Families of Solutions of Order 5 to the Johnson Equation Depending on 8 Parameters
Résumé
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 $a_i$ and $b_i$ for 1 ≤ i ≤ 4.