Nonlinear wave interaction problems in the three-dimensional case

Abstract

Abstract. Three dimensional nonlinear wave interactions have been analytically described. The procedure under interest can be applied to three dimensional quasilinear systems of first order, whose hydrodynamic reductions are homogeneous semi-Hamiltonian hydrodynamic type systems (i.e. possess diagonal form and infinitely many conservation laws). The interaction of N waves was studied. In particular we prove that they behave like simple waves and they distort after the collision region. The amount of the distortion can be analytically computed.

Solutions via double wave ansatz to the 1-D non-homogeneous gas-dynamics equations

“…As far as the exact description of nonlinear wave interaction processes is concerned, we remark that it is fully developed for 2 × 2 strictly hyperbolic models but, unfortunately, such an analysis cannot be in general applied to quasilinear hyperbolic systems involving more dependent and/or independent variables although special wave interaction problems were solved [23,35].…”

“…Moreover some further results concerning the non-homogeneous 3-D case have been obtained in [29] . Therefore, the main aim of the present paper is to develop, along the lines of the analysis carried on in [22], a reduction procedure for determing classes of exact double wave solutions to (1)-( 3) and, consequently, some nonlinear wave problems of relevant interest such as Riemann problems ( [30]- [32]) and nonlinear wave interactions ( [33]- [35]) are analysed. In particular, after reducing the full set of governing equations to a suitable 2 × 2 hyperbolic auxiliary system, following the idea developed in [17,18], a Riemann problem will be solved as well as an exact analitical description of nonlinear waves interaction admitted by the governing system under interest will be given.…”

Design of Commercial Building Complex Based on 3D Landscape Interaction

“…Literature [16] puts forward the idea of establishing the 3D model base of urban landscape to realize the rapid reconstruction of urban 3D landscape. Literature [17] mentioned in the research that, in the process of urban construction, we should maintain the existing urban history and culture, do a good job in the publicity and popularization of urban history and culture, and create a strong urban cultural atmosphere and noble urban humanistic spirit.…”

Riemann problem for rate‐type materials with nonconstant initial conditions

“…1,2 A variety of mathematical methods for finding exact solutions to such systems have been proposed over the years (see previous works [3][4][5][6] ). The approach based on the use of differential constraints, proposed by Janenko 7 (see also Rozhdestvenski and Janenko 8 and Sidorov et al 9 ), has been of considerable interest in recent years (see previous studies [10][11][12][13][14][15][16][17][18][19][20][21][22][23] ). Based on Lie symmetry analysis, an approximate rarefaction wave-type solution to the Riemann problem with nonclassical discontinuous initial data for a system of balance laws describing rate-type materials was presented in Conforto et al 6 ; here, the initial data for the variable u are discontinuous whereas the initial data for the variable v are constants.…”