arXiv subjects

Wolfgang Reichel

Publications and source records attributed to Wolfgang Reichel.

30 records · Page 2Linked to original sources

arXiv2016

A priori bounds and global bifurcation results for frequency combs modeled by the Lugiato-Lefever equation

In nonlinear optics $2π$-periodic solutions $a\in C^2([0,2π];\mathbb{C})$ of the stationary Lugiato-Lefever equation $-d a"= ({\rm i} -ζ)a +|a|^2a-{\rm i} f$ serve as a model for frequency combs, which are optical signals consisting of a superposition of modes with equally spaced frequencies. We prove that nontrivial frequency combs can only be observed for special ranges of values of the forcing and detuning parameters $f$ and $ζ$, as it has been previously documented in experiments and numerical simulations. E.g., if the detuning parameter $ζ$ is too large then nontrivial frequency combs do not exist, cf. Theorem 2. Additionally, we show that for large ranges of parameter values nontrivial frequency combs may be found on continua which bifurcate from curves of trivial frequency combs. Our results rely on the proof of a priori bounds for the stationary Lugiato-Lefever equation as well as a detailed rigorous bifurcation analysis based on the bifurcation theorems of Crandall-Rabinowitz and Rabinowitz. We use the software packages AUTO and MATLAB to illustrate our results by numerical computations of bifurcation diagrams and of selected solutions.

math.AP↗

arXiv2016

A breather construction for a semilinear curl-curl wave equation with radially symmetric coefficients

We consider the semilinear curl-curl wave equation $s(x) \partial_t^2 U +\nabla\times\nabla\times U + q(x) U \pm V(x) |U|^{p-1} U = 0 \mbox{ for } (x,t)\in \mathbb{R}^3\times\mathbb{R}$. For any $p>1$ we prove the existence of time-periodic spatially localized real-valued solutions (breathers) both for the $+$ and the $-$ case under slightly different hypotheses. Our solutions are classical solutions that are radially symmetric in space and decay exponentially to $0$ as $|x|\to \infty$. Our method is based on the fact that gradient fields of radially symmetric functions are annihilated by the curl-curl operator. Consequently, the semilinear wave equation is reduced to an ODE with $r=|x|$ as a parameter. This ODE can be efficiently analyzed in phase space. As a side effect of our analysis, we obtain not only one but a full continuum of phase-shifted breathers $U(x,t+a(x))$, where $U$ is a particular breather and $a:\mathbb{R}^3\to\mathbb{R}$ an arbitrary radially symmetric $C^2$-function.

math.AP↗

arXiv2016

Existence of cylindrically symmetric ground states to a nonlinear curl-curl equation with non-constant coefficients

We consider the nonlinear curl-curl problem $\nabla\times\nabla\times U + V(x) U=f(x,|U|^2)U$ in $\mathbb{R}^3$ related to the nonlinear Maxwell equations with Kerr-type nonlinear material laws. We prove the existence of a symmetric ground-state type solution for a bounded, cylindrically symmetric coefficient $V$ and subcritical cylindrically symmetric nonlinearity $f$. The new existence result extends the class of problems for which ground-state type solutions are known. It is based on compactness properties of symmetric functions due to Lions, new rearrangement type inequalities from Brock and the recent extension of the Nehari-manifold technique by Szulkin and Weth.

math.AP↗

arXiv2014

Ground States of a Nonlinear Curl-Curl Problem in Cylindrically Symmetric Media

We consider the nonlinear curl-curl problem $\nabla\times\nabla\times U + V(x) U= Γ(x)|U|^{p-1}U$ in $\mathbb{R}^3$ related to the nonlinear Maxwell equations for monochromatic fields. We search for solutions as minimizers (ground states) of the corresponding energy functional defined on subspaces (defocusing case) or natural constraints (focusing case) of $H(\mathrm{curl};\mathbb{R}^3)$. Under a cylindrical symmetry assumption on the functions $V$ and $Γ$ the variational problem can be posed in a symmetric subspace of $H(\mathrm{curl};\mathbb{R}^3)$. For a strongly defocusing case $\mathrm{esssup}\, Γ<0$ with large negative values of $Γ$ at infinity we obtain ground states by the direct minimization method. For the focusing case $\mathrm{essinf}\, Γ>0$ the concentration compactness principle produces ground states under the assumption that zero lies outside the spectrum of the linear operator $\nabla \times \nabla \times +V(x)$. Examples of cylindrically symmetric functions $V$ are provided for which this holds.

