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Moreover, this equation has been extensively studied as a model of relativistic field theory. There is a vast literature on the sine-Gordon equation. Its discussion and further references can be found in [14, 37, 129, 270]. 1. We will consider a perturbed sine-Gordon equation with small parameter which is related to large space and time scales of the process under study (£ = x/e, r —t / e y x , t ~ 1). 143) a e 2(utt ~ uxx) -I----- sin 7 ^ = 0 of the sine-Gordon equation with variable coefficients a = a{Xy t) > 0 , 7 = 7 (2;, t) > 0 48 1.

141) must be also completed with Cauchy conditions for t = 0, x e K 1 and for t = 0 , x < 0. 141) and the vector £ is transversal to this curve. 139) a well-posed problem. The subsequent derivations follow the scheme set out above. 141) in the domain {(#,£), x < 0} completed with conditions avf = <7j(t). = *»(»). 130). 5. Other examples In this section we consider two equations important for physical applications, namely, the sine-Gordon equation and the Benjamin-Bona-Mahogany equation (the regularized long waves equation).

62) while the orthogonality condition for the higher order correction entails the equation A~1(put + «l^ it + a2¥>i = G, where ol\ — —A. 1 A - 2{ A \ A - * I 2)u + (^4. uo)t 28 ii0 2uo®®, - 32SxVAjt + 36i(25u0 - 6M 2)}, iio = duo{

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A Geometrical Determination of the Canonical Quadric of Wilczynski by Stouffer E. B.


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