Reducible–KAM torus for two dimensional completely resonant reversible quintic Schrödinger systems

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We consider the two-dimensional completely resonant reversible quintic Schrödinger systems { ( i ∂ t − Δ ) u + | u | 4 u + ∂ u ¯ H 1 ( | u | 2 , | v | 2 ) = 0 ( i ∂ t − Δ ) v + | v | 4 v + ∂ v ¯ H 2 ( | u | 2 , | v | 2 ) = 0 , x ∈ T 2 , with periodic boundary conditions, where the nonlinearity H h = o ( | ( u , v ) | 6 ) , h = 1 , 2 are real analytic functions in a neighborhood of the origin. Using a KAM algorithm together with partial Birkhoff normal form method, we find the existence of quasi-periodic solutions for a class of completely resonant reversible coupled nonlinear Schrödinger systems on two dimensional torus.