many-body problem. Instead we are looking for the integrability behind FQHEs as well as the Hamiltonian expression of it. In order to do so we separate the excitations of FQHE into two simple classes: the perturbative class and nonperturbative one. The nonperturbative class dominates the states in Hilbert space, a.k.a. the basis, the perturbative class organizes those basis into physical states. So perturbations actually are provided as structure constants (or superposition coefficients). Interestingly, this idea is like in CFT, where correlation function is made by conformal block and structure constant (it encodes the multiplicity of the corresponding conformal block in the correlation function. ) Since the ground state should not change by perturbations it belongs to the nonperturbative class. Hence it describes a sort of wave without dissipation which implies that the ground state is a solitonic wave.

In this way we have related the FQHE theory to soliton theory, the other important area of many-body physics. The question now is to extract excitations from the solitonic wavefunction. The stable excitations from the soliton wavefunction, are those solutions of quantum mechanics equation for soliton wave[@das1989integrable]. In this quantum mechanics, the logarithmic of the soliton wavefunction is a scalar function, while 