scholarly journals The Johnson Equation, Fredholm and Wronskian Representations of Solutions, and the Case of Order Three

2018 ◽  
Vol 2018 ◽  
pp. 1-18
Author(s):  
Pierre Gaillard
Keyword(s):  
Rational Solutions ◽  
Fredholm Determinants ◽  
Infinite Hierarchy

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 a1, a2, b1, and b2.

scholarly journals Other Families of Rational Solutions to the KPI Equation

2021 ◽  
pp. 27-34
Author(s):  
Pierre Gaillard
Keyword(s):  
Positive Integer ◽  
Rational Solutions ◽  
Infinite Hierarchy ◽  
Petviashvili Equation ◽  
Two Peaks ◽  
Explicit Expressions

Aims / Objectives: We present rational solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of polynomials in x, y and t depending on several real parameters. We get an infinite hierarchy of rational solutions written as a quotient of a polynomial of degree 2N(N + 1) - 2 in x, y and t by a polynomial of degree 2N(N + 1) in x, y and t, depending on 2N - 2 real parameters for each positive integer N. Place and Duration of Study: Institut de math´ematiques de Bourgogne, Universit´e de Bourgogne Franche-Cont´e between January 2020 and January 2021. Conclusion: We construct explicit expressions of the solutions in the simplest cases N = 1 and N = 2 and we study the patterns of their modulus in the (x; y) plane for different values of time t and parameters. In particular, in the study of these solutions, we see the appearance not yet observed of three pairs of two peaks in the case of order 2.

scholarly journals Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes

2020 ◽  
pp. 1-14
Author(s):  
SHOTA OSADA
Keyword(s):  
Point Processes ◽  
Duke Math ◽  
Random Point ◽  
Determinantal Point Processes ◽  
Fredholm Determinants ◽  
Poisson Point Processes ◽  
Translation Invariant ◽  
Ergodic Properties ◽  
Tree Representations ◽  
Invariant Kernels

Abstract We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

scholarly journals Infinite hierarchy of solitons: Interaction of Kerr nonlinearity with even orders of dispersion

2021 ◽  
Vol 3 (1) ◽  
Author(s):  
Antoine F. J. Runge ◽  
Y. Long Qiang ◽  
Tristram J. Alexander ◽  
M. Z. Rafat ◽  
Darren D. Hudson ◽  
...  
Keyword(s):  
Kerr Nonlinearity ◽  
Infinite Hierarchy

scholarly journals Integrability of Riemann-Type Hydrodynamical Systems and Dubrovin’s Integrability Classification of Perturbed KdV-Type Equations

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1077
Author(s):  
Yarema A. Prykarpatskyy
Keyword(s):  
Conservation Laws ◽  
Hamiltonian Structure ◽  
Necessary Condition ◽  
Type Representation ◽  
Infinite Hierarchy ◽  
Polynomial Coefficients ◽  
Kdv Type Equations ◽  
Special Case

Dubrovin’s work on the classification of perturbed KdV-type equations is reanalyzed in detail via the gradient-holonomic integrability scheme, which was devised and developed jointly with Maxim Pavlov and collaborators some time ago. As a consequence of the reanalysis, one can show that Dubrovin’s criterion inherits important parts of the gradient-holonomic scheme properties, especially the necessary condition of suitably ordered reduction expansions with certain types of polynomial coefficients. In addition, we also analyze a special case of a new infinite hierarchy of Riemann-type hydrodynamical systems using a gradient-holonomic approach that was suggested jointly with M. Pavlov and collaborators. An infinite hierarchy of conservation laws, bi-Hamiltonian structure and the corresponding Lax-type representation are constructed for these systems.

scholarly journals ON PROPERTIES OF FINITE-ORDER MEROMORPHIC SOLUTIONS OF A CERTAIN DIFFERENCE PAINLEVÉ I EQUATION

2011 ◽  
Vol 85 (3) ◽  
pp. 463-475 ◽  
Author(s):  
MEI-RU CHEN ◽  
ZONG-XUAN CHEN
Keyword(s):  
Finite Order ◽  
Meromorphic Solutions ◽  
Rational Solutions ◽  
Polynomial Solutions ◽  
The Difference ◽  
Image Position

AbstractIn this paper, we investigate properties of finite-order transcendental meromorphic solutions, rational solutions and polynomial solutions of the difference Painlevé I equation where a, b and c are constants, ∣a∣+∣b∣≠0.

Rational solutions of the KdV equation with damping

1979 ◽  
Vol 24 (4) ◽  
pp. 97-100 ◽  
Author(s):  
F. Calogero ◽  
M. A. Olshanetsky ◽  
A. M. Perelomov
Keyword(s):  
Kdv Equation ◽  
Rational Solutions ◽  
The Kdv Equation
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