By V. Lakshmikantham
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Additional info for Nonlinear Equations in Abstract Spaces. Proceedings of an International Symposium on Nonlinear Equations in Abstract Spaces, Held at the University of Texas at Arlington, Arlington, Texas, June 8–10, 1977
Nonlinear Superposition be discussed. 49 The first of these concerns "exact shooting" which has been widely promoted by Klamkin  and the second concerns nonlinear eigenparameter problems. A. Exact Shooting (Ames and Adams ) A substantial number of problems in fluid mechanics and diffusion possess nonlinearities of "power" type - that is the nonlinearities appear as powers of the dependent variables and/or their derivatives. A substantial subset of these possess invari ant (similar) solutions and the "constants of superposition" property.
In reality it Here we give a few of these rela- tions and omit the details of Backlund transformation construction. A. 2) sin [(z + z1)/2] z l' generate z2 through a 3 and, finally, generate and also from z3 through a 2· °2 z4 This permutability was expressed by Bianchi  in the following diagram (Figure 1). 3) which is the classical theorem of permutability. The knowledge of three solutions enables one to recursively generate an infinite sequence of particular solutions to the Sine-Gordon equation.
Thus if U(x^y) is a solution of there exist constants of superposition u(x^y) is also a solution, = for^ all λ, μ, and Lu = 0 ν do such that \U(\}x^vy) λ, μ, v? By exactly the same procedure as that described in the preceding paragraph one could, for example, characterize all those diffusion equations which possess invariance under the transformation u = \U, X = μ"ν^ y = v"V In many examples the invariance requirement places restric tions on the parameters such as one is a function of the other.
Nonlinear Equations in Abstract Spaces. Proceedings of an International Symposium on Nonlinear Equations in Abstract Spaces, Held at the University of Texas at Arlington, Arlington, Texas, June 8–10, 1977 by V. Lakshmikantham