Solution of a second-order integro–differential equation which occurs in laser modelocking

We solve the following integro–differential equation for the eigenfunction u (x): su (x) ={d2/dx2−[1−6sech2(x)]}u (x) −εℱ∞−∞u (x′)sech(x′) dx′ (1+ν2d2/dx2)sech(x), where s is the eigenvalue and ε and ν are arbitrary parameters which need not be small. This equation occurs in laser modelocking theory in the analysis of pulse stability, and s is proportional to the rate of growth of perturbations. We expand the eigenfunction u (x) in terms of a convenient basis set Λ (x,k) satisfying −k2Λ (x,k) =[d2/dx2+6sech2(x)]Λ (x,k). We find two discrete eigenfunctions u0(x) and u1(x) and a continuum u (x,s). We find that the lowest eigenvalue s0(ε) is −3 at ε=0 for finite or zero ν, and that the point where s0 =0 the parameters ε and ν obey ε=2/(1−ν2). This is the zero growth point at which no eigenfunction u (x) has a positive eigenvalue s.