A Unique Continuation Result for a 2D System of Nonlinear Equations for Surface Waves

Abstract In this paper, we establish a result of unique continuation for special two-dimensional nonlinear system that models the evolution long water waves with small amplitude in presence surface tension. More precisely, will show if $$(\eta ,\Phi ) = (\eta (x,y, t),\Phi t))$$ ( η , Φ ) = x y t is solution system, suitable function space, and )$$ vanishes on an open subset $$\Omega $$ Ω $$\mathbb {R}^2 \times [-T,T],$$ R 2 × [ - T ] then )\equiv 0$$ ≡ 0 horizontal component .$$ . To state such property, use Carleman-type estimate differential operator $$\mathcal {L}$$ L related to system. We prove Carleman using particular version well known Treves’ inequality.