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We find a local (d+1)×(d+1) Riemann–Hilbert problem characterizing the skew-orthogonal polynomials associated to the partition function of orthogonal ensembles of random matrices with a potential function of degree d . Our Riemann–Hilbert problem is similar to a local d×d Riemann–Hilbert problem found by Kuijlaars and McLaughlin characterizing the bi-orthogonal polynomials. This gives more motivation for finding methods to compute asymptotics of high order Riemann–Hilbert problems, and brings us closer to finding full asymptotic expansions of the skew-orthogonal polynomials. 相似文献
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Sheehan Olver 《Numerische Mathematik》2012,122(2):305-340
A new, numerical framework for the approximation of solutions to matrix-valued Riemann?CHilbert problems is developed, based on a recent method for the homogeneous Painlevé II Riemann?CHilbert problem. We demonstrate its effectiveness by computing solutions to other Painlevé transcendents. An implementation in Mathematica is made available online. 相似文献
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Arno Kuijlaars 《Constructive Approximation》2014,39(1):197-222
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《Nonlinear Analysis: Theory, Methods & Applications》2005,62(6):1091-1107
In this paper, we study the nonlinear initial–boundary Riemann problem and the generalized nonlinear initial–boundary Riemann problem for quasilinear hyperbolic systems of conservation laws with nonlinear boundary conditions on the domain . Under the assumption that each positive eigenvalue is either linearly degenerate or genuinely nonlinear, we get the existence and uniqueness of the self-similar solution to the nonlinear initial–boundary Riemann problem and of the global piecewise solution containing only shocks and (or) contact discontinuities to the corresponding generalized nonlinear initial–boundary Riemann problem. It shows that the self-similar solution to the nonlinear initial–boundary Riemann problem possesses the global structural stability. 相似文献
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In this paper we consider the relations between the Riemann–Hilbert monodromy problem and the matrix Riemann–Hilbert boundary-value problem with piecewise continuous coefficient and construct the so-called canonical matrix for the boundary-value problem for a piecewise continuous matrix-function. The formula for the calculation of the index is also obtained. 相似文献
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The solvability of the fully nonlinear stationary Venttsel' problem is established. The equation and the boundary condition are assumed to be uniformly elliptic. Bibliography: 12 titles. Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 3–26. 相似文献
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Marta Calanchi Bernhard Ruf 《Calculus of Variations and Partial Differential Equations》2010,38(1-2):111-133
We discuss existence and non-existence of positive solutions for the following system of Hardy and Hénon type: $$\left\{\begin{array}{ll} {-\Delta v=|x|^{\alpha}u^{p},\,-\Delta u=|x|^{\beta}v^{q} \,\,{\rm in}\, \Omega,}\\ {u=v=0 \quad\quad\quad\quad\quad\quad\quad\quad\quad{\rm on}\, \partial \Omega}, \end{array}\right.$$ where ${\Omega\ni 0}$ is a bounded domain in ${\mathbb{R}^{N}}$ , N ≥ 3, p, q > 1, and α, β > ?N. We also study symmetry breaking for ground states when Ω is the unit ball in ${\mathbb{R}^{N}}$ . 相似文献
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In this article we use linear spline approximation of a non-linear Riemann–Hilbert problem on the unit disk. The boundary condition for the holomorphic function is reformulated as a non-linear singular integral equation A(u) = 0, where A : H 1(Γ) → H 1(Γ) is defined via a Nemytski operator. We approximate A by A n : H 1(Γ) → H 1(Γ) using spline collocation and show that this defines a Fredholm quasi-ruled mapping. Following the results of (A.I. ?nirel'man, The degree of quasi-ruled mapping and a nonlinear Hilbert problem, Math. USSR-Sbornik 18 (1972), pp. 373–396; M.A. Efendiev, On a property of the conjugate integral and a nonlinear Hilbert problem, Soviet Math. Dokl. 35 (1987), pp. 535–539; M.A. Efendiev, W.L. Wendland, Nonlinear Riemann–Hilbert problems for multiply connected domains, Nonlinear Anal. 27 (1996), pp. 37–58; Nonlinear Riemann–Hilbert problems without transversality. Math. Nachr. 