arXiv subjects

Wolfgang Reichel

Publications and source records attributed to Wolfgang Reichel.

At least 19 recordsLinked to original sources

arXiv2026

Homoclinics, rogue waves and breathers in nonlinear lattice wave equations

We prove the existence of rogue waves and breathers in nonlinear lattice wave equations including the nonlinear Klein--Gordon and the FPUT lattice (with added local forces). The main feature of our lattice wave equation is that the kinetic part $\frac{\mathrm{d}^2}{\mathrm{d}t^2}$ is multiplied by a non-constant function $\frac{1}{d(t)}$ such that gaps in the spectrum of $\frac{1}{d(t)}\frac{\mathrm{d}^2}{\mathrm{d}t^2}$ open wide enough to include the spectrum of the spatial linear operator. We find solutions as critical points of an indefinite functional using a saddle-point method combined with concentration-compactness arguments. In case of temporally $T$-periodic coefficients and under identical assumptions as for rogue waves, the same variational method also provides existence of breather solutions whose temporal period is an arbitrary prescribed integer multiple of $T$.

math.AP↗

arXiv2026

Optimal data-driven solutions for a stationary diffusive model of population growth

We study optimal data-driven solutions for the stationary diffusive population growth model $-Δu = r u$ in a bounded domain $Ω\subset\mathbb R^N$ with Neumann boundary conditions. Instead of prescribing a functional relation between the position $x$, the net per-capita growth rate $r$ and the population size $u$, we look for a pair $(u,r)\in H^1(Ω)\times L^\infty(Ω)$ that fits a given data set in an optimal way measured by a cost functional $I$ and an additional penalty term. We characterize the relaxed cost functional sc$^- I$ by showing that its density is given as the partial lower convex envelope with respect to the variable $r$, and prove the existence of optimal data-driven solutions. Furthermore, we establish a consistency result comparing conventional solutions of $-Δu = \varrho(x,u)u$ with optimal data-driven solutions where the data set stems from the functional relation $(x,u)\mapsto \varrho(x,u)$. Finally, as data sets evolve, we prove the convergence of optimal solutions via the $Γ$-convergence of the associated cost functionals.

math.AP↗

arXiv2026

Traveling and Dispersive Shock Waves in a Two-Dimensional Fermi-Pasta-Ulam-Tsingou Lattice

In the present work we analyze traveling and dispersive shock waves of a two-dimensional Fermi-Pasta-Ulam-Tsingou lattice. In the first part of the paper, using variational techniques we prove the existence of both periodic and solitary traveling waves for convex potentials. In the case of unimodal profiles we are able to remove the assumption of convexity. The variational formulation also provides a natural algorithm for the numerical computation of traveling waves, which we use to explore both solitary and periodic traveling waves. The numerical computations are compared with analytical approximations based on the derivation of the KdV equation for quasi-one-dimensional propagation. In the second part of the paper, we focus on dispersive shock waves (DSWs), which are expanding modulated waves that connect states of different amplitude. In particular, we focus on line DSWs, which are constant along one direction and propagate in the direction orthogonal to which it is constant. Such solutions form when subject to quasi-one-dimensional jump initial data. We find that while the shape of the DSW depends on the direction of travel, properties such as the speed and amplitude do not. The systematic numerical study of the line~DSWs is then compared to those predicted by the KdV equation along the line of propagation. Key characteristics of the DSWs, such as the speeds of the trailing and leading edges, are investigated for various jump heights, yielding good agreement between simulation and KdV approximation in the limit of vanishing jump height. Finally, we apply the DSW fitting method to study the trailing and leading edge characteristics of the DSW, finding even better agreement to the numerics when compared to the KdV prediction. The KdV prediction and DSW fitting predictions agree in the limit of small jump height.

nlin.PS↗

arXiv2025

Breather solutions for semilinear wave equations

We prove existence of real-valued, time-periodic and spatially localized solutions (breathers) of semilinear wave equations $V(x)u_{tt} - u_{xx} = Γ(x) |u|^{p-1} u$ on $\mathbb{R}^2$ for all values of $p\in (1,\infty)$. Using tools from the calculus of variations our main result provides breathers as ground states of an indefinite functional under suitable conditions on $V, Γ$ beyond the limitations of pure $x$-periodicity. Such an approach requires a detailed analysis of the wave operator acting on time-periodic functions. Hence a generalization of the Floquet-Bloch theory for periodic Sturm-Liouville operators is needed which applies to perturbed periodic operators. For this purpose we develop a suitable functional calculus for the weighted operator $-\frac{1}{V(x)}\frac{\mathrm{d}^2}{\mathrm{d}x^2}$ with an explicit control of its spectral measure. Based on this we prove embedding theorems from the form domain of the wave operator into $L^q$-spaces, which is key to controlling nonlinearities. We complement our existence theory with explicit examples of coefficient functions $V$ and temporal periods $T$ which support breathers.

