1. Introduction. Tiague Takongmo Guy 1, *, Jean Roger Bogning 2. American Journal of Circuits, Systems and Signal Processing.

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1 American Journal of Circuits, Systems Signal Processing Vol. 4, No., 08, pp ISSN: (Print; ISSN: (Online Construction of Solitary Wave Solutions of Higher-Order Nonlinear Partial Differential Equations Modeled in a Modified Nonlinear Noguchi Electrical Line Tiague Takongmo Guy, *, Jean Roger Bogning Departement of Physics, Faculty of Science, University of Yaoundé, Yaoundé, Cameroon Departement of Physics, Higher Teacher s Training College, University of Bamenda, Bamenda, Cameroon Abstract Nowadays, a soliton is considered as a future wave because it is a robust, a stable a non-dissipative solitary wave. If one can use a soliton as a transmission signal in electrical lines, this will have advantages in the economic, technology educative domains. Since the propagation of soliton is due to the interaction between nonlinearity dispersion, it necessitates that the transmission medium should be nonlinear dispersive. The physical system that one has chosen for this study is a Noguchi electrical line due to the fact that it is cheaper very easy to manufacture than other transmission lines; then one find out analytically the variation that must undergo the charge of capacitors of that electrical line so that its transmission medium accepts the propagation of solitary wave of the wished type. The objective of this research is to define the analytical expression of the charge of capacitors in the line, to model higher-order nonlinear partial differential equations which govern the dynamics of those solitary waves in the line to find out some exact solutions of solitary waves type of those equations. To attain our objective, one apply Kirchhoff laws to the circuit of a modified nonlinear Noguchi electrical line to model the higher-order nonlinear partial differential equations which describe the dynamics of those solitons. Then, one apply the direct effective Bogning-Djeumen Tchaho-Kofane method based on the identification of basic hyperbolic function coefficients to construct some exact soliton solutions of modeled equations. The results obtained are supposed to permit: The amelioration of signals that will propagate in those line, the reduction of amplification stations of those lines, the facilitation of the choice of the type of the line relative to the type of signal one wishes to transmit, to augment the mathematical field knowledge, the manufacturing of new capacitors new electrical lines susceptible of propagating those solitary waves. Keywords Noguchi Electrical Line, Construction, Model, Soliton Solution, Solitary Wave, Nonlinear Partial Differential Equation, Kink, Pulse Received: July 7, 08 / Accepted: August 0, 08 / Published online: September 4, 08 The Authors. Published by American Institute of Science. This Open Access article is under the CC BY license. Introduction During the last decades, Solitary waves have evolved from a simple water wave to the propagation of Pulse solitons in optical fibers []. From the definition of solitary wave which is a wave capable to move on a long distance maintaining its shape its velocity; it has come to mind that if such a signal is used in engineering of information through Noguchi electrical line, it will resist best on diverged dissipation * Corresponding author address:

2 American Journal of Circuits, Systems Signal Processing Vol. 4, No., 08, pp factors. One has in this regard decided to come up with two definitions of nonlinear charge of capacitors constituting networks of modified Noguchi electrical line apply them to model new higher-order nonlinear partial differential equations which describe the dynamics of solitary waves in the given line. The construction of solitary wave solutions of each modeled nonlinear partial differential equation by mathematical methods presented in [-5] new Bogning- Djeumen Tchaho-Kofane method presented in [6-] has enabled to obtain solitary wave solutions of type Kink Solitary wave solution of type pulse. The work presented in this paper is partitioned as follows: In part, one carries out the general presentation of Bogning-Djeumen Tchaho- Kofane methods, in part, one presents the general modeling of a modified Noguchi electrical line. In part 4, one constructs the solitary wave solution of type Kink of the obtained differential equation, in part 5, one constructs the solitary wave solution of type Pulse of the obtained i i i l i n m x i i i xi i xi mn, xi xi differential equation. Finally, the conclusion is presented in part 6.. General Presentation of Bogning-Djeumen Tchaho-Kofane Method Bogning, Djeumen Kofane have developed an analytical method for obtaining solution of shape sech n in certain class of nonlinear partial differential equations. This method is focused on the construction of solitary wave solution has been adopted to facilitate the resolution of certain type of nonlinear partial differential equations where the nonlinear terms dispersive terms coexist. This method of construction of the solitary wave solutions intends to look for the solutions of certain categories of nonlinear partial differential equations on the form l n m ( u u u u u γ + b c + d + f u, u = 0. ( Where γ i, b i, c i, d i are the constants; i, j, l, m,n the positive integer; f a linear arbitrary function of u u the unknown to be determine u ; u the magnitude of u. One looks for solution of equation ( under the shape of a linear combination of the hyperbolic functions as follows F ( a sinh ( αx + n ( + n cosh ( αx i ( αx ( αx sinh u = a j. ( cosh Where α is a constant which depends on the system parameters which model the nonlinear partial differential equation a, the constants to be determined. Thus the combination of equation ( equation ( permits to have an equation under the shape ( αx l ( ( k ( cosh G a H a x cosh + T a sinh x cosh x + W a = 0. This equation presents five ranges of equations of coefficients a which are: the range of the coefficients F ( a of power G( a of power ( ( cosh n x sinh cosh n xx H a of power cosh k ( α x T a of power sinh ( x cosh l ( x, the range of coefficients, the range of coefficients, the range of coefficients α α the range of coefficients w( a of power. In the ranges of coefficients of cosh n x sinh cosh n xx (, the equations that best seeks the solutions are those raised to the must elevated powers. In the ranges of coefficients of cosh k ( αx ( x cosh l ( x sinh α α priority is given to the equations of coefficients of low powers. The last range of equations of coefficients w( a is not very important because it is considered like a confused domain for the correct solutions obtainable. One can classify these equations of coefficients in F a, G( a, H ( a, T ( a w( a. Here the importance or priority makes reference order of decreasing priority (