183 (1997), pp. 73–89; Nonlinear Riemann–Hilbert problems for doubly connected domains and closed boundary data, Topol. Methods Nonlinear Anal. 17 (2001), pp. 111–124; Nonlinear Riemann–Hilbert problems with Lipschitz, continuous boundary data without transversality, Nonlinear Anal. 47 (2001), pp. 457–466; Nonlinear Riemann–Hilbert problems with Lipschitz-continuous boundary data: Doubly connected domains, Proc. Roy. Soc. London Ser. A 459 (2003), pp. 945–955.), we define a degree of mapping and show the existence of the spline solutions of the fully discrete equations A n (u) = 0, for n large enough. We conclude this article by discussing the solvability of the non-linear collocation method, where we shall need an additional uniform strong ellipticity condition for employing the spline approximation. 相似文献
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Shusen Yan Jianfu Yang 《Calculus of Variations and Partial Differential Equations》2013,48(3-4):587-610
We consider the following problem $$\left\{ \begin{array}{ll}-\Delta u = \mu |u|^\frac{4}{N-2}u + \frac{|u|^\frac{4-2s}{N-2}u}{|x|^{s}} + a(x)u, & x \in \Omega,\\ u=0, & {\rm on}\; \partial \Omega \end{array}\right.$$ where ${ \mu \ge 0, 0 < s < 2, 0 \in \partial \Omega}$ and Ω is a bounded domain in R N . We prove that if ${N \ge 7, a(0) > 0}$ and all the principle curvatures of ?Ω at 0 are negative, then the above problem has infinitely many solutions. 相似文献
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A simple exact formula is derived for the profile of the optical pulse propagating over a DM fiber with zero mean dispersion. The dissipation is neglected, and dispersion is assumed to be constant along the adjacent legs of the waveguide, thus providing the applicability of the integrable NLS models within each leg. The formula describes a class of solutions called dispersion-managed solitons (DM solitons), which are periodic along the waveguide and exponentially localized in time. The DM solitons are parameterized by a certain class of spectral data, specified from numerical simulations. Using a related Riemann–Hilbert problem, we reconstruct a profile of the DM soliton from the given spectral data. For sufficiently long legs, the leading term of DM soliton is found in explicit form by asymptotic undressing of the Riemann–Hilbert problem. The analytic results are compared with numerical simulations. 相似文献
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Meina Sun Wancheng Sheng 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2009,60(2):271-283
The generalized Riemann problem for a scalar Chapman–Jouguet combustion model in a neighborhood of the origin (t > 0) on the (x, t) plane is studied. Under the entropy conditions, we obtain the solutions constructively. It is found that, for some cases,
the perturbed Riemann solutions are essentially different from the corresponding Riemann solutions. The perturbation may transform
a combustion wave CJDT into SDT in the neighborhood of the origin. Especially, it can be observed that burning happens although
the corresponding Riemann solution doesn’t contain combustion waves, which exhibits the instability for unburnt states.
This work is supported by NSFC 10671120 相似文献
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《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2020,37(5):1143-1166
In this paper we show uniqueness of the conductivity for the quasilinear Calderón's inverse problem. The nonlinear conductivity depends, in a nonlinear fashion, of the potential itself and its gradient. Under some structural assumptions on the direct problem, a real-valued conductivity allowing a small analytic continuation to the complex plane induce a unique Dirichlet-to-Neumann (DN) map. The method of proof considers some complex-valued, linear test functions based on a point of the boundary of the domain, and a linearization of the DN map placed at these particular set of solutions. 相似文献
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《Nonlinear Analysis: Theory, Methods & Applications》2007,66(1):241-252
Let and be the critical Sobolev–Hardy exponents. Via variational methods and the analytic technique, we prove the existence of a nontrivial solution to the singular semilinear problem , for and suitable functions . 相似文献