math.AP↗

arXiv2013

Interfaces Supporting Surface Gap Soliton Ground States in the 1D Nonlinear Schroedinger Equation

We consider the problem of verifying the existence of $H^1$ ground states of the 1D nonlinear Schrödinger equation for an interface of two periodic structures: $$-u" +V(x)u -λu = Γ(x) |u|^{p-1}u \ {on} \R$$ with $V(x) = V_1(x), Γ(x)=Γ_1(x)$ for $x\geq 0$ and $V(x) = V_2(x), Γ(x)=Γ_2(x)$ for $x<0$. Here $V_1,V_2,Γ_1,Γ_2$ are periodic, $λ<\minσ(-\tfrac{d^2}{dx^2}+V)$, and $p>1$. The article [T. Dohnal, M. Plum and W. Reichel, "Surface Gap Soliton Ground States for the Nonlinear Schrödinger Equation," \textit{Comm. Math. Phys.} \textbf{308}, 511-542 (2011)] provides in the 1D case an existence criterion in the form of an integral inequality involving the linear potentials $V_{1},V_2$ and the Bloch waves of the operators $-\tfrac{d^2}{dx^2}+V_{1,2}-λ$. We choose here the classes of piecewise constant and piecewise linear potentials $V_{1,2}$ and check this criterion for a set of parameter values. In the piecewise constant case the Bloch waves are calculated explicitly and in the piecewise linear case verified enclosures of the Bloch waves are computed numerically. The integrals in the criterion are evaluated via interval arithmetic so that rigorous existence statements are produced. Examples of interfaces supporting ground states are reported including such, for which ground state existence follows for all periodic $Γ_ {1,2}$ with $\esssup Γ_{1,2}>0$.

math.AP↗

arXiv2011

Distributional solutions of the stationary nonlinear Schrödinger equation: singularities, regularity and exponential decay

We consider the nonlinear Schrödinger equation $-Δu + V(x) u = Γ(x) |u|^{p-1}u$ in $\R^n$ where the spectrum of $-Δ+V(x)$ is positive. In the case $n\geq 3$ we use variational methods to prove that for all $p\in (\frac{n}{n-2},\frac{n}{n-2}+\eps)$ there exist distributional solutions with a point singularity at the origin provided $\eps>0$ is sufficiently small and $V,Γ$ are bounded on $\R^n\setminus B_1(0)$ and satisfy suitable Hölder-type conditions at the origin. In the case $n=1,2$ or $n\geq 3,1<p<\frac{n}{n-2}$, however, we show that every distributional solution of the more general equation $-Δu + V(x) u = g(x,u)$ is a bounded strong solution if $V$ is bounded and $g$ satisfies certain growth conditions.

math.AP↗

arXiv2011

Surface Gap Soliton Ground States for the Nonlinear Schrödinger Equation

We consider the nonlinear Schrödinger equation $(-Δ+V(x))u = Γ(x) |u|^{p-1}u$, $x\in \R^n$ with $V(x) = V_1(x) χ_{\{x_1>0\}}(x)+V_2(x) χ_{\{x_1<0\}}(x)$ and $Γ(x) = Γ_1(x) χ_{\{x_1>0\}}(x)+Γ_2(x) χ_{\{x_1<0\}}(x)$ and with $V_1, V_2, Γ_1, Γ_2$ periodic in each coordinate direction. This problem describes the interface of two periodic media, e.g. photonic crystals. We study the existence of ground state $H^1$ solutions (surface gap soliton ground states) for $0<\min σ(-Δ+V)$. Using a concentration compactness argument, we provide an abstract criterion for the existence based on ground state energies of each periodic problem (with $V\equiv V_1, Γ\equiv Γ_1$ and $V\equiv V_2, Γ\equiv Γ_2$) as well as a more practical criterion based on ground states themselves. Examples of interfaces satisfying these criteria are provided. In 1D it is shown that, surprisingly, the criteria can be reduced to conditions on the linear Bloch waves of the operators $-\tfrac{d^2}{dx^2} +V_1(x)$ and $-\tfrac{d^2}{dx^2} +V_2(x)$.

math.AP↗

arXiv2009

Localized Modes of the Linear Periodic Schrödinger Operator with a Nonlocal Perturbation