math.AP↗

arXiv2025

Travelling breather solutions in waveguides for cubic nonlinear Maxwell equations with retarded material laws

For Maxwell's equations with nonlinear polarization we prove the existence of time-periodic breather solutions travelling along slab or cylindrical waveguides. The solutions are TE-modes which are localized in space directions orthogonal to the direction of propagation. We assume a magnetically inactive and electrically nonlinear material law with a linear $χ^{(1)}$- and a cubic $χ^{(3)}$-contribution to the polarization. The $χ^{(1)}$-contribution may be retarded in time or instantaneous whereas the $χ^{(3)}$-contribution is always assumed to be retarded in time. We consider two different cubic nonlinearities which provide a variational structure under suitable assumptions on the retardation kernels. By choosing a sufficiently small propagation speed along the waveguide the second order formulation of the Maxwell system becomes essentially elliptic for the $\mathbf{E}$-field so that solutions can be constructed by the mountain pass theorem. The compactness issues arising in the variational method are overcome by either the cylindrical geometry itself or by extra assumptions on the linear and nonlinear parts of the polarization in case of the slab geometry. Our approach to breather solutions in the presence of time-retardation is systematic in the sense that we look for general conditions on the Fourier-coefficients in time of the retardation kernels. Our main existence result is complemented by concrete examples of coefficient functions and retardation kernels.

math.AP↗

arXiv2024

Existence of traveling breather solutions to cubic nonlinear Maxwell equations in waveguide geometries

We consider the full set of Maxwell equations in a slab or cylindrical waveguide with a cubically nonlinear material law for the polarization of the electric field. The nonlinear polarization may be instantaneous or retarded, and we assume it to be confined inside the core of the waveguide. We prove existence of infinitely many spatially localized, real-valued and time-periodic solutions (breathers) propagating inside the waveguide by applying a variational minimization method to the resulting scalar quasilinear elliptic-hyperbolic equation for the profile of the breathers. The temporal period of the breathers has to be carefully chosen depending on the linear properties of the waveguide. As an example, our results apply if a two-layered linear axisymmetric waveguide is enhanced by a third core region with low refractive index where also the nonlinearity is located. In this case we can also connect our existence result with a bifurcation result. We illustrate our results with numerical simulations. Our solutions are polychromatic functions in general, but for some special models of retarded nonlinear material laws, also monochromatic solutions can exist. In this case the numerical simulations raise an interesting open question: are the breather solutions with minimal energy monochromatic or polychromatic?

math.AP↗

arXiv2023

Pinning in the extended Lugiato-Lefever equation

We consider a variant of the Lugiato-Lefever equation (LLE), which is a nonlinear Schrödinger equation on a one-dimensional torus with forcing and damping, to which we add a first-order derivative term with a potential $εV(x)$. The potential breaks the translation invariance of LLE. Depending on the existence of zeroes of the effective potential $V_\text{eff}$, which is a suitably weighted and integrated version of $V$, we show that stationary solutions from $ε=0$ can be continued locally into the range $ε\not =0$. Moreover, the extremal points of the $ε$-continued solutions are located near zeros of $V_\text{eff}$. We therefore call this phenomenon \emph{pinning} of stationary solutions. If we assume additionally that the starting stationary solution at $ε=0$ is spectrally stable with the simple zero eigenvalue due to translation invariance being the only eigenvalue on the imaginary axis, we can prove asymptotic stability or instability of its $ε$-continuation depending on the sign of $V_\text{eff}'$ at the zero of $V_\text{eff}$ and the sign of $ε$. The variant of the LLE arises in the description of optical frequency combs in a Kerr nonlinear ring-shaped microresonator which is pumped by two different continuous monochromatic light sources of different frequencies and different powers. Our analytical findings are illustrated by numerical simulations.

