Real Analytic Solutions Research Articles

Partial differential equations from matrices with orthogonal columns

We discuss a system of third order PDEs for strictly convex smooth functions on domains of Euclidean space. We argue that it may be understood as the closure of the first order prolongation of a family of PDEs. We describe explicitly its real analytic solutions and all the solutions which satisfy a genericity condition; we also describe a family of non-generic solutions which has an application to Poisson geometry and Kähler structures on toric varieties. Our methods are geometric: we use the theory of Hessian metrics and symmetric spaces to link the analysis of the system of PDEs with properties of the manifold of matrices with orthogonal columns.

Response solutions to harmonic oscillators beyond multi–dimensional brjuno frequency

This paper focuses on the quasi–periodically forced nonlinear harmonic oscillators \begin{document}$ \begin{equation*} \ddot{x}+\lambda^{2}x = \epsilon f(\omega t,x), \end{equation*} $\end{document} where $ \lambda \in \mathcal{O} $, a closed interval not containing zero, the forcing term $ f $ is real analytic, and the frequency vector $ \omega \in \mathbb{R}^d \, (d \geq 2) $ is beyond Brjuno frequency, which we call as Liouvillean frequency. For the given class of the frequency $ \omega\in\mathbb{R}^{d}, $ which will be given later, we prove the existence of real analytic response solutions (the response solution is the quasi–periodic solution with the same frequency as the forcing) for the above equation. The proof is based on a modified KAM (Kolmogorov–Arnold–Moser) theorem for finite–dimensional harmonic oscillator systems with Liouvillean frequency.

MULTIPERIODIC SOLUTIONS OF SECOND-ORDER SYSTEMS WITH DIFFERENTIATION OPERATOR ON THE LYAPUNOV'S VECTOR FIELD

The problem of the existence and integral representation of a unique multiperiodic solution of a second-order linear inhomogeneous system with constant coefficients and a differentiation operator on the direction of the main diagonal of the space of time variables and of the vector fields in the form of Lyapunov systems with respect to space variables were considered. The multiperiodicity of zeros of this operator and the condition for the absence of a nonzero multiperiodic and real-analytic solution of the homogeneous system corresponding to the given system are established. An integral representation of solutions of an inhomogeneous linear autonomous system that multiperiodic in time variables and realanalytic in space variables is obtained. The existence theorem of a unique multiperiodic in time variables and real-analytic in space variables solutions of the original linear system in terms of the Green's function under sufficiently general conditions is substantiated.

Choptuik’s Critical Spacetime Exists

About twenty years ago, Choptuik studied numerically the gravitational collapse (Einstein field equations) of a massless scalar field in spherical symmetry, and found strong evidence for a universal, self-similar solution at the threshold of black hole formation. We prove rigorously the existence of a real analytic solution, that we interpret as the solution observed by Choptuik. Our construction covers an open neighborhood of the past light cone of the singularity. The proof is computer assisted. Starting from an explicit approximate solution, we show that nearby there is a true solution. The source code and a high precision data file (about 80 significant decimal digits, with rigorous error bounds) are included. We do not study perturbations.

Whiskered Tori for Forced Beam Equations with Multi-dimensional Liouvillean Frequency

In this paper we develop a Kolmogorov–Arnold–Moser (KAM) theory close to two fixed points for quasi-periodically forced nonlinear beam equation $$\begin{aligned} \begin{aligned} y_{tt}+my+y_{xxxx}=y^{3}+ \varepsilon f(\omega t,x,y),\ \ x\in [0,\ \pi ], \end{aligned} \end{aligned}$$where the forcing frequency $$\omega $$ is a small dilatation of a fixed vector $${\overline{\omega }},$$ i.e., $$\omega =\xi {\overline{\omega }}\in {\mathbb {R}}^{d}$$ with $$\xi \in {\mathcal {O}}:=[1,2].$$ We will prove the existence of real analytic quasi-periodic solutions of the above equations under the hypothesis that the frequency $${\overline{\omega }}$$ is Liouvillean. The quasi-periodic solutions we obtain are around the equilibria, $$y(t,x)\equiv \pm \sqrt{m},\forall (t,x)\in {\mathbb {R}}\times [0,\ \pi ],$$ of the system $$\begin{aligned} \begin{aligned} y_{tt}+my+y_{xxxx}=y^{3}, \end{aligned} \end{aligned}$$and are whiskered, that is the linearized equation around $$\pm \,\sqrt{m}$$ owns the hyperbolic directions (the hyperbolic directions are finitely many, depending on m, and the elliptic directions are infinitely many in our case). The proof is based on a modified KAM iteration for infinite dimensional systems with finitely many hyperbolic directions, infinitely many elliptic directions and Liouvillean forcing frequency. We believe that the approach in this paper can be applied also to other integrable PDEs. For example, the same strategy should work for the non-linear wave equations and the non-linear Schrodinger equations.

