DNA Decipher Journal | December 2014 | Volume 4 | Issue 3 | pp. 199-202 
Christianto, V. & Umniyati, Y., A Graphic Plot for a Soliton Solution of Sine-Gordon model of DNA 

 
ISSN: 2159-046X DNA Decipher Journal 

Published by QuantumDream, Inc. 
www.dnadecipher.com 

 

199 

Report 
 

A Graphic Plot for a Soliton Solution of Sine-Gordon model of DNA 
 

Victor Christianto
*1

&Yunita Umniyati2
 

 
1
Malang Institute of Agriculture, Malang, Indonesia 

2
Swiss German University, Tangerang – Indonesia 

 

ABSTRACT 

There are many models of DNA, both the linear ones and the nonlinear ones. One interesting 

model in this regard is the sine-Gordon model of DNA as proposed by Salerno. It belongs to 

nonlinear model of DNA which is close to realistic model. Here we discuss a graphical plot of 

soliton solution of such a sine-Gordon model of DNA. 

 

Key Words: soliton solution, sine-Gordon, DNA, graphic. 
 

 

Introduction 
 

There are many models of DNA, both the linear ones and the nonlinear ones [1]. One interesting 

model in this regard is the sine-Gordon model of DNA as proposed by Salerno [2], see also 

Daniel and Vasumathi [3]. It belongs to nonlinear model of DNA which is close to realistic 

model. A review of physical significance of such a sine-Gordon model was given in [6]. 

 

Here we discuss a graphical plot of soliton solution of such a sine-Gordon model of DNA.  

 

 

Soliton solution of a sine-Gordon model of DNA 
 

Assuming the wavefunction Ψ to be a function of x and t, then the sine-Gordon model of DNA 

can be written as follows: [3, p.7] 
 

  𝛹𝑡𝑡 −𝛹𝑧𝑧 + sin⁡(𝛹) = 0                             (1) 
 

or in Mathematica expression: 
 

=U[x-c t]; 

pde=D[,x,x]-D[,t,t]-sin[]0 
 

 

Now we will use Mathematica 9.0 to simplify and give graphical plot [3, p.443].To simplify with 

Mathematica: 

                                                           
*
Correspondence:Victor Christianto, Malang Institute of Agriculture, Malang – Indonesia.  

  URL: http://researchgate.net/profile/Victor_Christianto.  Email: victorchristianto@gmail.com 

http://researchgate.net/profile/Victor_Christianto
mailto:victorchristianto@gmail.com


DNA Decipher Journal | December 2014 | Volume 4 | Issue 3 | pp. 199-202 
Christianto, V. & Umniyati, Y., A Graphic Plot for a Soliton Solution of Sine-Gordon model of DNA 

 
ISSN: 2159-046X DNA Decipher Journal 

Published by QuantumDream, Inc. 
www.dnadecipher.com 

 

200 

 

−sin[𝑈[𝑧]] + 𝑈′′[𝑧] − 𝑐2𝑈′′[𝑧] = 0                                                     (2) 
 

The result is known as kink soliton wave: [3, p.444] 

 

𝛷 = 4ArcTan[𝑐Sinh[𝑥/Sqrt[1 − 𝑐^2]]/Cosh[𝑐𝑡/Sqrt[1 − 𝑐^2]]]  (3) 
 

or in Mathematica: 

4ArcTan [𝑐Sech [
𝑐𝑡

√1 − 𝑐2
] Sinh [

𝑥

√1 − 𝑐2
]] 

 

Differentiating for t, it yields: 

∂𝑡 (4ArcTan [𝑐Sech [
𝑐𝑡

√1 − 𝑐2
] Sinh [

𝑥

√1 − 𝑐2
]]) 

−
4𝑐2Sech[

𝑐𝑡

√1−𝑐2
]Sinh[

𝑥

√1−𝑐2
]Tanh[

𝑐𝑡

√1−𝑐2
]

√1 − 𝑐2(1 + 𝑐2Sech[
𝑐𝑡

√1−𝑐2
]2Sinh[

𝑥

√1−𝑐2
]2)

 

 

Simplifying the above result, it yields: 

 

Simplify [−
4𝑐2Sech [

𝑐𝑡

√1−𝑐2
] Sinh [

𝑥

√1−𝑐2
] Tanh [

𝑐𝑡

√1−𝑐2
]

√1 − 𝑐2 (1 + 𝑐2Sech [
𝑐𝑡

√1−𝑐2
]
2
Sinh [

𝑥

√1−𝑐2
]
2
)
] 

 

−
8𝑐2Sinh[

𝑐𝑡

√1−𝑐2
]Sinh[

𝑥

√1−𝑐2
]

