This paper is a tutorial that demonstrates various methods from the Colombeau theory of generalized functions in the context of semilinear wave equations. The Colombeau generalized functions constitute differential algebras that contain the space of distributions. We solve the 1D-semilinear wave equation in these algebras, show how delta waves can be computed and then turn to linear and nonlinear regularity theory.
Summarizing basic facts from abstract topological modules over Colombeau generalized complex numbers we discuss duality of Colombeau algebras. In particular, we focus on generalized delta functionals and operator kernels as elements of dual spaces. A large class of examples is provided by pseudodifferential operators acting on Colombeau algebras. By a refinement of symbol calculus we review a new characterization of the wave front set for generalized functions with applications to microlocal analysis.
Generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is based on a theory of generalized oscillatory integrals (OIs) whose phase functions as well as amplitudes may be generalized functions of Colombeau type. The mapping properties of these FIOs are studied as the composition with a generalized pseudodifferential operator. Finally, the microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wave front sets are investigated. This theory of generalized FIOs is motivated by the need of a general framework for partial diffe…
We introduce different notions of wave front set for the functionals in the dual of the Colombeau algebra $\Gc(\Om)$ providing a way to measure the $\G$ and the $\Ginf$- regularity in $\LL(\Gc(\Om),\wt{\C})$. For the smaller family of functionals having a ``basic structure'' we obtain a Fourier transform-characterization for this type of generalized wave front sets and results of noncharacteristic $\G$ and $\Ginf$-regularity.
In \cite{bf} Brézis and Friedman prove that certain nonlinear parabolic equations, with the $δ$-measure as initial data, have no solution. However in \cite{cl} Colombeau and Langlais prove that these equations have a unique solution even if the $δ$-measure is substituted by any Colombeau generalized function of compact support. Here we generalize Colombeau and Langlais their result proving that we may take any generalized function as the initial data. Our approach relies on resent algebraic and topological developments of the theory of Colombeau generalized functions and results from \cite{A}.
The paper is devoted to regularity theory of generalized solutions to semilinear wave equations with a small nonlinearity. The setting is the one of Colombeau algebras of generalized functions. It is shown that in one space dimension, an initial singularity at the origin propagates along the characteristic lines emanating from the origin, as in the linear case. The proof relies on a fixed point theorem in the ultra-metric topology on the algebras involved. The paper takes up the initiating research of the 1970s on anomalous singularities in classical solutions to semilinear hyperbolic equation…
Regularity theory in generalized function algebras of Colombeau type is largely based on the notion of ${\mathcal G}^\infty$-regularity, which reduces to $C^\infty$-regularity when restricted to Schwartz distributions. Surprisingly, in the nonstandard version of the Colombeau algebras, this basic property of ${\mathcal G}^\infty$-regularity does not hold.
In global seismology Earth's properties of fractal nature occur. Zygmund classes appear as the most appropriate and systematic way to measure this local fractality. For the purpose of seismic wave propagation, we model the Earth's properties as Colombeau generalized functions. In one spatial dimension, we have a precise characterization of Zygmund regularity in Colombeau algebras. This is made possible via a relation between mollifiers and wavelets.
This paper is devoted to the investigation of propagation of singularities in hyperbolic equations with non-smooth oefficients, using the Colombeau theory of generalized functions. As a model problem, we study the Cauchy problem for the one-dimensional wave equation with a discontinuous coefficient depending on time. After demonstrating the existence and uniqueness of generalized solutions in the sense of Colombeau to the problem, we investigate the phenomenon of propagation of singularities, arising from delta function initial data, for the case of a piecewise constant coefficient. We also pr…
In distribution theory the pullback of a general distribution by a $C^{\infty}$-function is well-defined whenever the normal bundle of the $C^{\infty}$-function does not intersect the wavefront set of the distribution. However, the Colombeau theory of generalized functions allows for a pullback by an arbitrary c-bounded generalized function. It has been shown in previous work that in the case of multiplication of Colombeau functions (which is a special case of a $C^{\infty}$ pullback), the generalized wave front set of the product satisfies the same inclusion relation as in the distributional …
In this article, a theory of generalized oscillatory integrals (OIs) is developed whose phase functions as well as amplitudes may be generalized functions of Colombeau type. Based on this, generalized Fourier integral operators (FIOs) acting on Colombeau algebras are defined. This is motivated by the need of a general framework for partial differential operators with non-smooth coefficients and distribution data. The mapping properties of these FIOs are studied, as is microlocal Colombeau regularity for OIs and the influence of the FIO action on generalized wave front sets.
