摘要:

Lett= (t1, …,tn) be a point of ℝn. We shall write . We put by definitionRα(u) =u(α−n)/2/Kn(α); hereαis a complex parameter,nthe dimension of the space, andKn(α) is a constant. First we evaluate □Rα(u) =Rα(u), where □ the ultrahyperbolic operator. Then we obtain the following results:R−2k(u) = □kδ;R0(u) = δ; and □kR2k(u) =δ,k= 0, 1, …. The first result is then‐dimensional ultrahyperbolic correlative of the well‐known one‐dimensional formula . Equivalent formulas have been proved by Nozaki by a completely different method. The particul

ISSN:0022-2526    

DOI:10.1002/sapm1988793185

年代:1988

数据来源: WILEY

摘要:

Self‐similar solutions describing simple waves on shear flows in a finite compressible barotropic atmosphere are found. These include the simple waves on shear flows in water as a special case. By making use of a number of transformations it becomes possible to write these solutions in an exact form. This form, though not explicit, is similar to the incomplete Beta function which seems to characterize this class of nonlinear physical problem

ISSN:0022-2526    

DOI:10.1002/sapm1988793193

年代:1988

数据来源: WILEY

摘要:

In general, weakly nonlinear high frequency almost periodic wave trains for systems of hyperbolic conservation laws interact and resonate to leading order. In earlier work the first two authors and J. Hunter developed simplified asymptotic equations describing this resonant interaction. In the important special case of compressible fluid flow in one or several space dimensions, these simplified asymptotic equations are essentially two inviscid Burgers equations for the nonlinear sound waves, coupled by convolution with a known kernel given by the sum of the initial vortex strength and the derivative of the initial entropy. Here we develop some of the remarkable new properties of the solutions of this system for resonant acoustics. These new features include substantial almost periodic exchange of energy between the nonlinear sound waves, the existence of smooth periodic wave trains, and the role of such smooth wave patterns in eliminating or suppressing the strong temporal decay of sawtooth profile solutions of the decoupled inviscid Burgers equations. Our approach combines detailed numerical modeling to elucidate the new phenomena together with rigorous analysis to obtain exact solutions as well as other elementary properties of the solutions of this system.

ISSN:0022-2526    

DOI:10.1002/sapm1988793205

年代:1988

数据来源: WILEY

摘要:

Equations governing leading order wave amplitudes of resonating almost periodic wave trains in weakly nonlinear acoustics have been obtained by Majda and Rosales [Stud. Appl. Math.71:149–179 (1984)]. These equations consist of a pair of Burgers equations coupled through an integral term with a known kernel. Numerical experiments reported by Majda, Rosales, and Schonbek have suggested the existence of smooth solutions of this system whose components consist of traveling waves moving in opposite directions. For the simplest cosine kernel, explicit formulae are given here for such resonating wave solutions. There is a wave of maximum amplitude with a “peak.” For more general kernels, small amplitude resonating waves are constructed via bifurc

ISSN:0022-2526    

DOI:10.1002/sapm1988793263

年代:1988

数据来源: WILEY

摘要:

A new variational principle is proposed for determining the asymptotic expansion of the solution of the Ackerberg‐O'Malley resonance problem [Stud. Appl. Math.49:277–295 (1970)] to any order inε. The method used yields new higher‐order results not permitted by the technique of Grasman and Matkowsky [SIAM J. Appl. Math.32:588–597 (1977)]. Explicit results using the method are reported toO(ε) and confirmed with asymptotic expansions of the exact solution; theO(1) results agree with those reported in the literature. In the case where the coefficient functions are analytic, an exact solution is presented. It is not difficult to perform the higher‐order calculations using the proposed variational approach, in contrast to the current met

ISSN:0022-2526    

DOI:10.1002/sapm1988793271

年代:1988

数据来源: WILEY

ISSN:0022-2526    

DOI:10.1002/sapm1988793291

年代:1988

数据来源: WILEY

ISSN:0022-2526    

DOI:10.1002/sapm1988793293

年代:1988

数据来源: WILEY