A general solution of the elastic fields in 1D hexagonal quasicrystals with point groups 6mm, 62 h 2 h , 6m2 h and 6/m h mm is given in terms of four 'harmonic' functions F i (i = 1, 2, 3, 4). Then we consider the problem of a circular crack embedded in an infinite 1D hexagonal quasicrystal of point group 6mm. The results obtained in this paper automatically reduce to those in the classical elasticity theory when the phason field is absent.
A new (2 + 1)-dimensional KdV equation is constructed by using Lax pair generating technique. Exact solutions of the new equation are studied by means of the singular manifold method. Bäcklund transformation in terms of the singular manifold is obtained. And localized structures are also investigated.
The extended mapping method has been used to derive some new exact solutions of the KdV-mKdV equation as a combination of some Jacobian elliptic functions. Solutions in the limiting cases have also been studied.
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