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Register a new userThe parametrix problem for Toda equation with steplike initial data
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Authors
Anton Pryimak
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This paper is a continuation of arXiv:17.01.02867. We give here rigorous solution of the parametrix problem for Toda rarefaction problem and complete asymptotic analysis, justifying the asymptotics obtained in arXiv:17.01.02867.
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In this paper we study the asymptotics of the Korteweg--de Vries (KdV) equation with steplike initial data, which leads to shock waves, in the middle region between the dispersive tail and the soliton region, as $t rightarrow infty$. In our previous work we have dealt with this question, but failed to obtain uniform estimates in $x$ and $t$ because of the previously unknown singular behaviour of the matrix model solution. The main goal of this paper is to close this gap. We present an alternative approach to the usual argument involving a small norm Riemann--Hilbert (R-H) problem, which is based instead on Fredholm index theory for singular integral operators. In particular, we avoid the construction of a global model matrix solution, which would be singular for arbitrary large $x$ and $t$, and utilize only the symmetric model vector solution, which always exists and is unique.
We show that the Cauchy problem for the KdV equation can be solved by the inverse scattering transform (IST) for any initial data bounded from below, decaying sufficiently rapidly at plus infinity, but unrestricted otherwise. Thus our approach doesnt require any boundary condition at minus infinity.
In recent time, by working in a plane with the metric associated with wave equation (the Special Relativity non-definite quadratic form), a complete formalization of space-time trigonometry and a Cauchy-like integral formula have been obtained. In this paper the concept that the solution of a mathematical problem is simplified by using a mathematics with the symmetries of the problem, actuates us for studying the wave equation (in particular the initial values problem) in a plane where the geometry is the one generated by the wave equation itself. In this way, following a classical approach, we point out the well known differences with respect to Laplace equation notwithstanding their formal equivalence (partial differential equations of second order with constant coefficients) and also show that the same conditions stated for Laplace equation allow us to find a new solution. In particular taking as initial data for the wave equation an arbitrary function given on an arm of an equilateral hyperbola, a Poisson-like integral formula holds.
In the present manuscript we consider the Boltzmann equation that models a polyatomic gas by introducing one additional continuous variable, referred to as microscopic internal energy. We establish existence and uniqueness theory in the space homogeneous setting for the full non-linear case, under an extended Grad assumption on transition probability rate, that comprises hard potentials for both the relative speed and internal energy with the rate in the interval $(0,2]$, which is multiplied by an integrable angular part and integrable partition functions. The Cauchy problem is resolved by means of an abstract ODE theory in Banach spaces, for an initial data with finite and strictly positive gas mass and energy, finite momentum, and additionally finite $k_*$ polynomial moment, with $k_*$ depending on the rate of the transition probability and the structure of a polyatomic molecule or its internal degrees of freedom. Moreover, we prove that polynomially and exponentially weighted Banach space norms associated to the solution are both generated and propagated uniformly in time.
An inverse scattering problem for a quantized scalar field ${bm phi}$ obeying a linear Klein-Gordon equation $(square + m^2 + V) {bm phi} = J mbox{in $mathbb{R} times mathbb{R}^3$}$ is considered, where $V$ is a repulsive external potential and $J$ an external source $J$. We prove that the scattering operator $mathscr{S}= mathscr{S}(V,J)$ associated with ${bm phi}$ uniquely determines $V$. Assuming that $J$ is of the form $J(t,x)=j(t)rho(x)$, $(t,x) in mathbb{R} times mathbb{R}^3$, we represent $rho$ (resp. $j$) in terms of $j$ (resp. $rho$) and $mathscr{S}$.
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