Open Access
2015 Bilinear Form and Two Bäcklund Transformations for the (3+1)-Dimensional Jimbo-Miwa Equation
He Li, Yi-Tian Gao
Abstr. Appl. Anal. 2015: 1-5 (2015). DOI: 10.1155/2015/834521
Abstract

With Bell polynomials and symbolic computation, this paper investigates the (3+1)-dimensional Jimbo-Miwa equation, which is one of the equations in the Kadomtsev-Petviashvili hierarchy of integrable systems. We derive a bilinear form and construct a bilinear Bäcklund transformation (BT) for the (3+1)-dimensional Jimbo-Miwa equation, by virtue of which the soliton solutions are obtained. Bell-polynomial-typed BT is also constructed and cast into the bilinear BT.

References

1.

M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, UK, 1991. MR1149378 0762.35001 M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, UK, 1991. MR1149378 0762.35001

2.

M. P. Barnett, J. F. Capitani, J. von zur Gathen, and J. Gerhard, “Symbolic calculation in chemistry: selected examples,” International Journal of Quantum Chemistry, vol. 100, no. 2, pp. 80–104, 2004. M. P. Barnett, J. F. Capitani, J. von zur Gathen, and J. Gerhard, “Symbolic calculation in chemistry: selected examples,” International Journal of Quantum Chemistry, vol. 100, no. 2, pp. 80–104, 2004.

3.

X. Lü, H.-W. Zhu, X.-H. Meng, Z.-C. Yang, and B. Tian, “Soliton solutions and a Bäcklund transformation for a generalized nonlinear Schrödinger equation with variable coefficients from optical fiber communicationsčommentComment on ref. [3a?]: We split this reference to [3a,3b?]. Please check similar cases throughout.,” Journal of Mathematical Analysis and Applications, vol. 336, no. 2, pp. 1305–1315, 2007. MR2353015 10.1016/j.jmaa.2007.03.017 1128.35385 X. Lü, H.-W. Zhu, X.-H. Meng, Z.-C. Yang, and B. Tian, “Soliton solutions and a Bäcklund transformation for a generalized nonlinear Schrödinger equation with variable coefficients from optical fiber communicationsčommentComment on ref. [3a?]: We split this reference to [3a,3b?]. Please check similar cases throughout.,” Journal of Mathematical Analysis and Applications, vol. 336, no. 2, pp. 1305–1315, 2007. MR2353015 10.1016/j.jmaa.2007.03.017 1128.35385

4.

X. Lü, “Bright-soliton collisions with shape change by intensity redistribution for the coupled Sasa-Satsuma system in the optical fiber communications,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 11, pp. 3969–3987, 2014. MR3212791 10.1016/j.cnsns.2014.03.013 07175188 X. Lü, “Bright-soliton collisions with shape change by intensity redistribution for the coupled Sasa-Satsuma system in the optical fiber communications,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 11, pp. 3969–3987, 2014. MR3212791 10.1016/j.cnsns.2014.03.013 07175188

5.

B. Tian, Y. T. Gao, and H. W. Zhu, “Variable-coefficient higher-order nonlinear Schrödinger model in optical fibers: variable-coefficient bilinear form, Bäcklund transformation, brightons and symbolic computation,” Physics Letters A, vol. 366, pp. 223–229, 2007. 1203.81067 10.1016/j.physleta.2007.02.098 B. Tian, Y. T. Gao, and H. W. Zhu, “Variable-coefficient higher-order nonlinear Schrödinger model in optical fibers: variable-coefficient bilinear form, Bäcklund transformation, brightons and symbolic computation,” Physics Letters A, vol. 366, pp. 223–229, 2007. 1203.81067 10.1016/j.physleta.2007.02.098

6.

B. Tian and Y. T. Gao, “Symbolic computation on cylindrical-modified dust-ion-acoustic nebulons in dusty plasmas,” Physics Letters A, vol. 362, no. 4, pp. 283–288, 2007. MR2157461 1197.82028 10.1016/j.physleta.2006.10.094 B. Tian and Y. T. Gao, “Symbolic computation on cylindrical-modified dust-ion-acoustic nebulons in dusty plasmas,” Physics Letters A, vol. 362, no. 4, pp. 283–288, 2007. MR2157461 1197.82028 10.1016/j.physleta.2006.10.094

7.