Quasi-periodic solutions for a Schr\xf6dinger equation with a quintic nonlinear term depending on the time and space variables

This article is devoted to the study of a nonlinear Schrödinger equation with an x-periodic and t-quasi-periodic quintic nonlinear term. It is proved that the equation admits small-amplitude, linearly stable, real analytic, and quasi-periodic solutions for most values of frequency vector. By utilizing the measure estimation of infinitely many small divisors, we construct a real analytic, symplectic change of coordinates which can transform the Hamiltonian into some sixth order Birkhoff normal form. We show an infinite-dimensional KAM theorem for non-autonomous Schrödinger equations and apply the theorem to prove the existence of quasi-periodic solutions.

Isolated singularities of the prescribed mean curvature equation in Minkowski 3-space

We give a classification of non-removable isolated singularities for real analytic solutions of the prescribed mean curvature equation in Minkowski 3-space.

Well-posedness and continuity properties of the Fornberg–Whitham equation in Besov spaces

In this paper, we prove well-posedness of the Fornberg–Whitham equation in Besov spaces B2,rs in both the periodic and non-periodic cases. This will imply the existence and uniqueness of solutions in the aforementioned spaces along with the continuity of the data-to-solution map provided that the initial data belongs to B2,rs. We also establish sharpness of continuity on the data-to-solution map by showing that it is not uniformly continuous from any bounded subset of B2,rs to C([−T,T];B2,rs). Furthermore, we prove a Cauchy–Kowalevski type theorem for this equation that establishes the existence and uniqueness of real analytic solutions and also provide blow-up criterion for solutions.

Real Analytic Quasi-Periodic Solutions with More Diophantine Frequencies for Perturbed KdV Equations

In this paper, we consider the perturbed KdV equation with Fourier multiplier $$\begin{aligned} u_{t} =- u_{xxx} + \big (M_{\xi }u+u^3 \big )_{x},\quad u(t,x+2\pi )=u(t,x),\quad \int _0^{2\pi }u(t,x)dx=0, \end{aligned}$$ with analytic data of size \(\varepsilon \). We prove that the equation admits a Whitney smooth family of small amplitude, real analytic quasi-periodic solutions with \(\tilde{J}\) Diophantine frequencies, where the order of \(\tilde{J}\) is \(O(\frac{1}{\varepsilon })\). The proof is based on a conserved quantity \(\int _0^{2\pi } u^2 dx\), Toplitz–Lipschitz property and an abstract infinite dimensional KAM theorem. By taking advantage of the conserved quantity \(\int _0^{2\pi } u^2 dx\) and Toplitz–Lipschitz property, our normal form part is independent of angle variables in spite of the unbounded perturbation.

On the non-vanishing property for real analytic solutions of the $p$-Laplace equation

By using a nonassociative algebra argument, we prove that $u\equiv0$ is the only cubic homogeneous polynomial solution to the $p$-Laplace equation $\mathrm{div} |Du|^{p-2}Du(x)=0 $ in $\mathbb{R}^n$ for any $n\ge2$ and $p\not\in\{0,2\}$.

Periodic and quasi-periodic solutions of a derivative nonlinear Schrödinger equation

This paper is concerned with a one-dimensional (1D) derivative nonlinear Schrödinger equation with periodic boundary conditionsWe show that above equation admits a family of real analytic quasi-periodic solutions with two Diophantine frequencies and a family of real analytic periodic solutions. The proof is based on a partial Birkhoff normal form and KAM method.

Construction of Almost-Sharp Fronts for the Surface Quasi-Geostrophic Equation

In this paper we construct families of real analytic solutions of the Surface Quasi-Geostrophic equation (SQG) that are locally constant outside a thin neighborhood of a curve of arbitrarily small thickness. Despite the fact that only local existence results are known for SQG, and that our initial conditions have a arbitrarily large gradient we show that solutions exist for a time independent of the thickness of the neighborhood.

Modeling solute/contaminant transport in heterogeneous aquifers.

