相关论文: Numerical Approach to Painlev\'e Transcendents on …

We investigate some of the discrete Painleve equations (dPII, qPI and qPII) and the discrete KdV equation over finite fields. The first part concerns the discrete Painleve equations. We review some of the ideas introduced in our previous…

We study parameterized elliptic systems on symmetric domains with additional system symmetries. We prove the existence of continua of nontrivial solutions bifurcating from the constant branch determined by a critical point of the potential,…

We extend bifurcation results of nonlinear eigenvalue problems from real Banach spaces to any neighbourhood of a given point. For points of odd multiplicity on these restricted domains, we establish that the component of solutions through…

A new numerical method for an inverse problem for an elliptic equation with unknown potential is proposed. In this problem the point source is running along a straight line and the source-dependent Dirichlet boundary condition is measured…

We consider a connection problem of the first Painlev\'{e} equation ($\mathrm{P_I}$), trying to connect the local behavior (Laurent series) near poles and the asymptotic behavior as the variable $t$ tends to negative infinity for real…

We propose a spectral collocation method to approximate the exact boundary control of the wave equation in a square domain. The idea is to introduce a suitable approximate control problem that we solve in the finite-dimensional space of…

This article firstly develops a proximal explicit approach for the generalized method of lines. In such a method, the domain of the PDE in question is discretized in lines and the equation solution is written on these lines as functions of…

Recently, new adaptive techniques were developed that greatly improved the efficiency of solving PDEs using spectral methods. These adaptive spectral techniques are especially suited for accurately solving problems in unbounded domains and…

This paper analyzes the structure of the set of nodal solutions of a class of one-dimensional superlinear indefinite boundary values problems with an indefinite weight functions in front of the spectral parameter. Quite astonishingly, the…

The asymptotics of the generic second Painleve transcendent in the complex domain is found and justified via the direct asymptotic analysis of the associated Riemann-Hilbert problem based on the Deift-Zhou nonlinear steepest descent method.…

The aim of this paper is investigating the existence of solutions of some semilinear elliptic problems on open bounded domains when the nonlinearity is subcritical and asymptotically linear at infinity and there is a perturbation term which…

For more than a century, the Painlev\'e I equation has played an important role in both physics and mathematics. Its two-parameter family of solutions was studied in many different ways, yet still leads to new surprises and discoveries. Two…

The sixth Painlev\'e equation is a basic equation among the non-linear differential equations with three fixed singularities, corresponding to Gauss's hypergeometric differential equation among the linear differential equations. It is known…

This paper establishes the existence of infinitely many solutions for nonlinear problems without any symmetry, achieving three major advances. First, in the setting of semilinear elliptic PDEs, we introduce a refined variational truncation…

For a general solution of the third Painlev\'e equation of complete type we show the Boutroux ansatz near the point at infinity. It admits an asymptotic representation in terms of the Jacobi sn-function in cheese-like strips along generic…

This work introduces a sampling method capable of solving Bayesian inverse problems in function space. It does not assume the log-concavity of the likelihood, meaning that it is compatible with nonlinear inverse problems. The method…

A general class of transcendental equations in complex domain is considered for functions belonging to the Stieltjes cone. Under certain conditions each transcendental equation has no solution or one, at most, in the complex plane cut along…

We study the full asymptotic expansion of the monodromy data ({\it i.e.}, Stokes multipliers) for the first Painlev\'{e} transcendent (PI) with large initial data or large pole parameters. Our primary approach involves refining the complex…

We apply polynomial approximation methods -- known in the numerical PDEs context as spectral methods -- to approximate the vector-valued function that satisfies a linear system of equations where the matrix and the right hand side depend on…

We investigate the discrete Painleve II equation over finite fields. We treat it over local fields and observe that it has a property that is similar to the good reduction over finite fields. We can use this property, which seems to be an…