A low-regularity Fourier integrator for the Davey-Stewartson II system with almost mass conservation

In this work, we propose a low-regularity Fourier integrator with almost mass conservation to solve the Davey-Stewartson II system (hyperbolic-elliptic case). Arbitrary order mass convergence could be achieved by the suitable addition of correction terms, while keeping the first order accuracy in H^γ × H^γ+1 for initial data in H^γ+1 × H^γ+1 with γ > 1. The main theorem is that, up to some fixed time T, there exist constants τ₀ and C depending only on T and \(||u|{|_{{L^\infty }\left( {(0,T);{H^{\gamma + 1}}} \right)}}\) $$\mid \mid u\mid {\mid }_{L^{\infty }\left((0,T);H^{\gamma +1}\right)}$$ such that, for any 0 < τ ≤ τ₀, we have that \(||u({t_n}, \cdot ) - {u^n}|{|_{{H^\gamma }}} \le C\tau ,\,\,\,\,||v({t_n}, \cdot ) - {v^n}|{|_{{H^{\gamma + 1}}}} \le C\tau ,\) $$\mid \mid u(t_n,\cdot )-u^n\mid {\mid }_{H^{\gamma }}\le C\tau ,\mid \mid v(t_n,\cdot )-v^n\mid {\mid }_{H^{\gamma +1}}\le C\tau ,$$ where uⁿ and vⁿ denote the numerical solutions at t_n = nτ. Moreover, the mass of the numerical solution M(uⁿ) satisfies that \(|M({u^n}) - M({u_0})| \le C{\tau ^5}.\) $$\mid M(u^n)-M(u_0)\mid \le C{\tau }^5.$$