Z.-Y. Yan and H.-q. Zhang, “Symbolic computation and new families of exact soliton-like solutions to the integrable Broer-Kaup (BK) equations in $(2+1)$-dimensional spaces,” Journal of Physics A: Mathematical and General, vol. 34, no. 8, pp. 1785–1792, 2001. MR1818767 10.1088/0305-4470/34/8/320 Z.-Y. Yan and H.-q. Zhang, “Symbolic computation and new families of exact soliton-like solutions to the integrable Broer-Kaup (BK) equations in $(2+1)$-dimensional spaces,” Journal of Physics A: Mathematical and General, vol. 34, no. 8, pp. 1785–1792, 2001. MR1818767 10.1088/0305-4470/34/8/320

8.

X. Lü and M. Peng, “Systematic construction of infinitely many conservation laws for certain nonlinear evolution equations in mathematical physics,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 9, pp. 2304–2312, 2013. MR3042037 10.1016/j.cnsns.2012.11.006 1304.35030 X. Lü and M. Peng, “Systematic construction of infinitely many conservation laws for certain nonlinear evolution equations in mathematical physics,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 9, pp. 2304–2312, 2013. MR3042037 10.1016/j.cnsns.2012.11.006 1304.35030

9.

X. Lü and M. Peng, “Nonautonomous motion study on accelerated and decelerated solitons for the variable-coefficient Lenells-Fokas model,” Chaos, vol. 23, no. 1, Article ID 013122, 2013. 1319.37010 10.1063/1.4790827 X. Lü and M. Peng, “Nonautonomous motion study on accelerated and decelerated solitons for the variable-coefficient Lenells-Fokas model,” Chaos, vol. 23, no. 1, Article ID 013122, 2013. 1319.37010 10.1063/1.4790827

10.

C. Rogers and W. F. Shadwick, Bäacklund Transformations and Their Applications, Academic Press, New York, NY, USA, 1982. 0492.58002 C. Rogers and W. F. Shadwick, Bäacklund Transformations and Their Applications, Academic Press, New York, NY, USA, 1982. 0492.58002

11.

X. Lü, “New bilinear Bäcklund transformation with multisoliton solutions for the (2+1)-dimensional Sawada–Kotera model,” Nonlinear Dynamics, vol. 76, no. 1, pp. 161–168, 2014. MR3189162 10.1007/s11071-013-1118-y 1319.35222 X. Lü, “New bilinear Bäcklund transformation with multisoliton solutions for the (2+1)-dimensional Sawada–Kotera model,” Nonlinear Dynamics, vol. 76, no. 1, pp. 161–168, 2014. MR3189162 10.1007/s11071-013-1118-y 1319.35222

12.

X. Lü and J. Li, “Integrability with symbolic computation on the Bogoyavlensky-Konoplechenko model: BELl-polynomial manipulation, bilinear representation, and Wronskian solution,” Nonlinear Dynamics, vol. 77, no. 1-2, pp. 135–143, 2014. MR3215135 10.1007/s11071-014-1279-3 1314.37049 X. Lü and J. Li, “Integrability with symbolic computation on the Bogoyavlensky-Konoplechenko model: BELl-polynomial manipulation, bilinear representation, and Wronskian solution,” Nonlinear Dynamics, vol. 77, no. 1-2, pp. 135–143, 2014. MR3215135 10.1007/s11071-014-1279-3 1314.37049

13.

B. Tian, Y.-T. Gao, and W. Hong, “The solitonic features of a nonintegrable $(3+1)$-dimensional Jimbo-Miwa equation,” Computers & Mathematics with Applications, vol. 44, no. 3-4, pp. 525–528, 2002. MR1912847 10.1016/S0898-1221(02)00166-9 B. Tian, Y.-T. Gao, and W. Hong, “The solitonic features of a nonintegrable $(3+1)$-dimensional Jimbo-Miwa equation,” Computers & Mathematics with Applications, vol. 44, no. 3-4, pp. 525–528, 2002. MR1912847 10.1016/S0898-1221(02)00166-9

14.

