Linear systems, determinants, and solutions of the Kadomtsev–Petviashvili equation

Let left parenthesis minus upper A comma upper B comma upper C right parenthesis $(-A,B,C)$ ( − A , B , C ) be a linear system in continuous time t greater than 0 t > 0 t > 0 with input and output space upper C C C and state space upper H H H . The scattering (or impulse response) functions phi left parenthesis x right parenthesis left parenthesis t right parenthesis equals Ce minus left parenthesis t plus 2 x right parenthesis AB φ₍ₓ₎(t)=Ce-(t+2x)AB ϕ ( x ) ( t ) = C e − ( t + 2 x ) A B determines a Hankel integral operator upper Gamma phi left parenthesis x right parenthesis Γ_φ₍ₓ₎ Γ ϕ ( x ) ; if