3 8 Tiague Takongmo Guy Jean Roger Bogning: Construction of Solitary Wave Solutions of Higher-Order Nonlinear Partial Differential Equations Modeled in a Modified Nonlinear Noguchi Electrical Line to the range that permits to obtain good results or merely that which tends more to the exact value. While identifying the coefficients of equation ( to zero, one gets the first range ( 0 F a = (4 ( 0 G a =. (5 In the set of the above equations, priority is given to equations which are to the power sinh cosh n xx cosh n x from most elevated. But care must be taken because it is not the most elevated power which gives the best solution directly; it depends on the shape of solution considered from the onset, the symmetry of the equation to solve as well as from its nonlinearity degree. In these conditions, one moves directly to the equations of lower powers until the good equation to solve is obtained. In the case where the first two set of equations (4 (5 don t give a satisfactory solution, one moves to the set of equations of the following range ( 0 H a = (6 ( 0 T a =. (7 In the set of equation (6 (7, priority is given to the equations of low powers of cosh k ( α x ( x cosh l ( x sinh α α. In general, the first two set of equations (4 (5 permit to find the solution of the problem. In the case where that they don t give satisfactory solution, it would be cautious to change the shape of the solution or merely the form of solution we want to construct or simply the method. The ranges w( a is considered now as the one that brings no reliable information. It is important to mention that this method appears complicated in the case where the properties of transformations of hyperbolic functions are not mastered. A mastery of these transformations reduces the difficulties considerably as regard to the calculations.. General Modeling of a Modified Nonlinear Noguchi Electrical Line Let us consider in figure, a modified Noguchi electrical line constituting various identical networks numbered by positive integer numbers n. In network n, we have: an inductor with a constant inductance L connected in parallel with a capacitor having a constant capacitor C; i n is the current flowing through the set; another capacitor whose charge varies in nonlinear manner relative to the voltage u n across it. It is to be noted that the electrical line presented in figure is that of Noguchi which was modified by the fact that, the inductor with constant inductance that used to be connected in parallel with nonlinear capacitor was disconnected or has an elevated capacitance constituting an open branch. Figure. Presentation of a modified nonlinear Noguchi electrical line.

