Understanding of quasilinear hyperbolic systems through the investigation of their asymptotic solutions

Abstract

The main aim of the study is to deepen understanding of quasilinear hyperbolic systems through the investigation of their asymptotic solutions, with specific objectives related to theoretical analysis, wave dynamics characterization, and practical applications in physics and engineering. The methods of mathematical analysis employed include asymptotic analysis, Taylor series expansions, and the formulation of transfer equations. The paper considers systems of quasilinear hyperbolic equations in partial derivatives of the first order with two independent variables. The main results of the paper are: 1) high-frequency asymptotic solutions of small amplitude for quasilinear hyperbolic systems of the first order were obtained. For fixed values of t and , values of the modulus  are limited by р→∞, because the transfer equations depend on p. Thus, the moduli of the decomposition coefficients are bounded at p→∞ and at fixed u and ; 2) It has been established that for u_i⁰= const, , , independent of t,  the solution of the equation is greatly simplified because the coefficient а0 is constant. For the linear function ,  is also constant. Practical applications of the results lie in fields such as fluid dynamics, wave propagation, and materials science, where understanding dispersion phenomena is crucial.