A fissured aquifer may be considered as a dense network of fissures separated by low permeability matrix blocks. A conceptual modeling of such a system consists of an infinite number of parallel fractures separated by constant width matrix slabs. While the fissures are assumed to be main flow conduits, the fluid in the porous matrix blocks are considered to be virtually immobile. The mathematical model of the transport of a solute and/or contaminant which assumes a purely convective flow in fissures and diffusion into the matrix blocks consists of two coupled differential equations. An analytical solution of this model for the case of solute entering into the system at a constant concentration has been presented by Skopp and Warrick in Soil Sci Soc Am Proc 38:545-550, 1974. Note however, Skopp and Warrick (Soil Sci Soc Am Proc 38:545-550, 1974) have not considered the additional processes of adsorption and radioactive decay. Unfortunately, their solution had computational limitations as it involved numerical integration of a quite complex expression. Therefore, one had to turn to employing numerical Laplace transform inverters to compute the solutions. This work presents simple real space analytical solutions for the contaminant transport model described above including the adsorption and radioactive decay. The real space solutions have been developed using the method of double Laplace transform and binomial series approximation. An accurate approximate solution has also been presented which converges to the exact solution only after computing three terms in the series full solution. The developed model has been used for 1) assessment of the efficiency of numerical Laplace transform algorithms and 2) investigation of the degree and scale of contamination, and 3) designing remediation schemes for the already contaminated aquifers.

Real-analytic solutions of the nonlinear Schrödinger equation

Real-analytic solutions of the nonlinear Schrödinger equation

Abel’s Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions

We obtain full description of eigenvalues and eigenvectors of composition operators $${C_{\varphi}:\fancyscript{A}\mathbb{R}\to \fancyscript{A}\mathbb{R}}$$ for a real analytic self map $${\varphi:\mathbb{R} \rightarrow \mathbb{R} }$$ as well as an isomorphic description of corresponding eigenspaces. We completely characterize those $${\varphi}$$ for which Abel’s equation $${f\circ \varphi=f+1}$$ has a real analytic solution on the real line. We find cases when the operator $${C_{\varphi}}$$ has roots using a constructed embedding of $${\varphi}$$ into the so-called real analytic iteration semigroups.

Quasi-Periodic Solutions with Prescribed Frequency in Reversible Systems

In this paper, we obtain a family of small-amplitude real analytic quasi-periodic solutions for a class of derivative nonlinear Schrodinger equations, subject to Dirichlet boundary conditions, which correspond to infinite-dimensional reversible systems with critical unbounded perturbations. We prove that the frequencies of the quasi-periodic solutions, accordingly, the tangential frequencies of the invariant tori for these reversible systems can be in a fixed direction.

On the pressure transfer function for solitary water waves with vorticity

In this paper we analyse the role which the pressure function on the sea-bed plays in determining solitary waves with vorticity. We prove that the pressure function on the flat bed determines a unique, real analytic solitary wave solution to the governing equations, given a real analytic vorticity distribution. In particular, the pressure function on the flat bed prescribes a unique surface profile for the resulting solitary water wave.

Real analytic quasi-periodic solutions for the derivative nonlinear Schrödinger equations

In this paper, we show that one dimension derivative nonlinear Schrödinger equation admits a whitney smooth family of small amplitude, real analytic quasi-periodic solutions with two Diophantine frequencies. The proof is based on a partial Birkhoff normal form reduction and an abstract infinite dimensional Kolmogorov-Arnold-Moser (KAM) theorem.

Study on Finite Element Analysis Models of Slant Rig Working Course

Slant-hole drilling rig, which can be used to drill inclined well from the ground, is used widely. During designing and research, it is very important to meet working reliability and requirement under various loads and working conditions. As for the complex structure of slant-hole drilling rig, we cannot get the real analytical solutions at present condition. Therefore, the paper establish the three-dimensional finite element models which are about the lifting process and working conditions of slant-hole drilling rig, study the applied loads of each working condition and analyze the stress-strain regulation of slant-hole drilling rig and its influential factor. Based on the analytical results of deformation and stress, an improvement scheme for the slant-hole drilling rig’s structure can is put forward.

Approximate Analytical Solution for the Temperature Field in Welding

To determine the temperature fields associated with welding, significant efforts have been made to establish the relative merits of numerical approaches with variable material properties and the analytical approaches with constant material properties. Currently, analytical solutions are either based on the temperature field generated by a point source of heat or are developed for a finite domain derived approximately by using an infinite or semi-infinite heat kernel. Furthermore, the heat kernel applied in these solutions is derived from the Image method (for example, Nguyen’s book (Thermal Analysis of Welds, 2004)). The main problem with the heat kernels obtained from Image method is that they face the problem of singularity at and around the point where the heat source is located, and they do not satisfy the boundary condition accurately. That is why the Laplace transform method has been applied here instead of using the Image method to formulate a heat kernel that (1) converges rapidly, (2) avoids the problem of singularity, and (3) gives a good and robust approximation of the real analytic solution for the temperature field. The results obtained from the analytical solutions were compared with the results obtained from finite element method. The current work is believed to make a considerable contribution to the avoidance of previously mentioned problems by deriving a new approximate analytical solution for the temperature field on a three-dimensional finite body.