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International Journal of Application or Innovation in Engineering & Management (IJAIEM),Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com Volume 1, Issue 2, October 2012 ISSN 2319 - 4847 Analysis of the Solitary Wave Solution based on the Jacobi Elliptic Functions JIANG Xingfang 1, XU Songlin 2 , WEI Jianping 3 , LIN Jin-ling 4 and Tang Bin 5 1,2,3,4,5 School of Matheamtics and Physics, Changzhou University, Changzhou, 213164 Abstract For researching the dispersion problem of information transmission in optical fiber, the nonlinear Schrödinger equation had been constructed from the generalized high order nonlinear Schrodinger equation which was deducted from Maxwell equations. The nonlinear Schrödinger equation was included the group velocity dispersion, linear potential, gain and nonlinearity. The solitary wave solution had been obtained by trial method. One of the forms was Jacobi elliptic functions. The numerical calculation results were shown figures by Matlab. The results showed four points. The first one was that the high intensity of the solitary waves were occurred in (0, 1.2) for x direction. The second one was that the solitary waves were periodic paired up the time axis. The third one was that the dispersion of the solitary waves when the parameter k was larger. The scope of k was (0, 1). The fourth one was that the dispersion of the solitary waves was relevant of x direction and was irrelevant of time axis. The transmission of the solitary waves remained unchanged for long time. Keywords: Information Optics; dispersion; Jacobi elliptic function 1. INTRODUCTION The information communication volume presents exponential growth with people's living standard rise ceaselessly. In the process of the optical fiber replaced gradually the copper core cable, the signal attenuation and dispersion were two problems. The problems limited in optical fiber transmission of information. One of them the signal attenuation had been controlled effectively when the low loss transmission windows of transmission wavelength were found and the rare-earth doped fiber was developed successfully. With the large power high capacity information transmission in optical fiber, the signal transmission was limited by the dispersion such as the material dispersion, waveguide dispersion, polarization mode dispersion, and other restrictions. The solitary wave that discovered by John Scott Russell in 1834 at first had the characteristics for keeping the sharp and the amplitude. It got an idea from the characteristics of the solitary wave. In 1870 the Boussinseq equation had been built. In 1895 the KdV equation had been studied for the motion of the shallow water wave. In 1973, Hasegawa and Tappert first predicted the solitary wave in a nonlinear optical fiber and it had been confirmed in 1980 by Mollenauer et al in experiment [1]. The studied results were shown that the optical solitary wave pulses were produced in balance conditions. The movement of the optical solitary wave pulses was according to the nonlinear Schrodinger equation. The balance conditions were balance between the group velocity dispersion and the self phase modulation in anomalous dispersion region. The optical solitary wave pulses were in all kinds of systems such as in the photonic crystal fiber. Literatures [1-5] introduced initially Rogue wave that was found from the ocean. The solitary wave in optical fiber was similar to the Rogue wave in ocean and its movement was according to the nonlinear Schrödinger equation. It was an important issue for the optical solitary wave pulse keeping the spectral width in transmission process or producing as small as possible spectral broadening. The transmission capacity of the transmission achieved as high as possible. A very good scheme was that the ultra short optical pulse was distortionless transmission in optical fiber. At the same time the solitary wave pulse in nonlinear science was one of the important research fields in 21 Century. The researching of the optical solitary wave had important theory significance and great practical value. 