represents Hamilton's equations for the Hamiltonian system (M, Ω, H), where M = X s , Ω is the the canonical 2-form with respect to the…”

“…(These orbits are obtained as perturbations of explicit homoclinic solutions to suitably scaled equations at (δ, µ) = (0, 0).) Groves & Sandstede [15] have shown that for δ < 0 there are also infinitely many homoclinic solutions which resemble multiple copies of u k,k+1 and T u k,k+1 glued together in a strictly alternating sequence. These multipulse homoclinic solutions are obtained by the homoclinic Lyapunov-Schmidt theory which reduces the existence question to a bifurcation equation for N − 1 'times of flight' of orbits close to the primary homoclinic solutions u k,k+1 and T u k,k+1 ; the solvability of the bifurcation equation (which has a transparent structure due to the Hamiltonian structure and reversibility) is addressed using asymptotic information from the 'tails' of the homoclinic orbits.…”

Spots and stripes in nonlinear Schrödinger‐type equations with nearly one‐dimensional potentials

“…This idea goes back to Kirchgässner and has been extensively used for a wide range problems in extended domains (see e.g. and the references therein). The resulting dynamical system is typically ill‐posed, but the question of interest is that of existence of bounded solutions.…”