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.

本文提出用修正的\(1/G^{\prime}\)求解(2+1)维近轴波动方程的行波解- 扩展和修改的 Kudryashov 方法。已经使用这些方法产生了 (2+1) 维近轴波动方程的不同类型的行波解。结果和讨论部分讨论了由两种分析方法产生的解决方案的相似和不同方面。讨论了近轴波动方程关于衍射和色散现象的行波解，这在物理学中具有重要的地位。通过增加获得的行波解之一的频率，近轴波动方程（薛定谔型方程）对相频速度和波数的影响进行了数值研究。此外，还用孤立波的行为模拟了波频率并进行了详细讨论。3D，二维和等高线图形是通过为解析方法找到的解决方案中的常数赋予特殊值来呈现的。呈现的这些图形代表了任何给定时刻的驻波形状。计算机封装程序还用于求解复杂运算、绘制图形和代数方程组等运算。