Applied Mathematics and Computation 246 (2014) 460–463

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Solitary wave and shock wave solitons to the transmission linemodel for nano-ionic currents along microtubules

http://dx.doi.org/10.1016/j.amc.2014.08.0530096-3003/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author.E-mail addresses: younis.pu@gmail.com (M. Younis), safdarali.mu@gmail.com (S. Ali).

Muhammad Younis a,⇑, Safdar Ali b

a Centre for Undergraduate Studies, University of the Punjab, Lahore 54590, Pakistanb Department of Mathematics, Minhaj University, Lahore, Pakistan

a r t i c l e i n f o a b s t r a c t

Keywords:Solitary solitonsShock solitonsSolitary wave ansatzTransmission line model

In this letter, the solitary wave and shock wave solitons for nonlinear equation of specialinterest in nanobiosciences, namely the transmission line model for nano-ionic currentsalong microtubules, have been constructed successfully. The solitary wave ansatz is usedto carry out the solutions which shows the consistency.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Nonlinear phenomena is one of the basic and fundamental object of nature and a growing interest has been given to thepropagation of nonlinear waves in the dynamical system. It appears in almost every field of life such as electrochemistry,electromagnetic, fluid dynamics, acoustics, cosmology, astrophysics and plasma physics [1–4].

The study of exact solutions of the nonlinear equation plays a vital role in soliton theory. The available solutions of thesenonlinear equations facilitates the numerical solvers and aids in the stability analysis of the solutions. For the sack to find thesolutions of nonlinear equations a lot of techniques have been developed in recent past, i.e., the ðG0=GÞ-expansion method[5,6], the first integral method [7], the Adomian decomposition method [8,9], the generalized differential transform method[10], Jacobi elliptic method [11], the automated tanh-function method [12] and the modified simple equation method [13]etc. In recent times, the different researchers [14–18] have celebrated the soliton solutions of nonlinear equations and dis-cussed their importance. For more references see also [19–25].

In this letter, the solitary wave ansatz method [26,27] has been used, which is rather heuristic and has significant featuresthat make it practical for the determination of single soliton solutions for a wide class of nonlinear equations. The solitarywave and shock wave solitons, using solitary wave ansatz method, for nonlinear equation of special interest in nanobio-sciences (namely the transmission line model for nano-ionic currents along microtubules) have been constructedsuccessfully.

Microtubules are very important cytoskeletal structures implicated in different cellular activities. We should mention thecell division and traffic of organelles (mitochondria, vesicles and other cargos) by kinesin and dynein motor proteins. Theconditions enabling microtubules to act as nonlinear electrical transmission lines for ions flow along their cylinders aredescribed by Sataric et al. [28] and Sekulic et al. [29]. A model in which each tubulin dimmer protein is an electric elementwith a capacitive, resistive and negative incrementally resistive property due to polyelectrolyte nature of microtubules incytosol is described. The model reads

M. Younis, S. Ali / Applied Mathematics and Computation 246 (2014) 460–463 461

L2

3uxxx þ

Z32

LðvGo � 2dCoÞuut þ 2ux þ

ZCo

Lut þ

1LðRZ�1 � GoZÞu ¼ 0; ð1:1Þ

where R ¼ 0:34� 109X the resistance of the elementary rings (ER) L ¼ 8� 10�9 m; Co ¼ 1:8� 10�15 F is the total maximalcapacitance of the ER. Go ¼ 1:1� 10�13 Si is the conductance of pertaining nano-pores (NPS) and Z ¼ 5:5� 1010 X is thecharacteristic impedance of the system. The parameter d and v describe the nonlinearity of ER capacitor and conductanceof NPS in ER, respectively.

The rest of the article is organized as follows, in Section 2 the solitary wave solitons and in Section 3 the shock wave sol-itons for the model described in equation (1.1) have been constructed, respectively. In the last Section 4, the conclusion hasbeen drawn.

2. Solitary wave solitons

In this section, the solitary wave solitons (bright solitons or non-topological solutions) for the equation (1.1) have beenfound using the solitary wave ansatz. For this, we have

uðx; tÞ ¼ Acoshpn

and n ¼ Bðx� mtÞ: ð2:2Þ

Where A is the amplitude of the solitons, B is the inverse width of the solitons while m is the velocity of the solitary wave. Thevalue of the exponent p is determined later using the homogeneous balance. From equation (2.2), it can be followed

ux ¼�ABp tanh n

coshpn; ð2:3Þ

ut ¼ABmp tanh n

coshpn; ð2:4Þ

uut ¼A2Bmp tanh n

cosh2pn; ð2:5Þ

uxxx ¼�AB3p3 tanh n

coshpnþ AB3pðpþ 1Þðpþ 2Þ tanh n

coshpþ2n: ð2:6Þ

After the involvement of equations (2.3)–(2.6) and equation (1.1), the one can obtain the following equation