math.AP↗

arXiv2022

Breathers and rogue waves for semilinear curl-curl wave equations

We consider localized solutions of variants of the semilinear curl-curl wave equation $s(x) \partial_t^2 U +\nabla\times\nabla\times U + q(x) U \pm V(x) |U|^{p-1} U = 0$ for $(x,t)\in \mathbb{R}^3\times\mathbb{R}$ and arbitrary $p>1$. Depending on the coefficients $s, q, V$ we can prove the existence of three types of localized solutions: time-periodic solutions decaying to $0$ at spatial infinity, time-periodic solutions tending to a nontrivial profile at spatial infinity (both types are called breathers), and rogue waves which converge to $0$ both at spatial and temporal infinity. Our solutions are weak solutions and take the form of gradient fields. Thus they belong to the kernel of the curl-operator so that due to the structural assumptions on the coefficients the semilinear wave equation is reduced to an ODE. Since the space dependence in the ODE is just a parametric dependence we can analyze the ODE by phase plane techniques and thus establish the existence of the localized waves described above. Noteworthy side effects of our analysis are the existence of compact support breathers and the fact that one localized wave solution $U(x,t)$ already generates a full continuum of phase-shifted solutions $U(x,t+b(x))$ where the continuous function $b:\mathbb{R}^3\to\mathbb{R}$ belongs to a suitable admissible family.

math.AP↗

arXiv2022

Breather solutions for a semilinear Klein-Gordon equation on a periodic metric graph

We consider the nonlinear Klein-Gordon equation $\partial_t^2u(x,t)-\partial_x^2u(x,t)+αu(x,t)=\pm|u(x,t)|^{p-1}u(x,t)$ on a periodic metric graph (necklace graph) for $p>1$ with Kirchhoff conditions at the vertices. Under suitable assumptions on the frequency we prove the existence and regularity of infinitely many spatially localized time-periodic solutions (breathers) by variational methods. We compare our results with previous results obtained via spatial dynamics and center manifold techniques. Moreover, we deduce regularity properties of the solutions and show that they are weak solutions of the corresponding initial value problem. Our approach relies on the existence of critical points for indefinite functionals, the concentration compactness principle, and the proper set-up of a functional analytic framework. Compared to earlier work for breathers using variational techniques, a major improvement of embedding properties has been achieved. This allows in particular to avoid all restrictions on the exponent $p>1$ and to achieve higher regularity.

math.AP↗

arXiv2022

Global continua of solutions to the Lugiato-Lefever model for frequency combs obtained by two-mode pumping

We consider Kerr frequency combs in a dual-pumped microresonator as time-periodic and spatially $2π$-periodic traveling wave solutions of a variant of the Lugiato-Lefever equation, which is a damped, detuned and driven nonlinear Schrödinger equation given by $\mathrm{i}a_τ=(ζ-\mathrm{i})a - d a_{x x}-|a|^2a+\mathrm{i}f_0+\mathrm{i}f_1\mathrm{e}^{\mathrm{i}(k_1 x-ν_1 τ)}$. The main new feature of the problem is the specific form of the source term $f_0+f_1\mathrm{e}^{\mathrm{i}(k_1 x-ν_1 τ)}$ which describes the simultaneous pumping of two different modes with mode indices $k_0=0$ and $k_1\in \mathbb{N}$. We prove existence and uniqueness theorems for these traveling waves based on a-priori bounds and fixed point theorems. Moreover, by using the implicit function theorem and bifurcation theory, we show how non-degenerate solutions from the $1$-mode case, i.e. $f_1=0$, can be continued into the range $f_1\not =0$. Our analytical findings apply both for anomalous ($d>0$) and normal ($d<0$) dispersion, and they are illustrated by numerical simulations.

math.AP↗

arXiv2022

Bandwidth and Conversion-Efficiency Analysis of Kerr Soliton Combs in Dual-Pumped Resonators with Anomalous Dispersion

Kerr frequency combs generated in high-Q microresonators offer an immense potential in many applications, and predicting and quantifying their behavior, performance and stability is key to systematic device design. Based on an extension of the Lugiato-Lefever equation we investigate in this paper the perspectives of changing the pump scheme from the well-understood monochromatic pump to a dual-tone configuration simultaneously pumping two modes. For the case of anomalous dispersion we give a detailed study of the optimal choices of detuning offsets and division of total pump power between the two modes in order to optimize single-soliton comb states with respect to performance metrics like power conversion efficiency and bandwidth. Our approach allows also to quantify the performance metrics of the optimal single-soliton comb states and determine their trends over a wide range of technically relevant parameters.

physics.optics↗

arXiv2022

Wellposedness for a (1+1)-dimensional wave equation with quasilinear boundary condition

