In this paper, a new approach is devoted to find novel analytical and approximate solutions to the damped quadratic nonlinear Helmholtz equation (HE) in terms of the Weiersrtrass elliptic function. The exact solution for undamped HE (integrable case) and approximate/semi-analytical solution to the damped HE (non-integrable case) are given for any arbitrary initial conditions. As a special case, the necessary and sufficient condition for the integrability of the damped HE using an elementary approach is reported. In general, a new ansatz is suggested to find a semi-analytical solution to the non-integrable case in the form of Weierstrass elliptic function. In addition, the relation between the Weierstrass and Jacobian elliptic functions solutions to the integrable case will be derived in details. Also, we will make a comparison between the semi-analytical solution and the approximate numerical solutions via using Runge–Kutta fourth-order method, finite difference method, and homotopy perturbation method for the first-two approximations. Furthermore, the maximum distance errors between the approximate/semi-analytical solution and the approximate numerical solutions will be estimated. As real applications, the obtained solutions will be devoted to describe the characteristics behavior of the oscillations in RLC series circuits and in various plasma models such as electronegative complex plasma model.

在本文中，一种新的方法致力于根据Weiersrtrass椭圆函数找到阻尼二次非线性Helmholtz方程（HE）的新颖解析和近似解。对于任意任意初始条件，均给出了无阻尼HE（可积分情况）的精确解和阻尼HE（非可积分情况）的近似/半解析解。作为一种特殊情况，报告了使用基本方法获得阻尼HE的可积分性的充要条件。通常，建议使用新的ansatz来找到Weierstrass椭圆函数形式的不可积分情况的半解析解。另外，将详细推导Weierstrass和雅可比椭圆函数解与可积情况之间的关系。还，我们将使用Runge–Kutta四阶方法，有限差分方法和同伦摄动方法对前两个近似进行半解析解和近似数值解之间的比较。此外，将估计近似/半解析解与近似数值解之间的最大距离误差。作为实际应用，所获得的解决方案将专门描述振荡的特性行为。RLC系列电路以及各种等离子模型，例如电负性复杂等离子模型。