J. F. Zhang and F. M. Wu, “Bäcklund transformation and multiple soliton solutions for the $(3+1)$-dimensional Jimbo-Miwa equation,” Chinese Physics, vol. 11, no. 5, p. 425, 2002. J. F. Zhang and F. M. Wu, “Bäcklund transformation and multiple soliton solutions for the $(3+1)$-dimensional Jimbo-Miwa equation,” Chinese Physics, vol. 11, no. 5, p. 425, 2002.

15.

Q. L. Zha and Z. B. Li, “Multiple periodic-soliton solutions for (3+1)-dimensional Jimbo-Miwa equation,” Communications in Theoretical Physics, vol. 50, no. 5, pp. 1036–1040, 2008. 1392.35083 10.1088/0253-6102/50/5/04 Q. L. Zha and Z. B. Li, “Multiple periodic-soliton solutions for (3+1)-dimensional Jimbo-Miwa equation,” Communications in Theoretical Physics, vol. 50, no. 5, pp. 1036–1040, 2008. 1392.35083 10.1088/0253-6102/50/5/04

16.

X. Lü, “Soliton behavior for a generalized mixed nonlinear Schrödinger model with N-fold Darboux transformation,” Chaos, vol. 23, Article ID 033137, 2013. 1323.35163 10.1063/1.4821132 X. Lü, “Soliton behavior for a generalized mixed nonlinear Schrödinger model with N-fold Darboux transformation,” Chaos, vol. 23, Article ID 033137, 2013. 1323.35163 10.1063/1.4821132

17.

R. Hirota, “Exact solution of the korteweg–-de Vries equation for multiple collisions of solitons,” Physical Review Letters, vol. 27, no. 18, pp. 1192–1194, 1971. 1168.35423 10.1103/PhysRevLett.27.1192 R. Hirota, “Exact solution of the korteweg–-de Vries equation for multiple collisions of solitons,” Physical Review Letters, vol. 27, no. 18, pp. 1192–1194, 1971. 1168.35423 10.1103/PhysRevLett.27.1192

18.

R. Hirota, “A new form of Bäcklund transformations and its relation to the inverse scattering problem,” Progress of Theoretical Physics, vol. 52, no. 5, pp. 1498–1512, 1974. 1168.37322 10.1143/PTP.52.1498 R. Hirota, “A new form of Bäcklund transformations and its relation to the inverse scattering problem,” Progress of Theoretical Physics, vol. 52, no. 5, pp. 1498–1512, 1974. 1168.37322 10.1143/PTP.52.1498

19.

R. Hirota and J. Satsuma, “A variety of nonlinear network equations generated from the bäcklund transformation for the toda lattice,” Progress of Theoretical Physics Supplements, vol. 59, pp. 64–100, 1976. 1079.35536 10.1143/PTP.55.2037 R. Hirota and J. Satsuma, “A variety of nonlinear network equations generated from the bäcklund transformation for the toda lattice,” Progress of Theoretical Physics Supplements, vol. 59, pp. 64–100, 1976. 1079.35536 10.1143/PTP.55.2037

20.

R. Hirota, X.-B. Hu, and X.-Y. Tang, “A vector potential KdV equation and vector Ito equation: soliton solutions, bilinear Bäcklund transformations and Lax pairs,” Journal of Mathematical Analysis and Applications, vol. 288, no. 1, pp. 326–348, 2003. MR2019765 10.1016/j.jmaa.2003.08.046 1055.35100 R. Hirota, X.-B. Hu, and X.-Y. Tang, “A vector potential KdV equation and vector Ito equation: soliton solutions, bilinear Bäcklund transformations and Lax pairs,” Journal of Mathematical Analysis and Applications, vol. 288, no. 1, pp. 326–348, 2003. MR2019765 10.1016/j.jmaa.2003.08.046 1055.35100

21.

R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, UK, 2004. MR2085332 1099.35111 R. Hirota, The Direct Method in Soliton Theory, Cambridge University Press, Cambridge, UK, 2004. MR2085332 1099.35111

22.

