New Horizons in Mathematical Physics

4

Authors: Biswanath Rath

We propose a model PT - symmetry operator(H) in the form of (2x2) matrix and find corresponding charge operator (C) such that it stisfies all the necessary appropriate commutation relations [H,PT]=0;[H,C]=0,[C,PT]=0 and [H,CPT]=0 in order to justify a model in complex quantum system like that to preserve the necessary conditions in Hilbert space

9

Authors: Pierre Gaillard

We give different representations of the solutions of the Johnson equation with parameters. First, an expression in terms of Fredholm determinants is given; we give also a representation of the solutions written as a quotient of wronskians of order 2N. These solutions of order N depend on 2N − 1 parameters. When one of these parameters tends to zero, we obtain N order rational solutions expressed as a quotient of two polynomials of degree 2N(N + 1) in x, t and 4N(N + 1) in y depending on 2N − 2 parameters. Here, we explicitly construct the expressions of the rational solutions of order 5 depending on 8 real parameters and we study the patterns of their modulus in the plane (x, y) and their evolution according to time and parameters ai and bi for 1 ≤ i ≤ 4.