### A high-order numerical method for the nonlinear Helmholtz equation in multidimensional layered media

- JOURNAL OF COMPUTATIONAL PHYSICS
, 2009

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### REMARK ON MAGNETIC SCHRÖDINGER OPERATORS IN EXTERIOR DOMAINS

, 907

"... Abstract. We study the Schrödinger operator with a constant magnetic field in the exterior of a two-dimensional compact domain. Functions in the domain of the operator are subject to a boundary condition of the third type (Robin condition). In addition to the Landau levels, we obtain that the spectr ..."

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Abstract. We study the Schrödinger operator with a constant magnetic field in the exterior of a two-dimensional compact domain. Functions in the domain of the operator are subject to a boundary condition of the third type (Robin condition). In addition to the Landau levels, we obtain that the spectrum of this operator consists of clusters of eigenvalues around the Landau levels and that they do accumulate to the Landau levels from below. We give a precise asymptotic formula for the rate of accumulation of eigenvalues in these clusters, which appears to be independent from the boundary condition. 1.

### Localized Boundary-Domain Singular Integral Equations Based on Harmonic Parametrix for Divergence-Form Elliptic PDEs with Variable Matrix Coefficients

"... Abstract. Employing the localized integral potentials associated with the Laplace operator, the Dirichlet, Neumann and Robin boundary value problems for general variable-coefficient divergence-form second-order elliptic partial differential equations are reduced to some systems of localized boundary ..."

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Abstract. Employing the localized integral potentials associated with the Laplace operator, the Dirichlet, Neumann and Robin boundary value problems for general variable-coefficient divergence-form second-order elliptic partial differential equations are reduced to some systems of localized boundary-domain singular integral equations. Equivalence of the integral equations systems to the original boundary value problems is proved. It is established that the corresponding localized boundarydomain integral operators belong to the Boutet de Monvel algebra of pseudo-differential operators. Applying the Vishik-Eskin theory based on the factorization method, the Fredholm properties and invertibility of the operators are proved in appropriate Sobolev spaces.

### Potential Type Operators and Transmission Problems for Strongly Elliptic Second-Order Systems in Lipschitz Domains∗

, 2009

"... Dedicated to the memory of Solomon G. Mikhlin Abstract. We consider a strongly elliptic second-order system in a bounded n-dimensional do-main Ω+ with Lipschitz boundary Γ, n 2. The smoothness assumptions on the coefficients are minimized. For convenience, we assume that the domain is contained in ..."

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Dedicated to the memory of Solomon G. Mikhlin Abstract. We consider a strongly elliptic second-order system in a bounded n-dimensional do-main Ω+ with Lipschitz boundary Γ, n 2. The smoothness assumptions on the coefficients are minimized. For convenience, we assume that the domain is contained in the standard torus Tn. In previous papers, we obtained results on the unique solvability of the Dirichlet and Neumann problems in the spaces Hσp and B σ p without use of surface potentials. In the present paper, using the approach proposed by Costabel and McLean, we define surface potentials and discuss their properties assuming that the Dirichlet and Neumann problems in Ω+ and the complementing do-main Ω − are uniquely solvable. In particular, we prove the invertibility of the integral single layer operator and the hypersingular operator in Besov spaces on Γ. We describe some of their spectral properties as well as those of the corresponding transmission problems.

### Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions

, 2005

"... In [J. Comput. Phys. 171 (2001) 632–677] we developed a fourth-order numerical method for solving the nonlinear Helmholtz equation which governs the propagation of time-harmonic laser beams in media with a Kerr-type nonlinearity. A key element of the algorithm was a new nonlocal two-way artificial b ..."

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In [J. Comput. Phys. 171 (2001) 632–677] we developed a fourth-order numerical method for solving the nonlinear Helmholtz equation which governs the propagation of time-harmonic laser beams in media with a Kerr-type nonlinearity. A key element of the algorithm was a new nonlocal two-way artificial boundary condition (ABC), set in the direction of beam propagation. This two-way ABC provided for reflectionless propagation of the outgoing waves while also fully transmitting the given incoming beam at the boundaries of the computational domain. Altogether, the algorithm of [J. Comput. Phys. 171 (2001) 632–677] has allowed for a direct simulation of nonlinear self-focusing without neglecting nonparaxial effects and backscattering. To the best of our knowledge, this capacity has never been achieved previously in nonlinear optics. In the current paper, we propose an improved version of the algorithm. The principal innovation is that instead of using the Dirichlet boundary conditions in the direction orthogonal to that of the laser beam propagation, we now introduce Sommerfeld-type local radiation boundary conditions, which are constructed directly in the discrete framework. Numerically, implementation of the Sommerfeld conditions requires evaluation of eigenvalues and eigenvectors for a non-Hermitian matrix. Subsequently, the separation of variables, which is a key building block of the aforementioned nonlocal ABC, is implemented through an expansion with respect to the nonorthogonal basis of the eigenvectors. Numerical simulations show that the new algorithm offers a considerable improvement in its numerical