Y. Matsuno, Bilinear Transformation Method, vol. 174 of Mathematics in Science and Engineering, Academic Press, London, UK, 1984. MR759718 Y. Matsuno, Bilinear Transformation Method, vol. 174 of Mathematics in Science and Engineering, Academic Press, London, UK, 1984. MR759718

23.

C.-X. Li, W.-X. Ma, X.-J. Liu, and Y.-B. Zeng, “Wronskian solutions of the Boussinesq equation–-solitons, negatons, positons and complexitons,” Inverse Problems, vol. 23, no. 1, pp. 279–296, 2007. MR2302974 10.1088/0266-5611/23/1/015 1111.35044 C.-X. Li, W.-X. Ma, X.-J. Liu, and Y.-B. Zeng, “Wronskian solutions of the Boussinesq equation–-solitons, negatons, positons and complexitons,” Inverse Problems, vol. 23, no. 1, pp. 279–296, 2007. MR2302974 10.1088/0266-5611/23/1/015 1111.35044

24.

H. Aratyn, L. A. Ferreira, and A. H. Zimerman, “Exact static soliton solutions of (3+1)-dimensional integrable theory with nonzero Hopf numbers,” Physical Review Letters, vol. 83, no. 9, pp. 1723–1726, 1999. H. Aratyn, L. A. Ferreira, and A. H. Zimerman, “Exact static soliton solutions of (3+1)-dimensional integrable theory with nonzero Hopf numbers,” Physical Review Letters, vol. 83, no. 9, pp. 1723–1726, 1999.

25.

A. Wazwaz, “Integrable (2+1)-dimensional and (3+1)-dimensional breaking soliton equations,” Physica Scripta, vol. 81, no. 3, Article ID 035005, 2010. 1191.35226 A. Wazwaz, “Integrable (2+1)-dimensional and (3+1)-dimensional breaking soliton equations,” Physica Scripta, vol. 81, no. 3, Article ID 035005, 2010. 1191.35226

26.

Y. T. Gao and B. Tian, “New family of overturning soliton solutions for a typical breaking soliton equation,” Computers & Mathematics with Applications, vol. 30, no. 12, pp. 97–100, 1995. MR1360328 10.1016/0898-1221(95)00176-Y 0836.35140 Y. T. Gao and B. Tian, “New family of overturning soliton solutions for a typical breaking soliton equation,” Computers & Mathematics with Applications, vol. 30, no. 12, pp. 97–100, 1995. MR1360328 10.1016/0898-1221(95)00176-Y 0836.35140

27.

J. Yu and Z. Lou, “A $(3+1)$-dimensional Painlevé integrable model obtained by deformation,” Mathematical Methods in the Applied Sciences, vol. 25, no. 2, pp. 141–148, 2002. MR1879655 10.1002/mma.281 J. Yu and Z. Lou, “A $(3+1)$-dimensional Painlevé integrable model obtained by deformation,” Mathematical Methods in the Applied Sciences, vol. 25, no. 2, pp. 141–148, 2002. MR1879655 10.1002/mma.281

28.

J. B. Chen, “Finite-gap solutions of $2+1$ dimensional integrable nonlinear evolution equations generated by the Neumann systems,” Journal of Mathematical Physics, vol. 51, no. 8, Article ID 083514, 2010. MR2683556 1312.35148 10.1063/1.3462249 J. B. Chen, “Finite-gap solutions of $2+1$ dimensional integrable nonlinear evolution equations generated by the Neumann systems,” Journal of Mathematical Physics, vol. 51, no. 8, Article ID 083514, 2010. MR2683556 1312.35148 10.1063/1.3462249

29.

R. S. Ward, “Nontrivial scattering of localized solitons in a (2+1)-dimensional integrable system,” Physics Letters A, vol. 208, no. 3, pp. 203–208, 1995. MR1363151 10.1016/0375-9601(95)00782-X 1020.37537 R. S. Ward, “Nontrivial scattering of localized solitons in a (2+1)-dimensional integrable system,” Physics Letters A, vol. 208, no. 3, pp. 203–208, 1995. MR1363151 10.1016/0375-9601(95)00782-X 1020.37537

30.

