Chaos, Solitons & Fractals ( IF 7.8 ) Pub Date : 2020-10-04 , DOI: 10.1016/j.chaos.2020.110325 Nikolay A. Kudryashov
We consider the nonlinear fourth-order partial differential equation that can be used for describing solitary waves in nonlinear optics. The Cauchy problem for this equation is not solved by the inverse scattering transform. However we demonstrate that nonlinear ordinary differential equation for description of the wave packet envelope possesses the Painlevé property and is integrable. The Lax pair to this nonlinear ordinary differential equation is presented. Using the determinant for the Lax pair matrix, we find the first integrals of a nonlinear ordinary differential equation. The general solution of the fourth-order nonlinear differential equation is given via the ultraelliptic integrals. Special cases of exact solutions for the fourth-order equation are expressed in terms of the Jacobi elliptic sine. Optical solitons of the original partial differential equation are found.
中文翻译:
具有波包包络的可积方程模型的光学孤子
我们考虑了非线性四阶偏微分方程,该方程可用于描述非线性光学中的孤立波。逆散射变换不能解决该方程式的柯西问题。但是,我们证明用于描述波包包络线的非线性常微分方程具有Painlevé性质,并且是可积分的。给出了该非线性常微分方程的Lax对。使用Lax对矩阵的行列式,我们找到了非线性常微分方程的第一积分。通过超椭圆积分给出了四阶非线性微分方程的一般解。用Jacobi椭圆正弦表示四阶方程精确解的特殊情况。