# From Fredholm and Wronskian representations to rational solutions to the KPI equation depending on 2N − 2 parameters, the structure of the solutions and the case of fourth order

@inproceedings{Gaillard2017FromFA, title={From Fredholm and Wronskian representations to rational solutions to the KPI equation depending on 2N − 2 parameters, the structure of the solutions and the case of fourth order}, author={Pierre Gaillard}, year={2017} }

We have already constructed solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of Fredholm determinants and wronskians of order 2N. These solutions have been called solutions of order N and they depend on 2N − 1 parameters. We construct here N-order rational solutions. We prove that they can be written as a quotient of 2 polynomials of degree 2N (N + 1) in x, y and t depending on 2N − 2 parameters. We explicitly construct the expressions of the rational solutions of order 4…

## 3 Citations

### Fredholm and wronskian representations of solutions to the Johnson equation and the third order case

- Mathematics
- 2020

We construct solutions to the Johnson equation (J) by means of Fredholm determinants first, then by means of wronskians of order 2N giving solutions of order N depending on 2N−1 parameters. We obtain…

### Rational solutions to the KPI equation of order 7 depending on 12 parameters

- Mathematics
- 2018

We construct in this paper, rational solutions as a quotient of two determinants of order 2N = 14 and we obtain what we call solutions of order N = 7 to the Kadomtsev-Petviashvili equation (KPI) as a…

### Families of solutions to the KPI equation given by an extended Darboux transformation

- MathematicsPartial Differential Equations and Applications
- 2022

By means of a Darboux transform with particular generating function solutions to the Kadomtsev–Petviashvili equation (KPI) are constructed. We give a method that provides different types of solutions…

## References

SHOWING 1-10 OF 51 REFERENCES

### On a family of solutions of the Kadomtsev–Petviashvili equation which also satisfy the Toda lattice hierarchy

- Mathematics, Physics
- 2003

We describe the interaction pattern in the x–y plane for a family of soliton solutions of the Kadomtsev–Petviashvili (KP) equation, The solutions considered also satisfy the finite Toda lattice…

### Hierarchy of solutions to the NLS equation and multi-rogue waves

- Mathematics
- 2015

The solutions to the one dimensional focusing nonlinear Schrödinger equation (NLS) are given in terms of determinants. The orders of these determinants are arbitrarily equal to 2N for any nonnegative…

### Patterns of deformations of Peregrine breather of order 3 and 4 solutions to the NLS equation with multi parameters

- Mathematics
- 2016

In this article, one gives a classification of the solutions to the one dimensional nonlinear focusing Schrödinger equation (NLS) by considering the modulus of the solutions in the (x, t) plan in the…

### Two parameters wronskian representation of solutions of nonlinear Schrödinger equation, eighth Peregrine breather and multi-rogue waves

- Mathematics
- 2014

In this paper, we present a representation of solutions of the one dimensional focusing nonlinear Schrodinger equation as a quotient of two wronskians depending on two parameters. Here, we give the…

### Rational solutions of the Kadomtsev–Petviashvili hierarchy and the dynamics of their poles. I. New form of a general rational solution

- Mathematics
- 1994

A new approach to the construction of the rational solutions to the hierarchy of the Kadomtsev–Petviashvili equation is presented. The generalization of the ‘‘superposition formula’’ is found to…

### Other 2N − 2 parameters solutions of the NLS equation and 2N + 1 highest amplitude of the modulus of the N th order AP breather

- Mathematics
- 2015

In this paper, we construct new deformations of the Akhmediev-Peregrine (AP) breather of order N (or APN breather) with 2 N − 2 ?> real parameters. Other families of quasirational solutions of the…

### Degenerate determinant representation of solutions of the NLS equation, higher Peregrine breathers and multi-rogue waves.

- Mathematics
- 2012

We present a new representation of solutions of the focusing NLS equation as a quotient of two determinants. This work is based on a recent paper in which we have constructed a multi-parametric…

### Two-parameter determinant representation of seventh order rogue wave solutions of the NLS equation

- Mathematics
- 2013

We present a new representation of solutions of focusing nonlinear Schrödinger equation (NLS) equation as a quotient of two determinants. We construct families of quasi-rational solutions of the NLS…

### Higher order Peregrine breathers solutions to the NLS equation

- Mathematics
- 2015

The solutions to the one dimensional focusing nonlinear Schrodinger equation (NLS) can be written as a product of an exponential depending on t by a quotient of two polynomials of degree N(N + 1) in…