Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation

Yıl 2021, , 154 - 163, 30.12.2021

Pierre Gaillard

https://doi.org/10.32323/ujma.978875

Öz

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.

Anahtar Kelimeler

Kadomtsev Petviasvili eqaution, Fredholm determinants, Wronskians, Rational solutions

Destekleyen Kurum

NO

Proje Numarası

NO

Teşekkür

NO

Kaynakça

• [1] A.A. Albert, B. Muckenhoupt, On matrices of trace zero, Michigan Math. J. , 4 (1957), 1–3.
• [2] V. I. Arnold, On matrices depending on parameters, Russian Math. Surveys 26(2) (1971), 29–43.
• [3] H.J. Bernstein, A.V. Phillips, Fiber bundles and quantum theory, Scientific American 245(1) (1981), 122–137.
• [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)
• [5] Sh. Friedland, Simultaneous Similarity of Matrices, Adv. Math., 50 (1983), 189–265.
• [6] F. Gaines, A Note on Matrices with Zero Trace Amer. Math. Month., 73(6) (1966), 630–631.
• [7] M.I Garcia-Planas, On simultaneously and approximately simultaneously diagonalizable pairs of matrices Fundam. J. Math. Appl., 2 (2019), 50–55.
• [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.
• [9] R.A. Horn, C.R. Johnson, Matrix Analysis, 2nd ed. Cambridge University Press, Cambridge, 2013.
• [10] D. Husemoller. Fibre bundles (Vol. 5). McGraw-Hill, New York, 1966.
• [11] E. Lubkin, Geometric definition of gauge invariance, Ann. Physics, 23(2) (1963), 233-283.
• [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.
• [13] S. Okubo, Introduction to Octonion and Other Non-Associative Algebras in Physics. Cambridge University Press. 1995.
• [14] A. Trautman, Fiber bundles, gauge fields, and gravitation. General relativity and gravitation 1 (1980), 287–308.