Supersymmetric construction of self-consistent condensates in the large 
N
 Gross-Neveu model: Solitons on finite-gap potentials

In the present work, the set of stationary solutions of the Gross-Neveu model in 't Hooft limit is extended. Such extension is obtained by striving a hidden supersymmetry associated to disconnected sets of stationary solutions. How the supersymmetry arises from the Darboux-Miura transformations between Lax pairs of the stationary modified Korteweg-de Vries and the stationary Korteweg-de Vries hierarchies is shown, associating the correspondent superpotentials to self-consistent condensates for the Gross-Neveu model.


Abtract
In the present talk the set of stationary solutions of the Gross-Neveu model in t'Hooft limit is extended. Such extension is obtained striving a hidden supersymmetry associated to disconnected sets of stationary solutions.
It is shown how the supersymmetry arises from the Darboux-Miura transformations between Lax pairs of stationary modified Korteweg-de Vries and the stationary Korteweg-de Vries hierarchies, associating the correspondent supercharges to self-consistent con-densates for the Gross-Neveu model The Gross-Neveu model in Large N limit The Gross-Neveu model corresponds to a quantum field theory for nonlinear interacting fermions without mass. The model presents some interesting properties: dynamical mass generation, is asymptotically free and presents a spontaneous breaking of symmetry..
The GN model, is described by the Lagrangian: are N fermions of different flavors.
For this model a bonsonization is allowed, where the bosonic field corresponds to the fermionic condensate And trough path integral approach we can obtain an effective action for stationary ∆ in the form At the t'Hooft limit and , it is possible to use the saddle point method to ensure the convergence of the two-point propagator. The convergence happens for the minimas of effective action. In this direction the variation of such action yields the consistency equations The G¨ orkov diagonal resolvent of the Bogoliubov-de Gennes operator or Dirac Hamiltonian in 1+1D satisfies the following algebraic properties and also satisfies the Dickey-Eilenberger equation The Görkov resolvent power series expansion in energy variable can be truncated in order to find analytic solutions for the condensate ∆(x). In this case the resolvent takes the form under the truncation condition The truncation condition defines ∆(x) as a solution of the s-mKdVh. The first five equations in the hierarchy correspond to The coefficients are related with the edges of spectrum of Hamiltonian operator In the search of condensate solutions we introduce the Lax pair formulation of stationary sector of GN model or s-mKdV hierarchy where P is a 2×2 matrix differential operator of order n and take the role of the Lax-Novikov integral of the Dirac Hamiltonian. An important behavior of the Lax pair operators is a like Burchnall-Chaundy relationship between matrix differential operator, that relate potencies of the Lax pair operator in the following form that relates the eigenvalues and over a hyper-elliptic curve this relation is in the basis of algebro-geometric solution method of the s-mKdVh.
The Miura transformation is defined by

s-KdVh s-mKdVh Miura transformation
if v is any s-mKdVh solution then u is a s-KdVh solution where s-KdVh corresponds to . Note that the inverse affirmation is not correct.
Explicitly, one finds

How to obtain s-mKdV solutions from s-KdV solutions?
The s-mKdVh is invariant under the change v → −v, then the Miura Transformation of v allows to define where are both s-KdVh solutions dependent on v.
Let's assume now from another perspective that we have two functions and given in function of v(x), and suppose that both functions and satisfy the same equation in the s-KdVh. In this case v(x) must satisfy simultaneously for v and for −v. Adding these two equations we obtain , which implies that v must satisfy the s-mKdV equation

The Lax equation for KdV takes the form where the Lax pair corresponds to a Schrödinger operator and a Lax-Novikov integral
The KdV hierarchy is defined recursively in the form

Inverse Miura transformation and hidden supersymmetry
Using the function defined in Miura transformation we can construct a superalgebra in the form Were the fermionic integrals correspons to Note that these correscponds, to a unitary transformation of , and it is a extended Schrödinger Hamiltonian. This frame is known as Witten supersymmetric quantum mechanics.
It is natural to ask why we want to change the problem from the search of one solution of the mKdVh to the search of two connected solutions of the KdVh?
In the next we show how starting from an initial Schrödinger potential construct via Darboux transformation a second one . A characteristic of the Darboux transformation is that maintain the symmetries of , thus if have a Lax-Novikov integral then also has its respective Lax-Novikov integral.

Conclusion
Using an exotic supersymmetry between finite-gap systems with defects, we have constructed the set of stationary and analytical solutions for the GN model, observing the existence of inhomogeneous and nonperiodic condensates, but with bands structures and a finite number of bound states.
The Darboux transformation has allowed us recursively to construct infinite families of exactly solvable Scrödinger systems from a finite-gap potential in the Its-Matveev form.
The process of constructing solitary defects on finite-gap Schrödinger operators has generated Dirac or Bogoliubov-de Gennes operators with scalar potential with solitary defects on finite-gap background, these Dirac systems exhibit an integral of motion corresponding to a Lax operator of the s-mKdVh.
The Darboux dressing of the Lax operator of finite-gap systems have allowed to found the self-consistency equations for all the system, having as main characteristic the independence of the consistence of each defect, depending only in the data of finite-gap background.