4 American Journal of Circuits, Systems Signal Processing Vol. 4, No., 08, pp Applying Kirchhoff laws to the circuit of figure, one obtains the following equations Substituting below n n n = +, i i q ( u u in n n u = L LC + u i n n n. (8 (9 of (8 in (9 one obtains the equation ( u u in qn n n u = L + L LC + un. n (0 i Replacing L n of (9 evaluated on the next order in (0, t the following equation is obtained ( un+ un + un qn n+ n + n + = u u u LC L. ( derivative relative to x = nh which is the distance measured from the beginning of the line. h indicates the distance that separates two consecutives nodes is equivalent to the spatial sampling derivative period. As such we obtain the u + spatial partial derivative by using Taylor expansion of n un closely to u n considering the terms tills the fourth order in the following manner 4 4 h un h un h un h un n+ = n u u! x! x! x 4! x 4 4 h un h un h un h un n = n u u! x! x! x 4! x Equations ( ( permit us to have u 4 4 un h un n+ un + un = h + 4. x x ( ( (4 Using equation (4, equation ( can be rewritten as follows To obtain the continuum model, the left h member of equation ( must be approximated to a spatial partial un h un u n LCh u n qn h + + LCh + L = x x x t x (5 We can therefore present the general continuum model of a modified nonlinear Noguchi electrical line shown in figure by the partial differential equation written as follows h u( xt, u( xt, LCh u( xt, u( xt, qu ( ( xt, + h + LCh L = x x x t x (6 4. Construction of Solitary Wave Solutions of Type Kink Relative to the Partial Differential Equation (6 We define the nonlinear charge of the capacitor under the analytical shape given as follows: 5 u( xt, + A0 qu ( ( xt, = Au ( xt, + Au ( xt, + Au ( xt, + A4 ln. u( xt, A0 (7 Withu( xt, > A0. A ; A ; A A 4 are non-nil real numbers whose conditions of choice will be established. Replacing qu ( ( xt, of (7 in differential equation (6 we obtain higher order nonlinear partial differential equation given below

5 40 Tiague Takongmo Guy Jean Roger Bogning: Construction of Solitary Wave Solutions of Higher-Order Nonlinear Partial Differential Equations Modeled in a Modified Nonlinear Noguchi Electrical Line 4 LCh u ( xt, (, (, (, (, 4 (, ( LCh Au xt LCh u xt LCh A u xt u xt LCh A 4 0u xt x x 4 LCh A 0 ( ( LA LAA ( 0LAA 0 + 6LA u xt + 4 6LA A 4LAA 4 LA A u ( xt, LAA u ( xt, u( xt, + 8 (, 60 LAu ( xt, 4 LAA 0 4LA4A ( h u ( xt, h Au ( xt, h A ( h u ( xt, 4 h Au ( xt, h A 4 ( 7LAA LA4A0 u( xt, 4 ( 40LAA 0 44 LAA0 u ( xt + ( u( xt, u( xt, x u( xt, + +, = LAA 7 LA u ( xt, 40 LAu ( xt, x (8 Let us find out the solution of differential equation (8 under the analytical shape written as follow u( xt, = atanh( kx vt. (9 Where a, k v are non-nil real numbers that will be determined in terms of Noguchi electrical line parameters. Replacing (, u xt of (9 in equation (8, one obtains the following: h a k 40LAa v + 96LCh a k v 5 768LCh a A 9 5 0k v 48h a k 0La a v + 4h a k + 48LAa 4 Av LCh a k v 48LAa v A0 6h a A0k + 4h A0ak 4LAa v h a k 00LAa A0v 7LAa v A0 + LCh A0ak v LCh aa0k v + 480LAa A0v 9LCh a A0k v + 7LAa v sinh( kx vt LCh aa0k v + LCh a k v + 64h A0a k + 70LAa v cosh ( kx vt LCh aa 5 4 0k v + 88LAa v + 44LAa v A0 4LAa v A h A ak + 44LAa v A0 + 48LAa v A0 + 8h A0ak LAa 4 4 v A0 544LCh a k v 64LCh a A0k v 48LAav 4 A LAa v + 48h a A0k + 04LCh a k v 48h A0a k 0LAa A0v sinh( kx vt 5 cosh ( kx vt LAa A0v + 4h a k LCh a A0k v + 56c + 440LAa v LAa v 88LCh a k v 48h A0a k 4LAa v 60LAa A0v sinh( kx vt sinh( kx vt LAa 7 A0v 4h a k + 44LAa v 9 67LCh a k v + 70LCh aa0k v cosh ( kx vt cosh ( kx vt LCh a k v 440LCh a k v LAa A0v + 7LCh a k v LCh a A0k v 60LAa v sinh( kx vt + ( 60LAa v + 70LCh a k v = 0. cosh ( kx vt Equation (0 is valid if only if each of its basic hyperbolic function coefficients is zero. This permit to obtain the set of equations presented below (0