J. F. Zhang, “Multiple soliton-like solutions for $(2+1)$-dimensional dispersive Long-Wave equations,” International Journal of Theoretical Physics, vol. 37, no. 9, pp. 2449–2455, 1998. J. F. Zhang, “Multiple soliton-like solutions for $(2+1)$-dimensional dispersive Long-Wave equations,” International Journal of Theoretical Physics, vol. 37, no. 9, pp. 2449–2455, 1998.

31.

A. V. Mikhailov and R. I. Yamilov, “Towards classification of $(2+1)$-dimensional integrable equations,” Journal of Physics A, vol. 31, Article ID 6707, 1998. MR1643816 10.1088/0305-4470/31/31/015 A. V. Mikhailov and R. I. Yamilov, “Towards classification of $(2+1)$-dimensional integrable equations,” Journal of Physics A, vol. 31, Article ID 6707, 1998. MR1643816 10.1088/0305-4470/31/31/015

32.

M. Jimbo and T. Miwa, “Solitons and infinite dimensional Lie algebras,” Publications of the Research Institute for Mathematical Sciences, vol. 19, pp. 943–1001, 1983. MR723457 0557.35091 10.2977/prims/1195182017 M. Jimbo and T. Miwa, “Solitons and infinite dimensional Lie algebras,” Publications of the Research Institute for Mathematical Sciences, vol. 19, pp. 943–1001, 1983. MR723457 0557.35091 10.2977/prims/1195182017

33.

X.-Y. Tang and J. Lin, “Conditional similarity reductions of Jimbo-Miwa equations via the classical Lie group approach,” Communications in Theoretical Physics, vol. 39, no. 1, pp. 6–8, 2003. MR2002177 10.1088/0253-6102/39/1/6 X.-Y. Tang and J. Lin, “Conditional similarity reductions of Jimbo-Miwa equations via the classical Lie group approach,” Communications in Theoretical Physics, vol. 39, no. 1, pp. 6–8, 2003. MR2002177 10.1088/0253-6102/39/1/6

34.

B. Dorrizzi, B. Grammaticos, A. Ramani, and P. Winternitz, “Are all the equations of the KP hierarchy integrable?” Journal of Mathematical Physics, vol. 27, pp. 2848–2852, 1986. 0619.35086 10.1063/1.527260 B. Dorrizzi, B. Grammaticos, A. Ramani, and P. Winternitz, “Are all the equations of the KP hierarchy integrable?” Journal of Mathematical Physics, vol. 27, pp. 2848–2852, 1986. 0619.35086 10.1063/1.527260

35.

J. Weiss, M. Tabor, and G. Carnevale, “The Painlevé property for partial differential equations,” Journal of Mathematical Physics, vol. 24, no. 3, pp. 522–526, 1983. MR692140 10.1063/1.525721 0514.35083 J. Weiss, M. Tabor, and G. Carnevale, “The Painlevé property for partial differential equations,” Journal of Mathematical Physics, vol. 24, no. 3, pp. 522–526, 1983. MR692140 10.1063/1.525721 0514.35083

36.

C.-L. Bai and H. Zhao, “Some special types of solitary wave solutions for $(3+1)$-dimensional Jimbo-Miwa equation,” Communications in Theoretical Physics, vol. 41, no. 6, pp. 875–877, 2004. MR2105620 10.1088/0253-6102/41/6/875 C.-L. Bai and H. Zhao, “Some special types of solitary wave solutions for $(3+1)$-dimensional Jimbo-Miwa equation,” Communications in Theoretical Physics, vol. 41, no. 6, pp. 875–877, 2004. MR2105620 10.1088/0253-6102/41/6/875

37.

A.-M. Wazwaz, “Multiple-soliton solutions for the Calogero-Bogoyavlenskii-Schiff, Jimbo-Miwa and YTSF equations,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 592–597, 2008. MR2458974 10.1016/j.amc.2008.05.004 1154.65366 A.-M. Wazwaz, “Multiple-soliton solutions for the Calogero-Bogoyavlenskii-Schiff, Jimbo-Miwa and YTSF equations,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 592–597, 2008. MR2458974 10.1016/j.amc.2008.05.004 1154.65366

38.

