Traveling-Wave Solutions of Several Nonlinear Mathematical Physics Equations

Petar Popivanov and Angela Slavova ()
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Petar Popivanov: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
Angela Slavova: Institute of Mechanics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria

Mathematics, 2025, vol. 13, issue 6, 1-18

Abstract: This paper deals with several nonlinear partial differential equations (PDEs) of mathematical physics such as the concatenation model (perturbed concatenation model) from nonlinear fiber optics, the plane hydrodynamic jet theory, the Kadomtsev–Petviashvili PDE from hydrodynamic (soliton theory) and others. For the equation of nonlinear optics, we look for solutions of the form amplitude Q multiplied by e i Φ , Φ being linear. Then, Q is expressed as a quadratic polynomial of some elliptic function. Such types of solutions exist if some nonlinear algebraic system possesses a nontrivial solution. In the other five cases, the solution is a traveling wave. It satisfies Abel-type ODE of the second kind, the first order ODE of the elliptic functions (the Weierstrass or Jacobi functions), the Airy equation, the Emden–Fawler equation, etc. At the end of the paper a short survey on the Jacobi elliptic and Weierstrass functions is included.

Keywords: nonlinear Schrödinger equation; concatenation model from nonlinear fiber optics; Kadomtsev–Petviashvili equation; solitons; Airy function; Abel’s ODE of the second kind; blowup solutions; traveling-wave solution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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