2. NONLINEAR SCHRÖDINGER EQUATION AND ITS SOLUTION The component of the electric field could be expressed as equation (1) in the approximation condition of varying slowly envelope from passive Maxwell equations. 2 E (r , 0 ) ( )k02 E ( r , 0 ) 0 (1) Where E (r , 0 ) was Fourier transform of E ( r , t ) . The symbol was dielectric constant. For non ferromagnetic 2 material r 1 and k0 , 0 was wavelength in vacuum. 0 The equation (2) was obtained after the variables had been separated E ( r , 0 ) F ( x, y ) A( z , 0 )ei z 0 (2) Volume 1, Issue 2, October 2012 Page 116 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com Volume 1, Issue 2, October 2012 ISSN 2319 - 4847 2A The item had been ignored. Then z 2 2F 2F A 2 [ ( ) k 02 2 ]F 0 , 2i ( 2 02 ) A 0 (3) x 2 y z The equation (1) could be written as ax 2 E (r ,t ) [ F ( x, y ) A( z , t )e i ( 0t 0 z ) c.c.] (4) 2 The Fourier transform for the amplitude of varying slowly A( z , t ) was A( z , 0 ) . A i[ ( ) 0 ] A (5) z The approximation was 2 02 ( 0 )( 0 ) 2 0 ( 0 ) It noted ( ) ( ) and the Taylor expansion of ( ) was as equation (6) ( 0 ) 2 ( 0 ) 3 ( ) 0 ( 0 ) 1 2 3 ... (6) 2 6 dm Where m , | A |2 i and i could be replaced as ( 0 ) . The equation (5) could be written as d m 0 2 t equation (7) A in n n A i ( ) A i | A | 2 A (7) z 2 n 1 n! t Equation (7) was a high order nonlinear Schrödinger equation. If letting n=2, 2 , 2i 2 , and the variable z and the variable t was exchanged. The variable z was replaced as x, and the variable A was replaced as . Then (t ) 2 i ( x, t ) g (t ) | |2 i (t ) (8) t 2 x 2 The intensity of the wave function could be expressed as equation (9) in reference [3]. 2 2 2 2 2 t ( s ) ds {2[ (t ) x (t )] 4 (t ) 3} 64 (t ) 2 | ( x, t ) |2 a0 | (t ) | e 0 2 2 2 (9) {1 2[ (t ) x (t )] 4 (t )} 3. NUMERICAL CALCULATION AND ITS ILLUSTRATION sn (t , k ) The group velocity was selected as , dn (t , k ) 1 t dn (t , k ) 2 cn (t , k )dt 2 0 The functions sn(t , k ) , cn (t , k ) , and dn (t , k ) were Jacobi elliptic functions [9]. The integral of could be replaced as sum in numerical calculation. For k (0, 0.1) , the step of k was took as 0.001, x (0, 4) , the step of x was took as 0.005, t (0, 40) , the step of t was took as 0.05. For k [0.1, 0.5) , the step of k was took as 0.1, x (0, 4) , the step of x was took as 0.005, t (0, 40) , the step of t was took as 0.05. For k [5, 9) , the step of k was took as 1, x (0, 8) , the step of x was took as 0.01, t (0, 40) , the step of t was took as 0.05. For k [9, 10) , the step of k was took as 0.1, x (0, 16) , the step of x was took as 0.02, t (0, 40) , the step of t was took as 0.05. The figures of the relative intensity in k=0.001, 0.1, 0.5, 0.9, 0.95, 0.99 were shown in Figure 1~Figure 6. Volume 1, Issue 2, October 2012 Page 117 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com Volume 1, Issue 2, October 2012 ISSN 2319 - 4847 4. CONCLUSIONS The conclusions include four aspects. The first one was that the nonlinear Schrödinger equations had over one hundred kinds of expression forms and models [7]. The expression forms could be derived strictly from Maxwell equations. The second one was that the solutions of the nonlinear Schrödinger equations could be obtained by trial method. The two coupled equations could be got when the real part was zero and the imaginary part was zero. The third one was that the wave functions were normalized wave functions and the Jacobi elliptic function was one of them. The fourth one was that the results for the Jacobi elliptic function shown in three points. The first point was that that the solitary wave intensity focused on (0, 1.2) region in x direction. The second point was that the solitary waves were periodic paired up the time axis. The third point was that the dispersion of the solitary waves when the