6 American Journal of Circuits, Systems Signal Processing Vol. 4, No., 08, pp h a k 40LAa v + 96LCh a k v 0La a v + 4h a k LAa 4 Av 0 6h a A0k + 4h A0ak 4LAa v 7LAa v A LCh A0ak v 9LCh a A0k v + 7LAa v = LCh aa0k v + LCh a k v + 44LAa v A LAa v A0 + 48LAa v A0 + 8h A0ak LCh a A0k v 48LAav 4 A0 48h A0a k 0LAa A0v LCh a A0k v 48h a k 480LCh a k v 48LAa v A h a k 00LAa A0v 480LCh aa0k v + 480LAa A0v h A0a k + 70LAa v 88LCh aa0k v + 88LAa v = h A0ak + 44LAa v A0 4LAa v A0 544LCh a k v LAa v + 48h a A0k + 04LCh a k v LAa A0v + 4h a k 57LCh a A0k v + 56c + 440LAa v h A0a k 4LAa v 60LAa A0v + 67LCh a k v + 70LCh aa0k v = LAa A0v + 7LCh a k v 400LCh a A0k v 60LAa v LAa v 88LCh a k v 70LAa A0v 4h a k + 44LAa v = LCh a k v 440LCh a k v ( 60LAa v + 70LCh a k v = 0. ( Solving the set of equation (, one obtains the solution the conditions given in ( of higher-order nonlinear partial differential equation (8 which govern the dynamics of solitary wave of type Kink in the modeled Noguchi electrical line A 4 = a = A 0 ; A0 A k = ± h C 4 A0 A ; v = ± LC A CA + 5AA ( 0 4 AA 0 C AA0 A A0A 6A C + C A A = A A A C C A AA C A A A A A A ; ; A > 0 A + 5AA 0 > CA ; A0 A 4 A0 A u( xt, = A0 tanh ± x t. h C ± LC ( A CA + 5AA 0 ( 5. Construction of Solitary Wave Solutions of Type Pulse Relative to the Partial Differential Equation (6 We define the nonlinear charge of the capacitor under the analytical shape given below 5 qu ( ( xt, = Au( xt, + Au ( xt, + Au ( xt,. ( Where A ; A ; A 4 A are non-zeros real numbers whose conditions of choice will be established. Replacing ( (, qu xt of ( in differential equation (6 one obtains the nonlinear partial differential equation given as follow

7 4 Tiague Takongmo Guy Jean Roger Bogning: Construction of Solitary Wave Solutions of Higher-Order Nonlinear Partial Differential Equations Modeled in a Modified Nonlinear Noguchi Electrical Line h + + LCh 4 4 h u( xt, u( xt, LCh u( xt, u( xt, x x x x 4 ( LA LAu ( xt, 5 LAu ( xt, + ( LAu xt LAu xt u( xt, u( xt, + 6 (, 0 (, = 0. (4 Let us find the solution of equation (4 under the analytical shape ( kx vt u( xt, = asech. (5 Where a, k v are non-nil real numbers that will be determined in terms of Noguchi electrical line parameters. Replacing u( xt, of (5 in equation (4 yields following equation ( LAa v + 4LCh av k 5LAa v + 70LCh av k + h ak LAav + h ak + h ak + LCh av k + LCh av k cosh cosh ( kx vt ( kx vt h ak LCh av k 9LAa v 0LCh av k + LAav h ak 6 cosh ( LCh av k + 0LAa v = 0. 7 cosh kx vt ( ( kx vt (6 Equation (6 is valid if only if each of its basic hyperbolic function coefficients is nil. This permit us to obtain the set of four equations written below LAa v + 4LCh av k 5LAa v + 70LCh av k + h ak = 0, LAav h ak h ak LCh av k LCh av k 0, = h ak LCh av k 9LAa v 0LCh av k + LAav h ak = 0, LCh av k + 0LAa v = 0. (7 Solving the set of nonlinear equations (7, one obtains the solutions the conditions presented in (8 of nonlinear partial differential equation (4 which governs the dynamics of solitary waves of type Pulse in the modeled Noguchi electrical line: AC + 54 AC AL + 0AA C L 6LAC A 56AC L A C AL A v = ± L A C A + 4 AC A + 54AC + 50AA C A L A AC L A + AC L A A C AL AAC L A L AC A ;

