Despite the widespread use of transcendental functions in the modeling of the dynamics of DNA, most research efforts are limited in their analytical studies of this enthralling system to cubic order polynomial approximations of the corresponding equations of motion. In this paper, we present an investigation of waves in the Peyrard–Bishop–Dauxois model of DNA; while extending the polynomial approximation of the Morse potential up to the sixth order. We show that, within a generalized version of the reductive perturbation method that we have adopted, the equations governing the envelop consist of the standard cubic nonlinear Schrödinger equation and its non homogeneous linearizations. Exact and explicit analytical solutions that correspond to bright solitary waves are obtained for these coupled amplitude equations. A notable qualitative feature of these solutions is the dependence of their propagation speeds and frequencies on their amplitudes. Our approach additionally unveils that these solutions contain some harmonic terms; which are missed in existing works. A very good agreement is found between our analytical analysis and the numerical simulations of the full discrete nonlinear equation of the lattice which use these solutions as initial conditions.

中文翻译：

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Peyrard-Bishop-Dauxois DNA 模型中调制波的高阶研究

尽管超越函数在 DNA 动力学建模中得到广泛使用，但大多数研究工作仅限于对这个令人着迷的系统的分析研究，即相应运动方程的三次多项式近似。在本文中，我们对 DNA 的 Peyrard-Bishop-Dauxois 模型中的波进行了研究；同时将莫尔斯势的多项式近似扩展到六阶。我们证明，在我们采用的简化扰动方法的广义版本中，控制包络的方程由标准三次非线性薛定谔方程及其非齐次线性化组成。对于这些耦合幅度方程，获得了与明亮孤立波相对应的精确且显式的解析解。这些解决方案的一个显着的定性特征是它们的传播速度和频率对其幅度的依赖性。我们的方法还揭示了这些解决方案包含一些调和项；现有作品中遗漏了哪些内容。我们的分析分析和使用这些解作为初始条件的晶格完全离散非线性方程的数值模拟之间发现了非常好的一致性。