Research Group:
Speaker:
Alexander Minakov
Institution:
SISSA
Schedule:
Tuesday, May 29, 2018 - 16:15
Location:
A-136
Abstract:
An initial-boundary value problem for a model of stimulated Raman Scattering was considered in [Moscovchenko Kotlyarov 2010]. The authors showed that in the long-time range $t\to+\infty$ the $x > 0, t > 0$ quarter plane is divided into 3 regions with qualitatively different asymptotic behavior of the solution: a region of a finite amplitude plane wave, a modulated elliptic wave region and a vanishing dispersive wave region. The asymptotics in the modulated elliptic region was studied under an implicit assumption of the solvability of the corresponding Whitham type equations. Here we establish the existence of these parameters, and thus justify the results in [Moscovchenko Kotlyarov 2010].
The problem reduces to proving the positivity of the polynomial of two variables $\alpha, x$
$\alpha^4+(4x^3+2\beta x^3-6\beta x^5)\alpha^3+(3\beta x^4+6x^6-14\beta x^6+3\beta x^8)\alpha^2$
$+(-6\beta x^7+4x^9+2\beta x^9)\alpha+x^{12}>0$
in the rectangle $x>1, 1<\alpha<\alpha_0,$ where $\alpha_0=\frac{27-18\beta-\beta^2+(9-\beta)\sqrt{(1-\beta)(9-\beta)}}{8\beta\sqrt{\beta}}$ and $0<\beta<1.$