Isaac Scientific Publishing

New Horizons in Mathematical Physics

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

Download PDF (1755.2 KB) PP. 53 - 61 Pub. Date: December 1, 2018

DOI: 10.22606/nhmp.2018.24001

Author(s)

  • Pierre Gaillard*
    Université de Bourgogne, Institut de mathématiques de Bourgogne, 9 avenue Alain Savary BP 47870 - 21078 Dijon Cedex, France

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 ai and bi for 1 ≤ i ≤ 4.

Keywords

Johnson equation, Fredholm determinants, wronskians, rational solutions, rogue waves.

References

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