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

Author

Listed:
  • Petar Popivanov

    (Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
    These authors contributed equally to this work.)

  • Angela Slavova

    (Institute of Mechanics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
    These authors contributed equally to this work.)

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.

Suggested Citation

  • Petar Popivanov & Angela Slavova, 2024. "Some Non-Linear Evolution Equations and Their Explicit Smooth Solutions with Exponential Growth Written into Integral Form," Mathematics, MDPI, vol. 12(7), pages 1-24, March.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:7:p:1003-:d:1365226
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    References listed on IDEAS

    as
    1. Kudryashov, Nikolai A., 2005. "Simplest equation method to look for exact solutions of nonlinear differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 24(5), pages 1217-1231.
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