We consider the existence of localized modes corresponding to eigenvalues of the periodic Schrödinger operator $-\partial_x^2+ V(x)$ with an interface. The interface is modeled by a jump either in the value or the derivative of $V(x)$ and, in general, does not correspond to a localized perturbation of the perfectly periodic operator. The periodic potentials on each side of the interface can, moreover, be different. As we show, eigenvalues can only occur in spectral gaps. We pose the eigenvalue problem as a $C^1$ gluing problem for the fundamental solutions (Bloch functions) of the second order ODEs on each side of the interface. The problem is thus reduced to finding matchings of the ratio functions $R_\pm=\frac{ψ_\pm'(0)}{ψ_\pm(0)}$, where $ψ_\pm$ are those Bloch functions that decay on the respective half-lines. These ratio functions are analyzed with the help of the Prüfer transformation. The limit values of $R_\pm$ at band edges depend on the ordering of Dirichlet and Neumann eigenvalues at gap edges. We show that the ordering can be determined in the first two gaps via variational analysis for potentials satisfying certain monotonicity conditions. Numerical computations of interface eigenvalues are presented to corroborate the analysis.

math.SP↗

arXiv2009

Existence of solutions to nonlinear, subcritical higher-order elliptic Dirichlet problems

We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $Ω\subset\R^N$ with Dirichlet boundary conditions on $\partialΩ$. The operator $L$ is a uniformly elliptic linear operator of order $2m$ whose principle part is of the form $\big(-\sum_{i,j=1}^N a_{ij}(x) \frac{\partial^2}{\partial x_i\partial x_j}\big)^m$. We assume that $f$ is superlinear at the origin and satisfies $\lim \limits_{s\to\infty}\frac{f(x,s)}{s^q}=h(x)$, $\lim \limits_{s\to-\infty}\frac{f(x,s)}{|s|^q}=k(x)$, where $h,k\in C(\overlineΩ)$ are positive functions and $q>1$ is subcritical. By combining degree theory with new and recently established a priori estimates, we prove the existence of a nontrivial solution.

math.AP↗

arXiv2009

Large solutions to semilinear elliptic equations with Hardy potential and exponential nonlinearity

On a bounded smooth domain we study solutions of a semilinear elliptic equation with an exponential nonlinearity and a Hardy potential depending on the distance to the boundary of the domain. We derive global a priori bounds of the Keller-Osserman type. Using a Phragmen-Lindelof alternative for generalized sub and super-harmonic functions we discuss existence, nonexistence and uniqueness of so-called large solutions, i.e., solutions which tend to infinity at the boundary. The approach develops the one used by the same authors for a problem with a power nonlinearity instead of the exponential nonlinearity.

math.AP↗

arXiv2007

A priori bounds and a Liouville theorem on a half-space for higher order elliptic Dirichlet problems

We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $Ω$ in $R^N$ with Dirichlet boundary conditions. The operator $L$ is a uniformly elliptic operator of order $2m$. We assume that for $s\to \pm\infty$ the nonlinearity $f(x,s)$ behaves like $|s|^q$ multiplied by a continuous and positive function of $x$. Here the exponent $q$ is subcritical, i.e., $q>1$ if $N<=2m$, $1 2m$. We prove a priori bounds, i.e, we show that the $L^\infty$-norm of every solution $u$ is bounded by a constant independent of $u$. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on the following new Liouville-type theorem on a half-space: if $u$ is a classical, bounded, non-negative solution of $(-Δ)^m u = u^q$ in a half-space with Dirichlet boundary conditions and if $q>1$ is subcritical then $u$ vanishes identically.

math.AP↗

arXiv2007

"Boundary blowup" type sub-solutions to semilinear elliptic equations with Hardy potential

Semilinear elliptic equations which give rise to solutions blowing up at the boundary are perturbed by a Hardy potential. The size of this potential effects the existence of a certain type of solutions (large solutions): if the potential is too small, then no large solution exists. The presence of the Hardy potential requires a new definition of large solutions, following the pattern of the associated linear problem. Nonexistence and existence results for different types of solutions will be given. Our considerations are based on a Phragmen-Lindelof type theorem which enables us to classify the solutions and sub-solutions according to their behavior near the boundary. Nonexistence follows from this principle together with the Keller-Osserman upper bound. The existence proofs rely on sub- and super-solution techniques and on estimates for the Hardy constant derived in Marcus, Mizel and Pinchover.

math.AP↗