We consider the linear wave equation $V(x) u_{tt}(x, t) - u_{xx}(x, t) = 0$ on $[0, \infty)\times[0, \infty)$ with initial conditions and a nonlinear Neumann boundary condition $u_x(0, t) = (f(u_t(0,t)))_t$ at $x=0$. This problem is an exact reduction of a nonlinear Maxwell problem in electrodynamics. In the case where $f\colon\mathbb{R}\to\mathbb{R}$ is an increasing homeomorphism we study global existence, uniqueness and wellposedness of the initial value problem by the method of characteristics and fixed point methods. We also prove conservation of energy and momentum and discuss why there is no wellposedness in the case where $f$ is a decreasing homeomorphism. Finally we show that previously known time-periodic, spatially localized solutions (breathers) of the wave equation with the nonlinear Neumann boundary condition at $x=0$ have enough regularity to solve the initial value problem with their own initial data.

math.AP↗

arXiv2022

Traveling waves for a quasilinear wave equation

We consider a 2+1 dimensional wave equation appearing in the context of polarized waves for the nonlinear Maxwell equations. The equation is quasilinear in the time derivatives and involves two material functions $V$ and $Γ$. We prove the existence of traveling waves which are periodic in the direction of propagation and localized in the direction orthogonal to the propagation direction. Depending on the nature of the nonlinearity coeffcient $Γ$ we distinguish between two cases: (a) $Γ\in L^\infty$ being regular and (b) $Γ=γδ_0$ being a multiple of the delta potential at zero. For both cases we use bifuraction theory to prove the existence of nontrivial small-amplitude solutions. One can regard our results as a persistence result which shows that guided modes known for linear wave-guide geometries survive in the presence of a nonlinear constitutive law. Our main theorems are derived under a set of conditions on the linear wave operator. They are subsidised by explicit examples for the coefficients $V$ in front of the (linear) second time derivative for which our results hold.

math.AP↗

arXiv2021

Travelling waves for Maxwell's equations in nonlinear and nonsymmetric media

We look for travelling wave fields $$ E(x,y,z,t)= U(x,y) \cos(kz+ωt)+ \widetilde U(x,y)\sin(kz+ωt),\quad (x,y,z)\in\mathbb{R}^3,\, t\in\mathbb{R} $$ satisfying Maxwell's equations in a nonlinear medium which is not necessarily cylindrically symmetric. The nonlinearity of the medium enters Maxwell's equations by postulating a nonlinear material law $D=\varepsilon E+χ(x,y, \langle |E|^2\rangle)E$ between the electric field $E$, its time averaged intensity $\langle |E|^2\rangle$ and the electric displacement field $D$. We derive a new semilinear elliptic problem for the profiles $U,\widetilde U:\mathbb{R}^2\to\mathbb{R}^3$ $$Lu-V(x,y)u=f(x,y,u)\quad\hbox{with }u=\begin{pmatrix} U \\ \widetilde U \end{pmatrix}, \hbox{ for }(x,y)\in\mathbb{R}^2,$$ where $f(x,y,u)=ω^2χ(x,y, |u|^2)u$. Solving this equation we can obtain exact travelling wave solutions of the underlying nonlinear Maxwell equations. We are able to deal with super quadratic and subcritical focusing effects, e.g. in the Kerr-like materials with the nonlinear susceptibility of the form $χ(x,y,\langle |E^2|\rangle E) = χ^{(3)}(x,y)\langle |E|^2\rangle E$. A variational approach is presented for the semilinear problem. The energy functional associated with the equation is strongly indefinite, since $L$ contains an infinite dimensional kernel. The methods developed in this paper may be applicable to other strongly indefinite elliptic problems and other nonlinear phenomena.

math.AP↗

arXiv2021

Breather Solutions for a Quasilinear $(1+1)$-dimensional Wave Equation

We consider the $(1+1)$-dimensional quasilinear wave equation $g(x)w_{tt}-w_{xx}+h(x) (w_t^3)_t=0$ on $\mathbb{R}\times\mathbb{R}$ which arises in the study of localized electromagnetic waves modeled by Kerr-nonlinear Maxwell equations. We are interested in time-periodic, spatially localized solutions. Here $g\in L^{\infty}(\mathbb{R})$ is even with $g\not\equiv 0$ and $h(x)=γ\,δ_0(x)$ with $γ\in{\mathbb{R}}\backslash\{0\}$ and $δ_0$ the delta-distribution supported in $0$. We assume that $0$ lies in a spectral gap of the operators $L_k=-\frac{d^2}{dx^2}-k^2ω^2g$ on $L^2(\mathbb{R})$ for all $k\in 2\mathbb{Z}+1$ together with additional properties of the fundamental set of solutions of $L_k$. By expanding $w$ into a Fourier series in time we transfer the problem of finding a suitably defined weak solution to finding a minimizer of a functional on a sequence space. The solutions that we have found are exponentially localized in space. Moreover, we show that they can be well approximated by truncating the Fourier series in time. The guiding examples, where all assumptions are fulfilled, are explicitly given step potentials and periodic step potentials $g$. In these examples we even find infinitely many distinct breathers.