The purpose of this paper is to construct and to study algebras of generalized Gevrey ultardistributions. We define the generalized Gevrey wave front and give its main properties. As a fundamental application, the well known Hormander's theorem on the product of two distributions is extended to the case of generalized Gevrey ultradistributions
We introduce an intrinsic notion of Hoelder-Zygmund regularity for Colombeau generalized functions. In case of embedded distributions belonging to some Zygmund-Hoelder space this is shown to be consistent. The definition is motivated by the well-known use of Littlewood-Paley decomposition in characterizing Hoelder-Zygmund regularity for distributions. It is based on a simple interplay of differentiated convolution-mollification with wavelet transforms, which directly translates wavelet estimates into properties of the regularizations. Thus we obtain a scale of new subspaces of the Colombeau al…
The paper is devoted to proving an existence and uniqueness result for generalized solutions to semilinear wave equations with a small nonlinearity in space dimensions 1, 2, 3. The setting is the one of Colombeau algebras of generalized functions. It is shown that for a nonlinearity of arbitrary growth and sign, but multiplied with a small parameter, the initial value problem for the semilinear wave equation has a unique solution in the Colombeau algebra of generalized functions of bounded type. The proof relies on a fixed point theorem in the ultra-metric topology on the algebras involved. In…
This article addresses linear hyperbolic partial differential equations with non-smooth coefficients and distributional data. Solutions are studied in the framework of Colombeau algebras of generalized functions. Its aim is to prove upper and lower bounds for the singular support of generalized solutions for wave equations with discontinuous coefficients. New existence results with weaker assumptions on the representing families are required and proven. The program is carried through for various types of one- and multidimensional wave equations and hyperbolic systems.
We give the regularization for fractional integral by delta sequence and apply it to obtain existence-uniqueness theorems in Colombeau algebras for nonlinear equations with singularities: nonlinear system of integral equations with polar kernel and nonlinear parabolic equations (of ordinary type, with nonlinear conservative term and with Schrödinger kernel) with strongly singular initial data and non-Lipschitz nonlinearities. In a case of nonlinear parabolic equations we do in fact regularization of heat semigroup with delta sequence with respect to the time variable $t.$ We do the same for li…
We consider wave equations on Lorentzian manifolds in case of low regularity. We first extend the classical solution theory to prove global unique solvability of the Cauchy problem for distributional data and right hand side on smooth globally hyperbolic space-times. Then we turn to the case where the metric is non-smooth and present a local as well as a global existence and uniqueness result for a large class of Lorentzian manifolds with a weakly singular, locally bounded metric in Colombeau's algebra of generalized functions.
We analyse an algorithm of transition between Cauchy problems for second-order wave equations and first-order symmetric hyperbolic systems in case the coefficients as well as the data are non-smooth, even allowing for regularity below the standard conditions guaranteeing well-posedness. The typical operations involved in rewriting equations into systems are then neither defined classically nor consistently extendible to the distribution theoretic setting. However, employing the nonlinear theory of generalized functions in the sense of Colombeau we arrive at clear statements about the transfer …
We characterize microlocal regularity of Colombeau generalized functions by an appropriate extension of the classical notion of micro-ellipticity to pseudodifferential operators with slow scale generalized symbols. Thus we obtain an alternative, yet equivalent, way to determine generalized wave front sets, which is analogous to the original definition of the wave front set of distributions via intersections over characteristic sets. The new methods are then applied to regularity theory of generalized solutions of (pseudo-)differential equations, where we extend the general noncharacteristic re…
In this article we investigate the existence of a solution to a semilinear, elliptic, partial differential equation with distributional coefficients and data. The problem we consider is a generalization of the Lichnerowicz equation that one encounters in studying the constraint equations in general relativity. Our method for solving this problem consists of solving a net of regularized, semilinear problems with data obtained by smoothing the original, distributional coefficients. In order to solve these regularized problems, we develop a priori pointwise bounds and sub- and super-solutions and…
We present an extension of the methods of classical Lie group analysis of differential equations to equations involving generalized functions (in particular: distributions). A suitable framework for such a generalization is provided by Colombeau's theory of algebras of generalized functions. We show that under some mild conditions on the differential equations, symmetries of classical solutions remain symmetries for generalized solutions. Moreover, we introduce a generalization of the infinitesimal methods of group analysis that allows to compute symmetries of linear and nonlinear differential…
We derive a degenerate quasilinear Schrödinger equation that describes the resonant reflection of very weak, nonlinear sound waves off a weak sawtooth entropy wave.
We study the instability of standing waves for nonlinear Schrödinger equations. Under a general assumption on nonlinearity, we prove that linear instability implies orbital instability in any dimension. For that purpose, we establish a Strichartz type estimate for the propagator generated by the linearized operator around standing wave.
In this paper, the Lie symmetry method is performed for the nonlinear evolution equation for description nonlinear waves in a viscoelastic tube. we will find one and two-dimensional optimal system of Lie subalgebras. Furthermore, preliminary classification of its group-invariant solutions are investigated.
The well-known Stokes waves refer to periodic traveling waves under the gravity at the free surface of a two dimensional full water wave system. In this paper, we prove that small-amplitude Stokes waves with infinite depth are nonlinearly unstable under long-wave perturbations. Our approach is based on the modulational approximation of the water wave system and the instability mechanism of the focusing cubic nonlinear Schrödinger equation.
In this paper, we study the global existence and regularity of Hölder continuous solutions for a series of nonlinear partial differential equations describing nonlinear waves.
