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

Year 2021, , 154 - 163, 30.12.2021
https://doi.org/10.32323/ujma.978875

Abstract

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.

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References

  • [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.
Year 2021, , 154 - 163, 30.12.2021
https://doi.org/10.32323/ujma.978875

Abstract

Project Number

NO

References

  • [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.
There are 14 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Pierre Gaillard 0000-0002-7073-8284

Project Number NO
Publication Date December 30, 2021
Submission Date August 4, 2021
Acceptance Date December 16, 2021
Published in Issue Year 2021

Cite

APA Gaillard, P. (2021). Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation. Universal Journal of Mathematics and Applications, 4(4), 154-163. https://doi.org/10.32323/ujma.978875
AMA Gaillard P. Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation. Univ. J. Math. Appl. December 2021;4(4):154-163. doi:10.32323/ujma.978875
Chicago Gaillard, Pierre. “Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation”. Universal Journal of Mathematics and Applications 4, no. 4 (December 2021): 154-63. https://doi.org/10.32323/ujma.978875.
EndNote Gaillard P (December 1, 2021) Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation. Universal Journal of Mathematics and Applications 4 4 154–163.
IEEE P. Gaillard, “Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation”, Univ. J. Math. Appl., vol. 4, no. 4, pp. 154–163, 2021, doi: 10.32323/ujma.978875.
ISNAD Gaillard, Pierre. “Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation”. Universal Journal of Mathematics and Applications 4/4 (December 2021), 154-163. https://doi.org/10.32323/ujma.978875.
JAMA Gaillard P. Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation. Univ. J. Math. Appl. 2021;4:154–163.
MLA Gaillard, Pierre. “Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation”. Universal Journal of Mathematics and Applications, vol. 4, no. 4, 2021, pp. 154-63, doi:10.32323/ujma.978875.
Vancouver Gaillard P. Multi-Parametric Families of Real and Non Singular Solutions of the Kadomtsev-Petviasvili I Equation. Univ. J. Math. Appl. 2021;4(4):154-63.

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