math.AP↗

arXiv2020

Equilibrium measures and equilibrium potentials in the Born-Infeld model

In this paper, we consider the electrostatic Born-Infeld model \begin{equation*} \tag{$\mathcal{BI}$} \left\{ \begin{array}{rcll} -\operatorname{div}\left(\displaystyle\frac{\nabla ϕ}{\sqrt{1-|\nabla ϕ|^2}}\right)&=& ρ& \hbox{in }\mathbb{R}^N, \\[6mm] \displaystyle\lim_{|x|\to \infty}ϕ(x)&=& 0 \end{array} \right. \end{equation*} where $ρ$ is a charge distribution on the boundary of a bounded domain $Ω\subset \mathbb{R}^N$. We are interested in its equilibrium measures, i.e. charge distributions which minimize the electrostatic energy of the corresponding potential among all possible distributions with fixed total charge. We prove existence of equilibrium measures and we show that the corresponding equilibrium potential is unique and constant in $\overline Ω$. Furthermore, for smooth domains, we obtain the uniqueness of the equilibrium measure, we give its precise expression, and we verify that the equilibrium potential solves ($\mathcal{BI}$). Finally we characterize balls in $\mathbb{R}^N$ as the unique sets among all bounded $C^{2,α}$-domains $Ω$ for which the equilibrium distribution is a constant multiple of the surface measure on $\partialΩ$. The same results are obtained also for Taylor approximations of the electrostatic energy.

math.AP↗

arXiv2018

The Lugiato-Lefever equation with nonlinear damping caused by two photon absorption

In this paper we investigate the effect of nonlinear damping on the Lugiato-Lefever equation $$ ı\partial_t a = -(ı-ζ) a - da_{xx} -(1+ıκ)|a|^2a +ıf $$ on the torus or the real line. For the case of the torus it is shown that for small nonlinear damping $κ>0$ stationary spatially periodic solutions exist on branches that bifurcate from constant solutions whereas all nonconstant solutions disappear when the damping parameter $κ$ exceeds a critical value. These results apply both for normal ($d<0$) and anomalous ($d>0$) dispersion. For the case of the real line we show by the Implicit Function Theorem that for small nonlinear damping $κ>0$ and large detuning $ζ\gg 1$ and large forcing $f\gg 1$ strongly localized, bright solitary stationary solutions exists in the case of anomalous dispersion $d>0$. These results are achieved by using techniques from bifurcation and continuation theory and by proving a convergence result for solutions of the time-dependent Lugiato-Lefever equation.

math.AP↗

arXiv2018

Real-valued, time-periodic localized weak solutions for a semilinear wave equation with periodic potentials

We consider the semilinear wave equation $V(x) u_{tt} -u_{xx}+q(x)u = \pm f(x,u)$ for three different classes (P1), (P2), (P3) of periodic potentials $V,q$. (P1) consists of periodically extended delta-distributions, (P2) of periodic step potentials and (P3) contains certain periodic potentials $V,q\in H^r_{\per}(\R)$ for $r\in [1,3/2)$. Among other assumptions we suppose that $|f(x,s)|\leq c(1+ |s|^p)$ for some $c>0$ and $p>1$. In each class we can find suitable potentials that give rise to a critical exponent $p^\ast$ such that for $p\in (1,p^\ast)$ both in the "+" and the "-" case we can use variational methods to prove existence of time-periodic real-valued solutions that are localized in the space direction. The potentials are constructed explicitely in class (P1) and (P2) and are found by a recent result from inverse spectral theory in class (P3). The critical exponent $p^\ast$ depends on the regularity of $V, q$. Our result builds upon a Fourier expansion of the solution and a detailed analysis of the spectrum of the wave operator. In fact, it turns out that by a careful choice of the potentials and the spatial and temporal periods, the spectrum of the wave operator $V(x)\partial_t^2-\partial_x^2+q(x)$ (considered on suitable space of time-periodic functions) is bounded away from $0$. This allows to find weak solutions as critical points of a functional on a suitable Hilbert space and to apply tools for strongly indefinite variational problems.

math.AP↗