方程∂2u/∂x1∂x1+∂2u/∂x2∂x2=f(u)的Bäcklund变换
Bäcklund Transformations for the Equation ∂2u/∂x1∂x1+∂2u/∂x2∂x2=f(u)
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摘要: 本文利用Wahlquist-Estabrook过程(WEP)研究了方程∂2u/∂x1∂x1+∂2u/∂x2∂x2=f(u)(这里f是任意函数)的Bäcklund变换.我们发现该方程存在Bäcklund变换的充分条件是d2f/du2=λf.我们所得到的结果的一个特殊情况就是Leibbrandt[1,2]的结论.Abstract: Bäcklund transformations for the equation ∂2u/∂x1∂x1+∂2u/∂x2∂x2=f(u) is an arbitrary function) is studied in this paper, using the procedure of Wahlquist and Estabrook (WEP). We conclude that the condition d2f/du2=λf is sufficient for the existence of Bäcklund transformations for the equation of our interest. A special case of our results leads to the conclusion of Leibbrandt[1,2].
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[1] Leibbrandt,G.,Exact solutions of the elliptic sine equation in two space dimensions with applications to the Josephson effect,Phys.Rev.,B 15(1977),3353-3361. [2] Leibbrandt,G.,Soliton-like solutions of the elliptic sinecosine equation by means of harmonic functions,J.Math Phys.,19(1978),960-966. [3] Rogers,C.and W.F.Shadwick,Backlund Transformations and Their Applications,Academic Press(1982). [4] Segur,H.,Some open problems,Physica,18D(1986),1-12. -
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