Coverings and nonlocal symmetries as well as fundamental solutions of nonlinear equations derived from the nonisospectral AKNS hierarchy

Abstract

In the paper, we apply a new scheme to generate nonisospectral integrable hierarchies of evolution equations. First of all, we work out the positive-order and the negative-order nonisospectral AKNS hierarchies. Next we introduce a kind of loop algebra from which an expanding nonisospectral integrable model is derived. Specially, the expanding integrable model reduces to the well-known bond pricing equation and the interest rate modeling. By using the Lie group theory we investigate the Lie representations and characteristic solutions of a generalized bond pricing equation from the positive-order nonisospectral integrable equation, and with the help of the characteristic solutions and the Laplace transformation, we again obtain the fundamental solutions of the generalized bond pricing equation. Besides, we apply the Lie group analysis to deduce the invariant solutions and the nonlinear self-adjointness as well as conservation laws, non-invariant solutions of the generalized bond pricing equation. Some reductions of the negative-order nonisospectral integrable hierarchy are also produced. Two of them are the Sine-Gordan equation and the Sinh-Gordan equation. Finally, we investigate the coverings and the nonlocal symmetries of the generalized bond pricing equation and a generalized KdV equation, respectively, although the bond equation is linear which has many applications in the aspect of finance field, by applying the classical Frobenius theorem and the coordinates of a infinitely-dimensional manifold in the form of Cartesian product.