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We investigate the duality between minimal surfaces in Euclidean space and maximal surfaces in Lorentz–Minkowski space in the framework of rotational surfaces. We study if the dual surfaces of two congruent rotational minimal (or maximal) surfaces are congruent. We analyze the duality process when we deform a rotational minimal (maximal) surface by a one-parametric group of rotations. In this context, the family of Bonnet minimal (maximal) surfaces and the Goursat transformations play a remarkable role. [ABSTRACT FROM AUTHOR]

We study a special class of operators T satisfying the transmutation relation ( d 2 d x 2 − q ) T u = T d 2 d x 2 u in the sense of distributions, where q is a locally integrable function, and u belongs to a suitable space of distributions depending on the smoothness properties of q . A method which allows one to construct a fundamental set of transmutation operators of this class in terms of a single particular transmutation operator is presented. Moreover, following [27] , we show that a particular transmutation operator can be represented as a Volterra integral operator of the second kind. We study the boundedness and invertibility properties of the transmutation operators, and use these to obtain a representation for the general distributional solution of the equation d 2 u d x 2 − q u = λ u , λ ∈ C , in terms of the general solution of the same equation with λ = 0 . [ABSTRACT FROM AUTHOR]

Abstract: This work is concerned with the asymptotic behavior of global solutions of the Goursat problem for quasilinear hyperbolic systems. We prove that when tends to the infinity, the solutions approach a combination of traveling wave solutions at algebraic rate , provided that the boundary data decay at the rate , where is a positive constant. Finally, we give an application to the equation for motion of an elastic string. [Copyright &y& Elsevier]

In this paper we establish the existence of global continuous solutions of gas expansion into a vacuum for the two-dimensional pressure-gradient equations in gas dynamics. Under irrotational condition, By hodograph transformation, the flow is governed by the equation (p-p2v)puu+2pupvpuv+(p-p2u)pvv=0, which can be further reduced to a inhomogeneous linearly degenerate system of three equations. Then the problem of the expansion of a wedge of gas into a vacuum is solved in the same way. [Copyright &y& Elsevier]