# Families of Solutions of Order 5 to the Johnson Equation Depending on 8 Parameters

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-01963626
Contributor : Imb - Université de Bourgogne <>
Submitted on : Friday, December 21, 2018 - 2:21:00 PM
Last modification on : Sunday, December 23, 2018 - 1:07:41 AM

### Citation

Pierre Gaillard. Families of Solutions of Order 5 to the Johnson Equation Depending on 8 Parameters. New Horizons in Mathematical Physics, 2018, 2 (4), ⟨10.22606/nhmp.2018.24001⟩. ⟨hal-01963626⟩

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