8 American Journal of Circuits, Systems Signal Processing Vol. 4, No., 08, pp ( ( ( 0 + v ALC 0Lv C + AC + A a = A Lv C ; a A k = ± h C 4 ; u xt a ( kx vt (, = sech. (8 Let s recall that a; v; K must be evaluated with real numbers A ; A A chosen in sort a way that the expressions in the roots should be positive. 6. Conclusion At the end of this work where one has modeled a modified Noguchi electrical line by two different higher-order nonlinear partial differential equations which have permitted us to construct solitary wave solutions; it is necessary to point out that the results obtained will first of all enable in physics engineering of telecommunication, the manufacturing of two new transmission lines notably those of Noguchi electrical line presented in figure whose charge of nonlinear capacitors of its networks varies for one in nonlinear manner defined in (7 the other in nonlinear manner defined in (4. In addition, these results will permit an amelioration of the quality of signals that will be propagated in the new lines. In fact, those signals are solitary waves of type Kink obtained in ( type Pulse obtained in (8 which by their definition are propagated on a long distance without changing their shape velocity by resisting best on different dissipation factors. Finally, in a typical mathematical domain, the results obtained has permitted to define in (8 in (4 two new higher-order nonlinear partial differential equations which have respectively exact solitary wave solutions ( (8; this by increasing the field of mathematical knowledge. In order to bring new ideas on the stability of the obtained solitary waves; it is necessary for us to study next their modulational instability before carrying out a practical study where we will experiment the applicability the perfection of those two new lines. Acknowledgements We are grateful to all those who have been a source of inspiration in our studies; to the University of Yaoundé the ministry of higher education of Cameroon whom through their support has enabled us to achieve this work. References [] J. R. Bogning, A. S. Tchakoutio Nguetcho, T. C. Kofané, Gap solitons coexisting with bright soliton in nonlinear fiber arrays. International Journal of nonlinear science numerical simulation, Vol. 6 (4, ( [] Ekici, M., Zhou, Q., Sonmezoglu, A., Moshokoa, S. P., Zaka Ullah, M., Biswas, A., Belic, M.: Solitons in magneto-optic waveguides by extended trial function scheme. Superlattices Microstruct. 07, 97 8 (07c. [] Jawad, A. J. M., Mirzazadeh, M., Zhou, Q., Biswas, A.: Optical solitons with anti-cubic nonlinearity using three integration schemes. Superlattices Microstruct. 05, 0 (07. [4] Mirzazadeh, M., Ekici, M., Sonmezoglu, A., Eslami, M., Zhou, Q., Kara, A. H., Milovic, D., Majid, F. B., Biswas, A., Belic, M.: Optical solitons with complex Ginzburg Lau equation. Nonlinear Dyn. 85 (, (06. [5] Seadawy, A. R., Lu, D.: Bright dark solitary wave soliton solutions for the generalized higher order nonlinear Schrodinger equation its stability. Results Phys. 7, 4 48 (07. [6] A. M. Wazwaz, The tanh-coth method for new compactons solitons solutions for the K(n,n K(n+, n+ equations, persive K(m,n equations, Appl. Math. Comput, 88, ( [7] M. L. Wang, Exact solutions for a compound Käv- Burgers equation, Phys. Lett. A., ( [8] E. Fan, Extended tanh-function method its applications to nonlinear equations, Phys. Lett. A, 77, ( [9] E. Fan, J. Zhang, A note on the homogeneous balance method, Phys. Lett. A, 05, ( [0] Y. B. Zhou, M. L. Wang, Y. M. Wang, Periodic wave solutions to a coupled KdV equations with variable coefficients, Phys. Lett. A, 08, (00-7. [] A. M. Wazwaz, Solutions of compact non compact structures for nonlinear Klein-Gordon type equation, Appl. Math. Comput, 4, ( [] A. M. Wazwaz, Traveling wave solutions of generalized forms of Burgers KdV Burgers Huxly equations, Appl. Math. Comput, 69, ( [] Z. Feng, The first integral method to study the Burgers Korteweg-de-Vries equation, J. Phys. Lett. A. 5, (00, [4] Z. Feng, On explicit exat solutions for the Lienard equation its applications, J. Phys. Lett. A. 9, (00, [5] Z. Feng, On explicit exact solutions to compound Burgers- KdV equation, J. Math. Anal. Appl. 8, (007, [6] J. R. Bogning, C. T. Djeumen Tchaho T. C. Kofané, Construction of the soliton solutions of the Ginzburg-Lau equations by the new Bogning-Djeumen Tchaho-Kofané method, Phys. Scr, Vol. 85, (0, [7] J. R. Bogning, C. T. Djeumen Tchaho T. C. Kofané, Generalization of the Bogning- Djeumen Tchaho-Kofane Method for the construction of the solitary waves the survey of the instabilities, Far East J. Dyn. Sys, Vol. 0, No., (0, 0-9.

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