�L2

3AB3p3 tanh n

coshpnþ L2

3AB3pðpþ 1Þðpþ 2Þ tanh n

coshpþ2nþ Z

32

LðvGo � 2dCoÞ

A2Bmp tanh n

cosh2pn� 2

ABp tanh n

coshpn

þ ZCo

LABmp tanh n

coshpnþ 1

LðRZ�1 � GoZÞ A

coshpn¼ 0 ð2:7Þ

It may be noted that p ¼ 2 is being calculated when exponents pþ 2 and 2p are equated equal. Furthermore, set the coeffi-cients of the linearly independent terms to zero. Thus, we can write

L2

3AB3pðpþ 1Þðpþ 2Þ þ Z

32

LðvGo � 2dCoÞA2Bmp ¼ 0;

�L2

3AB3p3 � 2ABpþ ZCo

LABmp ¼ 0:

Solving the above system of equations for p ¼ 2, and then we get the amplitude A of the soliton as

A ¼ 3ð2L� ZComÞZ

32mðvGo � 2dCoÞ

; ð2:8Þ

while the inverse width of the soliton is given by

B ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZCom� 2L

4L3

s: ð2:9Þ

Moreover, the soliton width given in (2.9) provokes a constraint condition that is given by

ðZCom� 2LÞðL3Þ > 0: ð2:10Þ

462 M. Younis, S. Ali / Applied Mathematics and Computation 246 (2014) 460–463

Hence, the solitary wave solution of the nanobiosciences equation (1.1) is given by

uðx; tÞ ¼ 3ð2L� ZComÞZ

32mðvGo � 2dCoÞ

sech2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiZCom� 2L

4L3

sðx� mtÞ

( ): ð2:11Þ

3. Shock wave solitons

In this section, the shock wave solutions (dark solutions or topological solutions) for equation (1.1) have been found usingthe solitary wave ansatz. For this, we have

uðx; tÞ ¼ A tanhpn and n ¼ Bðx� mtÞ for p > 0: ð3:12Þ

Where A and B are free parameters and are the amplitude and inverse width of the soliton, respectively, while m is the veloc-ity of the soliton. The value of the exponent p is determined later. It can, thus, be written from equation (3.12) as

uxxx ¼ AB3pðp� 1Þðp� 2Þ tanhp�3n� AB3pð3p2 � 3pþ 2Þ tanhp�1nþ AB3pð3p2 þ 3pþ 2Þ tanhpþ1n

� AB3pðpþ 1Þðpþ 2Þ tanhpþ3n; ð3:13Þ

uut ¼ �A2Bmp tanh2p�1nþ A2Bmp tanh2pþ1n; ð3:14Þ

ux ¼ ABp tanhp�1n� ABp tanhpþ1n; ð3:15Þ

ut ¼ �ABmp tanhp�1nþ ABmp tanhpþ1n: ð3:16Þ

After substituting equations (3.13)–(3.16) into (1.1), the following equation is obtained

L2

3AB3pðp� 1Þðp� 2Þ tanhp�3n� L2

3AB3pð3p2 � 3pþ 2Þ tanhp�1n

þ L2

3AB3pð3p2 þ 3pþ 2Þ tanhpþ1n� L2

3AB3pðpþ 1Þðpþ 2Þ tanhpþ3n� Z

32

LðvGo � 2dCoÞA2Bmp tanh2p�1n

þ Z32

LðvGo � 2dCoÞA2Bmp tanh2pþ1nþ 2ABp tanhp�1n� 2ABp tanhpþ1n� ZCo

LABmp tanhp�1n

þ ZCo

LABmp tanhpþ1nþ 1

LðRZ�1 � GoZÞA tanhpn ¼ 0:

It may be noted that p ¼ 2 is being calculated when exponents pþ 3 and 2pþ 1 are to be set equal to each other. Further-more, set the coefficients of the linearly independent terms to zero. It can, thus, be written as

� L2

3AB3pðpþ 1Þðpþ 2Þ þ Z

32

LðvGo � 2dCoÞA2Bmp ¼ 0;

� L2

3AB3pð3p2 � 3pþ 2Þ þ 2ABp� ZCo

LABmp ¼ 0:

Solving the above system of equations and also set p ¼ 2, then it can be written

A ¼ 3ð2L� ZComÞ2Z

32mðvGo � 2dCoÞ

; m ¼ m; B ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð2L� ZComÞ

8L3

s:

Hence, the shock wave solution of the nanobiosciences equation (1.1) is given by

uðx; tÞ ¼ 3ð2L� ZComÞ2Z

32mðvGo � 2dCoÞ

tanh2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3ð2L� ZComÞ

8L3

sðx� mtÞ

!: ð3:17Þ

It may be noted that the soliton width B also provokes a constraint condition, which is given as

ð2L� ZComÞðL3Þ > 0: ð3:18Þ

4. Conclusion

The solitary wave ansatz method processes significant features that makes it practical for the determination of solitonsolutions for a wide class of nonlinear evolution equations. The solitary wave and shock wave solitons have been con-structed, using the solitary wave ansatz method, for the nonlinear equation of special interest in nanobiosciences, namely

M. Younis, S. Ali / Applied Mathematics and Computation 246 (2014) 460–463 463

the transmission line model for nano-ionic currents along microtubules and we clearly see that the consistency, which hasbeen applied successfully.

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