TY  - JOUR
T1  - Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation
AU  - Gaillard, Pierre
PY  - 2021
DA  - December
Y2  - 2021
DO  - 10.32323/ujma.978875
JF  - Universal Journal of Mathematics and Applications
JO  - Univ. J. Math. Appl.
PB  - Emrah Evren KARA
WT  - DergiPark
SN  - 2619-9653
SP  - 154
EP  - 163
VL  - 4
IS  - 4
LA  - en
AB  - Multi-parametric solutions to the Kadomtsev-Petviashvili equation (KPI) in terms of Fredholm determinants are constructed in function of exponentials. A representation of these solutions as a quotient of wronskians of order $2N$ in terms of trigonometric functions is deduced. All these solutions depend on $2N-1$ real parameters. A third representation in terms of a quotient of two real polynomials depending on $2N-2$ real parameters is given; the numerator is a polynomial of degree $2N(N+1)-2$ in $x$, $y$ and $t$ and the denominator is a polynomial of degree $2N(N+1)$ in $x$, $y$ and $t$.  The maximum absolute value is equal to $2(2N+1)^{2}-2$. We explicitly construct the expressions for the first third orders and we study the patterns of their absolute value in the plane $(x,y)$ and their evolution according to time and parameters.\\ It is relevant to emphasize that all these families of solutions are real and non singular.
KW  - Kadomtsev Petviasvili eqaution
KW  - Fredholm determinants
KW  - Wronskians
KW  - Rational solutions
CR  - [1] A.A. Albert, B. Muckenhoupt, On matrices of trace zero, Michigan Math. J. , 4 (1957), 1–3.
CR  - [2] V. I. Arnold, On matrices depending on parameters, Russian Math. Surveys 26(2) (1971), 29–43.
CR  - [3] H.J. Bernstein, A.V. Phillips, Fiber bundles and quantum theory, Scientific American 245(1) (1981), 122–137.
CR  - [4] M.L.A. Flores, Espacios Fibrados, Clases Carater´ısticas y el Isomorfismo de Thom. Pontificia Universidad Cat´olica del Peru-CENTRUM Catolica
(Peru), (2013)
CR  - [5] Sh. Friedland, Simultaneous Similarity of Matrices, Adv. Math., 50 (1983), 189–265.
CR  - [6] F. Gaines, A Note on Matrices with Zero Trace Amer. Math. Month., 73(6) (1966), 630–631.
CR  - [7] M.I Garcia-Planas, On simultaneously and approximately simultaneously diagonalizable pairs of matrices Fundam. J. Math. Appl., 2 (2019), 50–55.
CR  - [8] M.I. Garcia-Planas, T. Klymchuk, Differentiable families of traceless matrix triples RACSAM Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat, 114
(2019), 1–8.
CR  - [9] R.A. Horn, C.R. Johnson, Matrix Analysis, 2nd ed. Cambridge University Press, Cambridge, 2013.
CR  - [10] D. Husemoller. Fibre bundles (Vol. 5). McGraw-Hill, New York, 1966.
CR  - [11] E. Lubkin, Geometric definition of gauge invariance, Ann. Physics, 23(2) (1963), 233-283.
CR  - [12] J.M. Maillard, F.Y. Wu, C.K. Hu, Thermal transmissivity in discrete spin systems: Formulation and applications. J. Physics A: Math. Gen., 25(9) (1992),
2521.
CR  - [13] S. Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics. Cambridge University Press. 1995.
CR  - [14] A. Trautman, Fiber bundles, gauge fields, and gravitation. General relativity and gravitation 1 (1980), 287–308.
UR  - https://doi.org/10.32323/ujma.978875
L1  - https://dergipark.org.tr/en/download/article-file/1909237
ER  -