√1 − 𝑐2(1 − 𝑐2 + Cosh[
2𝑐𝑡

√1−𝑐2
] + 𝑐2Cosh[

2𝑥

√1−𝑐2
])

 

 

The 3D plot is given below for c= 0.72 

 



DNA Decipher Journal | December 2014 | Volume 4 | Issue 3 | pp. 199-202 
Christianto, V. & Umniyati, Y., A Graphic Plot for a Soliton Solution of Sine-Gordon model of DNA 

 
ISSN: 2159-046X DNA Decipher Journal 

Published by QuantumDream, Inc. 
www.dnadecipher.com 

 

201 

Figure 1. Mathematica plot of soliton solution on sine-Gordon equation for c=0.72 

 

 

Perturbed Sine-Gordon Equation (SGE) 
 

Perturbed SGE come in a variety of forms. One common form is a damped and driven SGE [7, 

p.17]: 

 

 𝛹𝑡𝑡 +𝛷𝛹𝑡 −𝛹𝑧𝑧 + sin⁡(𝛹) = 𝐹    (4) 

 

In addition, the following two versions of the perturbed SGE have been studied in the literature, 

including: 

 

 

a. Directly forced SGE: [7, p.19] 

 

  𝛹𝑡𝑡 −𝛹𝑧𝑧 + sin⁡(𝛹) = 𝑀𝑓(𝜔𝑡)     (5) 

  

b. Damped and drived SGE: 

 

  𝛹𝑡𝑡 −𝛹𝑧𝑧 + sin(𝛹) = 𝑀𝑓(𝜔𝑡) − 𝛼𝛹𝑡 + 𝜂         (6) 
 

 

In the meantime, (2+1)D SGE with additional spatial coordinate (y) is defined as [7,p.21]: 

 

  𝛹𝑡𝑡 = 𝛹𝑥𝑥 +𝛹𝑦𝑦 − sin⁡(𝛹)            (7) 

 

In their in-depth review of SGE, Ivancevic and Ivancevic [7] discuss potential applications of 

SGE solitons in DNA, protein folding, microtubules, neural impulse conduction and muscular 

contraction soliton. New insights may be expected in the near future in these biological fields, 

based on sine-Gordon equation soliton. 

 

 

Conclusion 
 

There are many models of DNA, both the linear ones and the nonlinear ones [1]. One interesting 

model in this regard is the sine-Gordon model of DNA as proposed by Salerno [2]. It belongs to 

nonlinear model of DNA which is close to realistic model. Here we have discussed a graphical 

plot of soliton solution of such a sine-Gordon model of DNA. 

 

Considering that sine-Gordon equation has been used extensively by particle physicists, it would 

be interesting to study possibility to improve or alter DNA using electromagnetic field/pulse 

such as laser. This may be considered as a DNA enhancement method.  New insights may be 

expected in the near future in these biological fields, based on sine-Gordon equation soliton. 
 



DNA Decipher Journal | December 2014 | Volume 4 | Issue 3 | pp. 199-202 
Christianto, V. & Umniyati, Y., A Graphic Plot for a Soliton Solution of Sine-Gordon model of DNA 

 
ISSN: 2159-046X DNA Decipher Journal 

Published by QuantumDream, Inc. 
www.dnadecipher.com 

 

202 

 

References 
 
[1] LudmilaV. Yakushevich.Nonlinear Physics of DNA. Second, rev. ed. Berlin: Wiley-VCH Verlag 

GmBH & Co., 2004. 

[2] M. Salerno, Phys. Rev. A 44 (1991) 5292  

[3] M. Daniel & V. Vasumathi. Soliton-like base pair opening in a helicoidal DNA: An analogy with 

helimagnet and cholesterics. arXiv:0812.4536 [nlin.PS], 2008. 

[4] Richard H. Enns & George C. McGuire.Nonlinear Physics with Mathematica for Scientists and 

Engineers. Berlin: Birkhäuser, 2001, p. 443-445. 

[5] Sadri Hassani. Mathematical Methods using Mathematica: For Students of Physics and Related 

Fields. New York: Springer-Verlag New York, Inc., 2003.  

[6] Sara Cuenda, Angel Sanchez, & Niurka R. Quintero. Does the dynamics of sine-Gordon solitons 

predict active regions of DNA? Physica D 223 (2006) 214-221. 

[7] Vladimir G. Ivancevic and Tijana T. Ivancevic. Sine-Gordon solitons, Kinks and Breathers as 

Physical Models of Nonlinear Excitations in Living Cellular Structures. arXiv:1305.0613 [q-bio.OT]