We construct solutions to the Johnson equation (J) first by means of Fredholm determinants and then by means of Wronskians of order 2N giving solutions of order N depending on 2N - 1 parameters. We obtain N order rational solutions that can be written 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. This method gives an infinite hierarchy of solutions to the Johnson equation. In particular, rational solutions are obtained. The solutions of order 3 with 4 parameters are constructed and studied in detail by means of their modulus in the (x, y) plane in function of time t and parameters a(1), a(2), b(1), and b(2).