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Discussions on diffraction and the dispersion for traveling wave solutions of the (2+1)-dimensional paraxial wave equation

Abstract

This article proposes to solve the traveling wave solutions of the (2+1)-dimensional paraxial wave equation by using modified \(1/G^{\prime}\)-expansion and modified Kudryashov methods. Different types of traveling wave solutions of the (2+1)-dimensional paraxial wave equation have been produced using these methods. Similar and different aspects of the solutions produced by both analytical methods are discussed in the results and discussion section. The discussion has been made on the resulting traveling wave solutions of the paraxial wave equation on diffraction and the dispersion phenomena, which have an important place in physics. The effect of the paraxial wave equation, which is a Schrödinger type equation, on the phase-frequency velocity and wave number by increasing the frequency in one of the traveling wave solutions obtained is examined numerically. In addition, the wave frequency is simulated with the behavior of the solitary wave and discussed in detail. 3D, 2D and contour graphics are presented by giving special values to the constants in the solutions found with analytical methods. These graphs presented represent the shape of the standing wave at any given moment. Computer package program is also used in operations such as solving complex operations, drawing graphics and systems of algebraic equations.

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Correspondence to Hülya Durur.

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Durur, H., Yokuş, A. Discussions on diffraction and the dispersion for traveling wave solutions of the (2+1)-dimensional paraxial wave equation. Math Sci (2021). https://doi.org/10.1007/s40096-021-00419-z

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Keywords

  • The modified \(1/G^{\prime}\)-expansion method
  • Modified Kudryashov method
  • The (2+1)-dimensional paraxial wave equation
  • Traveling wave solution