In this research, we study analytically the double-chain model. The model consists of two long elastic homogeneous strands (or rods), which represent two polynucleotide chains of the deoxyribonucleic acid molecule, connected with each other by an elastic membrane (or some linear springs) representing the hydrogen bonds between the base pairs of the two chains. The new extended direct algebraic method and the generalized Kudryashov method are successfully utilized to discuss the exact soliton solutions to the double-chain model of deoxyribonucleic acid that plays an important role in biology. The solutions obtained by these mechanisms can be divided into solitary, singular, kink, single wave, combine behavior as well as hyperbolic, plane wave, and trigonometric solutions with arbitrary parameters. Some solutions have been exemplified by graphics to understand the physical meaning of the DNA model. The accomplished solutions seem with all essential constraint conditions, which are obligatory for them to subsist. Hence, our techniques via fortification of symbolic computations provide an active and potent mathematical implement for solving diverse benevolent nonlinear wave problems. The results show that the system theoretically has extremely rich exact wave structures of biological relevance.

中文翻译：

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脱氧核糖核酸双链模型中精确孤子解的动力学

在这项研究中，我们对双链模型进行了分析研究。该模型由两条长的弹性同质链（或棒）组成，它们代表脱氧核糖核酸分子的两条多核苷酸链，通过弹性膜（或一些线性弹簧）相互连接，代表两者碱基对之间的氢键链。新的扩展直接代数方法和广义 Kudryashov 方法成功地用于讨论在生物学中发挥重要作用的脱氧核糖核酸双链模型的精确孤子解。通过这些机制获得的解可以分为孤立、奇异、扭结、单波、组合行为以及具有任意参数的双曲、平面波和三角解。一些解决方案已通过图形举例说明，以了解 DNA 模型的物理意义。已完成的解决方案似乎具有所有必要的约束条件，这些条件是它们存在的必要条件。因此，我们通过强化符号计算的技术为解决各种仁慈的非线性波问题提供了一种积极有效的数学工具。结果表明，该系统理论上具有极其丰富的生物相关性精确波结构。