Abstract
We consider a model called the coupled sine-Gordon equation for DNA dynamics by introducing two double helix structures. The second double helix structure is unilaterally influenced by the first one. The completely integrable coupled sine-Gordon equation admits kink-antikink solitons with increased width representing a wide base pair opening configuration in DNA. Also we propose another coupled sine-Gordon model with variable coefficients for DNA dynamics under an inhomogeneous background. We find that the inhomogeneous DNA model has many interesting localized nonrational rogue wave solutions. We can find that the appearance of the rogue waves (possibly means the genetic mutation) in the nonlinear DNA model is highly related to the inhomogeneity.
Highlights
The sine-Gordon equation is an important integrable equation which describes many interesting phenomena including the DNA double helix molecule [1], dynamics of coupled pendulums, Josephson junction arrays [2], living cellular structures [3], and the crystalline lattices [4]
The scalar sine-Gordon equation originates in differential geometry and has profound applications in physics which can be seen from a recent review [6]
In [7], the nonrational rogue waves can be induced by the inhomogeneity background of the sine-Gordon equation
Summary
The sine-Gordon equation is an important integrable equation which describes many interesting phenomena including the DNA double helix molecule [1], dynamics of coupled pendulums, Josephson junction arrays [2], living cellular structures [3], and the crystalline lattices [4]. We give the Backlund transformations of the Zn-sine-Gordon and the nonlinear superposition formula and Lax pairs. We will construct the Backlund transformation of the inhomogeneous coupled sine-Gordon equation and, from the Backlund transformation, we derive many solutions including the rogue wave solutions of some special inhomogeneous coupled sine-Gordon equation
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