Asymptotic Analysis - Volume 16, issue 1

Authors: Frank, Leonid S.

Article Type: Research Article

Abstract: A simple method is presented for explicitly finding the solitons’ wavetrain \eta for the Korteweg–de Vries equation in terms of the wave numbers of the solitons, their number N being fully determined by the initial disturbance \eta_0 \in L_{1/2} ({\NBbbR}) . The Boussinesq system is asymptotically examined and explicit formulae are given for the slightly deformed corresponding oscillating waves, the small parameter being \varepsilon = h_0 , where h_0 is the dimonsionless depth of the water layer at rest. The existence of such oscillatory waves is rigorously proved for the Boussinesq system. …A different Boussinesq system, yielding solitons, is also revisited. Show more

Citation: Asymptotic Analysis, vol. 16, no. 1, pp. 1-14, 1998

Authors: Levendorski\u{\i}, S.Z.

Article Type: Research Article

Abstract: The acoustic wave propagator in a perturbed periodic layer 0< z< 1 is given by H=- c^{2}(x, y)\rho(x, y)\Big[\nabla \frac{1}{\rho(x, y)}\nabla\Big], where the speed of sound c and density \rho are short‐range perturbations of functions periodic in (x, y) . A ‘limiting absorption principle’ for H is established between suitably weighted L_{2} ‐spaces. The limiting values of the resolvent are shown to be Hölder continuous at generic points of the real axis.

Citation: Asymptotic Analysis, vol. 16, no. 1, pp. 15-24, 1998

Authors: Alabau, F. | Hamdache, K. | Peng, Y.J.

Article Type: Research Article

Abstract: We consider, in the nonsteady‐state case, the problem of emitted charged particles in a plane diode. This problem is described by the Vlasov–Poisson system with singular data converging to some Dirac measures in the velocity space concentrated at t = 0 . Under a sufficient condition on the data, which implies that the particles emitted from the cathode are all collected by the anode, we prove that the solutions converge to a measure solution, with support on velocity, of the Vlasov–Poisson system.

Citation: Asymptotic Analysis, vol. 16, no. 1, pp. 25-48, 1998

Authors: Claudi, S. | Guarguaglini, F.R.

Article Type: Research Article

Abstract: We study the large time behaviour of nonnegative solutions of the Cauchy problem \cases{u_t+(\tfrac{u^m}{m})_x = u_{xx} - u^p& ${\rm in}\ {\EBbbR}\times{\EBbbR}^+,$\cr u=u_0 &${\rm in}\ {\EBbbR}\times\{0\},$} where m\Egeq 2 , p>1 and u_0 \in L^1({\EBbbR}) . We prove that in this range for m the diffusion term prevails over the convection one, thus obtaining the asymptotic profile of the solution in L^q({\EBbbR}) (1\Eleq q\Eleq +\infty ).

Citation: Asymptotic Analysis, vol. 16, no. 1, pp. 49-63, 1998

Authors: Amar, M.

Article Type: Research Article

Abstract: We extend the notion of two‐scale convergence introduced by G. Nguetseng and G. Allaire to the case of sequences of bounded Radon measures. We prove a compactness result for two‐scale convergence. We then apply it to the study of the asymptotic behaviour of sequences of positively 1‐homogeneus and periodically oscillating functionals with linear growth, defined on the space BV of the functions with bounded total variation.

Keywords: 2‐scale convergence, measures, BV‐functions, homogenization

Citation: Asymptotic Analysis, vol. 16, no. 1, pp. 65-84, 1998