Bi-solitons for a class of non-integrable non-linear partial differential equations with quadratic non-linearity

Investigating the bi-solitons of the class of equations (e + a?_x + b?_t + c?²_xt + d?²_xx)K = K², e ≠ 0, we report that they exist if and only if the operator of the linear part for K or e + K can be factorized.     

Temperature overshoots for a 4-velocity unidimensional discrete Boltzmann model

We propose a 4-velocity unidimensional discrete Boltzmann model with two different speeds 2, 1 and two different masses 1, 2. With the three conservation laws of mass, momentum, and energy satisfied, we can introduce a nontrivial temperature. First, we determine the similarity shock waves satisfying physical properties: positivity, shock stability, inequalities of the subsonic and supersonic flows, increase or decrease of both mass and temperature across the shock. It results that either the speed of the shock front is higher than the speed 1 of the slow particles and the shocks are compressive or less than 1 and the shocks are rarefactive. We observe overshoots of the temperature, across the shock, with bumps higher and higher as the shock front speed increases. Second, we study the (1+1)-dimensional shock waves. They represent the superposition and collision of two compressive shocks traveling in opposite directions and we observe temperature overshoots for not too large times.     

Exact solutions for the discrete Boltzmann models with specular reflection

We construct exact (1+1)-dimensional solutions (space x, time t), in the presence of a purely reflecting well, for both the four velocity discrete Boltzmann model and the Broadwell model. These exact solutions, sums of two similarity shock waves, are positive for x0, t0.     

Bi-solitons for a class of non-linear equations with quadratic non-linearity. I

We consider the class of non-integrable, non-linear equations,

$\text{L}{}_{\mathrm{q}}\text{K=K}{}^2\text{, L}{}_{\mathrm{q}}\text{=? +}\text{∑}\text{1?i+j?q}\text{aij?}{}^{\mathrm{i}}{}_{\mathrm{x}}{}^{\mathrm{i}}\text{?}{}^{\mathrm{j}}{}_{\mathrm{t}}{}^{\mathrm{j}}\text{, ?≠0,}$

in 1+1 dimensions. We seek rational solutions K(ω₁,ω₂), which we call bi-solitons, with exponential type variables ω_i = exp(γ_ix + ρ_it). In this paper, we restrict to q = 2 and 3, and investigate the general q case in the following paper. We find that these bi-solitons exist when the operator L_q (with ± ?) can be factorized as the product of smaller order differential operators. Besides the trivial factorized bi-solitons, we show that there exist non-trivial ones whenever K may be written as Σ^lmaxx ω^l₂F_l(Z = ω₁ + ω₂). In order to understand the origin of the factorization property, to any polynomial K = Σω^l₂F_l(Z) we associate a linear transformation such that L_qK has only the power ω^l₂ of K². For q = 2 and 3, we find that there exist particular polynomials of this type restraining L_q to be a product of smallr order operators. For the full non-linear equations we verify that all the bi-solitons can be obtained from these particular polynomials.     

Scaling of the diffraction peak in collisions of particles of arbitrary spins

We propose sufficient conditions which, together with some positivity properties of scattering amplitudes valid for arbitrary spins, allow one to show the existence of a scaling limit for the differential cross section for an elastic reaction in the nearly forward direction. These conditions appear to be very natural and in agreement with experiment.     

Bi-solitons for a class of non-linear equations with quadratic non-linearity. II

We generalize to any order q, the methods developed in a companion paper for q = 2,3 for finding bi-solitons, solutions of the class of non-integrable non-linear equations L_qK = K²; L_q = ? + Σ_i+j≤qa_ij?_xⁱ?_lⁱ, ? ≠ 0 in 1 + 1 dimensions. We call bi-solitons K(ω₁,ω₂) of the exponential type variables ω_i = exp(γ_ix + ρ_it), i = 1,2 and deal only with the so-called “non trivial” solutions which may be written as a finite sum K = Σ^lmax₀ω¹₂F_i(Z)_, F₁ rational function of Z = ω₁Z = ω₁ + ω₁. To any such polynomial K, we associate a linear transformation such that L_qK has only the power ω¹₂ of K² and we find that there are particular polynomialswhere the above restriction provide a factorization of the linear operator L_q in the product of smaller order differential operators. After this linear phase, we show in a second step that these forms yield solutions for the full non linear equation which can be derived in an intrinsic manner. Examples in the monomial and binomial cases are given.     

Enhancement of L-3-hydroxybutyryl-CoA dehydrogenase activity and circulating ketone body levels by pantethine. Relevance to dopaminergic injury

Background  

The administration of the ketone bodies hydroxybutyrate and acetoacetate is known to exert a protective effect against metabolic disorders associated with cerebral pathologies. This suggests that the enhancement of their endogenous production might be a rational therapeutic approach. Ketone bodies are generated by fatty acid beta-oxidation, a process involving a mitochondrial oxido-reductase superfamily, with fatty acid-CoA thioesters as substrates. In this report, emphasis is on the penultimate step of the process, i.e. L-3-hydroxybutyryl-CoA dehydrogenase activity. We determined changes in enzyme activity and in circulating ketone body levels in the MPTP mouse model of Parkinson's disease. Since the active moiety of CoA is pantetheine, mice were treated with pantethine, its naturally-occurring form. Pantethine has the advantage of being known as an anti-inflammatory and hypolipidemic agent with very few side effects.     

Power-like decreasing solutions of the non-linear Boltzman equation corresponding to Maxwellian interaction

We consider, in the Maxwell interaction model, nondiscrete eigensolutions of the linearized Boltzman equation decreasing as inverse powers of the energy. These new solutions provide for the nonlinear case, a nonlinear time dependent system which can be recursively determined and they lead to acceptable physical solutions which belong to a Hilbert space larger than the usual one.     

Erratum

We prove that the high-energy differential cross section for an elastic process has a maximum exactly in the forward direction and that the slope of the diffraction peak is at most (log s)². We compare the width of the diffraction peaks defined by the absorptive part and the differential cross section. Our assumptions are that the amplitude is dominated by the even signature amplitude and that the total cross section, if it decreases, decreases less fast than $\text{s}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$. Strictly speaking, our results hold only for a sequence of energies approaching infinity. The proofs are given for the spin-0-spin-0 case, but it is not unreasonable to hope that they can be generalized to arbitrary spins.     

Construction of positive exact (2+1)-dimensional shock wave solutions for two discrete Boltzmann models

It is proved that (2+1)-dimensional (spacex, y; timet) positive exact shock wave solutions of two discrete Boltzmann models exist. For each densityN _i, these solutions are linear combinations of three similarity shock waves,N _i =n _0i + _j n _ji /[1+d _j exp( _j y+y _j x+ _j t)],j=1,2,3. Two models with four independent densities are investigated: the square discrete-velocity Boltzmann model and the model with eight velocities oriented toward the eight corners of a cube.The positivity problem for the densities is nontrivial. Two classes of solutions are considered for which the two first similarity shock wave components depend on only one spatial dimension, _j=const· _j ,j=1,2. For the positivity, if ₁₂>0, it is sufficient to prove that the 16 asymptotic shock limitsn _0i ,n _0i +n _3i , _j=0 ² n _ji , _j=0 ³ n _ji are positive. The density solutions are built up with five arbitrary parameters and we prove that there exist subdomains of the arbitrary parameter space in which the 16 shock limits are positive. We study numerically two explicit shock wave solutions. We are interested in the movement of the shock front when the time is growing and in the possible appearance of bumps. In the space, at intermediate times, these bumps represent populations of particles which are larger than at initial time or at equilibrium time.