G. Q. Xu, “The soliton solutions, dromions of the Kadomtsev-Petviashvili and Jimbo-Miwa equations in $(3+1)$-dimensions,” Chaos, Solitons and Fractals, vol. 30, no. 1, pp. 71–76, 2006. MR2230893 10.1016/j.chaos.2005.08.089 G. Q. Xu, “The soliton solutions, dromions of the Kadomtsev-Petviashvili and Jimbo-Miwa equations in $(3+1)$-dimensions,” Chaos, Solitons and Fractals, vol. 30, no. 1, pp. 71–76, 2006. MR2230893 10.1016/j.chaos.2005.08.089

39.

S.-H. Ma, J.-P. Fang, B.-H. Hong, and C.-L. Zheng, “New exact solutions and interactions between two solitary waves for (3+1)-dimensional jimbo–miwa system,” Communications in Theoretical Physics, vol. 49, no. 5, pp. 1245–1248, 2008. MR2489658 10.1088/0253-6102/49/5/36 1392.35075 S.-H. Ma, J.-P. Fang, B.-H. Hong, and C.-L. Zheng, “New exact solutions and interactions between two solitary waves for (3+1)-dimensional jimbo–miwa system,” Communications in Theoretical Physics, vol. 49, no. 5, pp. 1245–1248, 2008. MR2489658 10.1088/0253-6102/49/5/36 1392.35075

40.

C. Gilson, F. Lambert, J. J. Nimmo, and R. Willox, “On the combinatorics of the Hirota D-operators,” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 452, no. 1945, pp. 223–234, 1996. MR1375891 0868.35101 10.1098/rspa.1996.0013 C. Gilson, F. Lambert, J. J. Nimmo, and R. Willox, “On the combinatorics of the Hirota D-operators,” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol. 452, no. 1945, pp. 223–234, 1996. MR1375891 0868.35101 10.1098/rspa.1996.0013

41.

F. Lambert and J. Springael, “On a direct procedure for the disclosure of Lax pairs and Bäcklund transformations,” Chaos, Solitons & Fractals, vol. 12, no. 14-15, pp. 2821–2832, 2001. MR1857659 10.1016/S0960-0779(01)00096-0 1005.37043 F. Lambert and J. Springael, “On a direct procedure for the disclosure of Lax pairs and Bäcklund transformations,” Chaos, Solitons & Fractals, vol. 12, no. 14-15, pp. 2821–2832, 2001. MR1857659 10.1016/S0960-0779(01)00096-0 1005.37043

42.

X. Lü, B. Tian, K. Sun, and P. Wang, “Bell-polynomial manipulations on the Bäcklund transformations and Lax pairs for some soliton equations with one tau-function,” Journal of Mathematical Physics, vol. 51, no. 11, Article ID 113506, 2010. MR2759487 1314.35129 10.1063/1.3504168 X. Lü, B. Tian, K. Sun, and P. Wang, “Bell-polynomial manipulations on the Bäcklund transformations and Lax pairs for some soliton equations with one tau-function,” Journal of Mathematical Physics, vol. 51, no. 11, Article ID 113506, 2010. MR2759487 1314.35129 10.1063/1.3504168

43.

X. Lü, F. H. Lin, and F. H. Qi, “Analytical study on a two-dimensional Korteweg-de Vries model with bilinear representation, Bäcklund transformation and soliton solutions,” Applied Mathematical Modelling, vol. 39, pp. 3221–3226, 2015. \endinput MR3344585 10.1016/j.apm.2014.10.046 1443.35135 X. Lü, F. H. Lin, and F. H. Qi, “Analytical study on a two-dimensional Korteweg-de Vries model with bilinear representation, Bäcklund transformation and soliton solutions,” Applied Mathematical Modelling, vol. 39, pp. 3221–3226, 2015. \endinput MR3344585 10.1016/j.apm.2014.10.046 1443.35135
Copyright © 2015 Hindawi
He Li and Yi-Tian Gao "Bilinear Form and Two Bäcklund Transformations for the (3+1)-Dimensional Jimbo-Miwa Equation," Abstract and Applied Analysis 2015(none), 1-5, (2015). https://doi.org/10.1155/2015/834521
Published: 2015
Vol.2015 • 2015
Back to Top