parameter k was large. The scope of k was (0, 1). The fourth point was that the dispersion of the solitary waves was relevant of x direction and was irrelevant of time axis for fixed k. The transmission of the solitary waves remained unchanged for long time. The results were benefit to discuss the physics problem of the optical solitary wave transmission and to build the trans- mission model for optical solitary wave pulse in single mode optical fiber. It had important theory significance, and has great practical value. (a) Intensity of solitary wave ~ x, t (b) Intensity of solitary wave in plane of x~t Figure 1 k=0.001 (a) Intensity of solitary wave ~ x, t (b) Intensity of solitary wave in plane of x~t Figure 2 k=0.1 Volume 1, Issue 2, October 2012 Page 118 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com Volume 1, Issue 2, October 2012 ISSN 2319 - 4847 (a) Intensity of solitary wave ~ x, t (b) Intensity of solitary wave in plane of x~t Figure 3 k=0.5 (a) Intensity of solitary wave ~ x, t (b) Intensity of solitary wave in plane of x~t Figure 4 k=0.9 (a) Intensity of solitary wave ~ x, t (b) Intensity of solitary wave in plane of x~t Figure 5 k=0.95 Volume 1, Issue 2, October 2012 Page 119 International Journal of Application or Innovation in Engineering & Management (IJAIEM) Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com Volume 1, Issue 2, October 2012 ISSN 2319 - 4847 (a) Intensity of solitary wave ~ x, t (b) Intensity of solitary wave in plane of x~t Figure 6 k=0.99 ACKNOWLEDGMENT This paper authors express their sincere thanks to the reviewer for suggestions and comments. This work is supported by Open Issues of Jiangsu Key Lab of Modern Optical Technology, Soochow University (KJS1004). The authors acknowledge support from the open issues of State Key laboratory of Satellite Ocean Environment Dynamics (No. SOED1201). REFERENCES [1] L. F. Mollenauer, R. H. Stolen, J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett., 45, pp. 1095-1098, 1980. [2] Z. Y. Yan, “Rogon-like solutions excited in the two-dimensional nonlocal nonlinear Schrödinger equation,” Journal of Mathematical Analysis and Applications, 380, pp. 689-696, 2011. [3] Z. Y. Yan, “Nonautonomous ‘Rogons’ in the inhomogeneous nonlinear Schrödinger equation with variable coefficients,” Physics Letters A., 374, pp. 672-679, 2010. [4] A. Ankiewicz, N. Devine, N. Akhmediev, “Are rogue waves robust against perturbations?,” Physics Letters A, 373, pp. 3997-4000, 2009. [5] C. Q. Dai, Y. Y. Wang, G. Q. Zhou, “The realization of controllable three dimensional rogue waves in nonlinear inhomogeneous system, ” Chaos, Solitons & Fractals, 45, pp. 1291-1300, 2012. [6] N. Akhmediev, J. M. Soto-crespo, A. Ankiewicz, et al, “Early detection of rogue waves in a chaotic wave field, ” Physics Letters A, 375, pp. 2999-3001, 2011. [7] R. M. Caplan, Q. E. Hoq, R. Carretero-Gonzalez, et al, “Azimuthal modulational instability of vortics in the nonlinear Schrödinger equation, ” Optics Communications, 282, pp. 1399-1405, 2009. [8] V. N. Serkin, A. Hasegawa, “Exactly intergrable nonlinear Schrödinger equation modes with varying dispersion, nonlinearity and gain: application for soliton dispersion managements, ” Journal of Selected Topics in Quantum Electronics, 8(3), pp. 418-431, 2002. [9] Jacobi elliptic functions. [Online]. Available: http://en.wikipedia.org/wiki/Jacobi_ elliptic_ functions. [Accessed: Sept. 12, 2012]. AUTHOR JIANG Xingfang (1963- ), Male, Han nationality, Professor, Doctor, Optical Engineering. He received the B. S. degree in Physics in 1985 from Department of physics, Nanjing University. He received the M. S. degree in Physics in 2001 from Department of physics, East China Normal University. He received Ph. D. in Optical Engineering in 2007 from Nanjing University of Science and Technology. He is a professor in Changzhou University. His research interests are in Optical Engineering and Computer Application. Volume 1, Issue 2, October 2012 Page 120