Binding energy of soliton molecules in time-dependent harmonic potential and nonlinear interaction

We calculate the binding energy of soliton molecules of an integrable nonlinear Schrödinger equation with time-dependent harmonic potential and cubic nonlinearity. Through a scaling transformation, an exact formula for the binding energy can be derived from that of the free soliton molecules in a homogeneous background. In the special case of oscillatory time dependence, sharp resonances occur at some integer and fractional multiples of the natural frequency of the molecule. Enhanced binding is obtained at these resonances and over some finite continuous range of low frequencies.