Xuncheng Huang1* and Edward H. C. Chang2

1 Yangzhou Polytechnic University 12-903 HaoYue Yuan, Moon ParkYangzhou, Jiangsu 225012 China
2 National Taipei College of Business321 Section 1, Chi-Nan Road, Taipei, Taiwan, R.O.C.


 

Download Citation: |
Download PDF


ABSTRACT


The Bäcklund transformation (BT) for a three-dimensional nonlinear wave equation and its nonlinear superposition formula are studied in this note. We prove that the three dimensional Bäcklund transformation obtained by Leibbrandt, et al. can be decomposed into three two-dimensional BTs. Some results on the N-dimensional Liouville equation are also discussed in the article.


Keywords: Bäcklund transformation; Liouville equation; superposition formula


Share this article with your colleagues

 


REFERENCES


[1] Ockendon, J. R. (Ed.) 2003. “AppliedPartial Differential Equations - OxfordTexts in Applied and EngineeringMathematics”, Oxford University Press,Oxford.

[2] Gesztesy, F., Holden H., Bollobas B.,Fulton W., Katok A., Kirwan F., and SarnakP. 2003. Soliton Equations and theirAlgebro-Geometric Solutions: Vol.,(1+1)-Dimensional Continuous Models“Cambridge Studies in AdvancedMathematics”, Cambridge UniversityPress.

[3] Leibbrandt, G., Wang S. S., and ZamaniN. 1982. Bäcklund generated solutionsof Liouville’s Equation and their graphicalrepresentations in three spatial dimensions,Journal of MathematicalPhysics, 23, 9: 1566-1572.

[4] Leibbrandt, G. 1980. Nonlinear superpositionfor Liouville’s equation in threespatial dimensions, Letters in MathematicalPhysics, 4: 317-321.

[5] Huang, X. C. 1983. Another method ofderiving auto-Bäcklund transformationsfor nonlinear evolution equations, Journalof Physics A-Mathematical and General,16: 891-892.

[6] Huang, X. C. 1983. A one-parameterBäcklund transformation for the EuclideanLiouville and wave equations,Physica Scripta, 27: 321-322.

[7] Tu, M. H. 2001. On the Bäcklund transformationfor the Moyal Korteweg-deVries hierarchy, Journal of PhysicsA-Mathematical and General, 34:623-629.

[8] Bai, C. L. 2004. Bäcklund transformationand multiple soliton solutions for the(3+1)-dimensional Nizhnik– Novikov–Veselov equation, Chinese Physics.,13: 1-4.

[9] Hu, X. B. and Springael, J. 2002. ABäcklund transformation and nonlinearsuperposition formula for the Lotka-Volterra hierarchy, ANZLAMLotka-Volterra hierarchy, ANZLAMJournal. 44: 121-128.

[10] Rourke, D. E. 2004. Elementary Bäcklundtransformations for a discreteAblowitz–Ladik eigenvalue problem,Journal of Physics A-Mathematical andGeneral, 37,7: 2693-2708.

[11] Tu, G. Z. and Huang, X. C. 2005. FromNewton’s law to generalized Hamiltoniansystem (I)——Some results on linearskew-symmetric operator, Journal ofYangzhou Polytechnic University, 9,2:36-46.

[12] Huang, X. C. and Tu, G. Z. 2006. A newhierarchy of integrable system of1+2-dimensions From Newton’s law togeneralized Hamiltonian system (II), InternationalJournal of Mathematics andMathematical Sciences. (to be published)


ARTICLE INFORMATION




Accepted: 2006-04-03
Available Online: 2006-08-25


Cite this article:

Huang, X., Chang, Edward H. C. 2006. A note on a Three-Dimensional bäcklund transformation. International Journal of Applied Science and Engineering, 4, 215–220. https://doi.org/10.6703/IJASE.2006.4(2).215