Some Non-Linear Evolution Equations and Their Explicit Smooth Solutions with Exponential Growth Written into Integral Form

Petar Popivanov and Angela Slavova ()
Additional contact information
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, 2024, vol. 12, issue 7, 1-24

Abstract: In this paper, exact solutions of semilinear equations having exponential growth in the space variable x are found. Semilinear Schrödinger equation with logarithmic nonlinearity and third-order evolution equations arising in optics with logarithmic and power-logarithmic nonlinearities are investigated. In the parabolic case, the solution u is written as u = b e − a x 2 , a < 0 , a , b being real-valued functions. We are looking for the solutions u of Schrödinger-type equation of the form u = b e − a x 2 2 , respectively, for the third-order PDE, u = A e i Φ , where the amplitude b and the phase function a are complex-valued functions, A > 0 , and Φ is real-valued. In our proofs, the method of the first integral is used, not Hirota’s approach or the method of simplest equation.

Keywords: semilinear parabolic equation; semilinear Schrödinger equation; logarithmic nonlinearity; parabolic equations with solutions of exponential growth; solutions into explicit form; special functions of Jacobi type; hyperbolic functions; Radhakrishnan–Kundu–Lakshmanan optic equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/12/7/1003/pdf (application/pdf)
https://www.mdpi.com/2227-7390/12/7/1003/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:12:y:2024:i:7:p:1003-:d:1365226

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().