We consider nonlinear elastic wave equations generalizing Gol'dberg's five constants model. We analyze the nonlinear interaction of two distorted plane waves and characterize the possible nonlinear responses. Using the boundary measurements of the nonlinear responses, we solve the inverse problem of determining elastic parameters from the displacement-to-traction map.
We consider nonlinear Schrödinger equations with either power-type or Hartree nonlinearity in the presence of an external potential. We show that for long-range nonlinearities, solutions cannot exhibit scattering to solitary waves or more general localized waves. This extends the well-known results concerning non-existence of non-trivial scattering states for long-range nonlinearities.
The global attraction to stationary states is established for solutions to 3D wave equations with concentrated nonlinearities: each finite energy solution converges as $t\to\pm\infty$ to stationary states. The attraction is caused by nonlinear energy radiation.
Global smooth solutions to the initial value problem for systems of nonlinear wave equations with multiple propagation speeds will be constructed in the case of small initial data and nonlinearities satisfying the null condition.
For scalar semilinear wave equations, we analyze the interaction of two (distorted) plane waves at an interface between media of different nonlinear properties. We show that new waves are generated from the nonlinear interactions, which might be responsible for the observed nonlinear effects in applications. Also, we show that the incident waves and the nonlinear responses determine the location of the interface and some information of the nonlinear properties of the media. In particular, for the case of a jump discontinuity at the interface, we can determine the magnitude of the jump.
In this paper we prove the existence of finite traveling-wave type solutions to the nonlinear double degenerate parabolic equation of turbulent filtration with absorption.
In this paper, a nonlinear Schrödinger equation with an attractive (focusing) delta potential and a repulsive (defocusing) double power nonlinearity in one spatial dimension is considered. It is shown, via explicit construction, that both standing wave and equilibrium solutions do exist for certain parameter regimes. In addition, it is proved that both types of wave solutions are orbitally stable under the flow of the equation by minimizing the charge/energy functional.
This paper is concerned with the final value problem for a system of nonlinear wave equations. The main issue is to solve the problem for the case where the nonlinearity is of a long range type. By assuming that the solution is spherically symmetric, we shall show global solvability of the final value problem around a suitable final state, and hence the generalized wave operator and long range scattering operator can be constructed.
In this paper we consider a semilinear parabolic equation in an infinite cylinder. The spatially varying nonlinearity is such that it connects two (spatially independent) bistable nonlinearities in a compact set in space. We prove that, given such a setting, a traveling wave obeying the equation with the one bistable nonlinearity and starting at the respective side of the cylinder, will converge to a traveling wave solution prescribed by the nonlinearity on the other side.
We extend the notion of orbital stability to systems of nonlinear Schrodinger equations, then we prove this property under suitable assumptions of the local nonlinearity involved.
In this paper we introduce a finite-parameters feedback control algorithm for stabilizing solutions of various classes of damped nonlinear wave equations. Specifically, stabilization the zero steady state solution of initial boundary value problems for nonlinear weakly and strongly damped wave equations, nonlinear wave equation with nonlinear damping term and some related nonlinear wave equations, introducing a feedback control terms that employ parameters, such as, finitely many Fourier modes, finitely many volume elements and finitely many nodal observables and controllers. In addition, we a…
In this paper, we consider the nonlinear fractional Schrödinger equations with Hartree type nonlinearity. We obtain the existence of standing waves by studying the related constrained minimization problems by applying the concentration-compactness principle. Via symmetric decreasing rearrangements, we also show that the standing waves, up to a translations and phases, are positive symmetric nonincreasing functions. Moreover, we prove that the set of minimizers is a stable set for the initial value problem of the equations, that is, a solution whose initial data is near the set will remain near…
Consider a nonlinear wave equation for a massless scalar field with self-interaction in the spatially flat de Sitter spacetime. We show that blow-up in a finite time occurs for the equation with arbitrary power nonlinearity as well as upper bounds of the lifespan of blow-up solutions.The blow-up condition is the same as in the accelerated expanding Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime. We also show the same results for the space derivative nonlinear term.
In this article, we study the local existence of solutions for a wave equation with a nonlocal in time nonlinearity. Moreover, a blow-up results are proved under some conditions on the dimensional space, the initial data and the nonlinear forcing term.
We study traveling wave solutions of the nonlinear variational wave equation. In particular, we show how to obtain global, bounded, weak traveling wave solutions from local, classical ones. The resulting waves consist of monotone and constant segments, glued together at points where at least one one-sided derivative is unbounded. Applying the method of proof to the Camassa--Holm equation, we recover some well-known results on its traveling wave solutions.
Fifth-order 1D nonlinear dispersion equations are shown to admit blow-up formation of shock waves as well as rarefaction waves. The concepts of smooth deformations are applied to distinguish "entropy" shocks from smooth rarefaction waves. Single point gradient catastrophe is shown to lead to nonuniqueness after blow-up.
We use a geometric approach to prove the existence of smooth travelling wave solutions of a nonlinear diffusion-reaction equation with logistic kinetics and a convex nonlinear diffusivity function which changes sign twice in our domain of interest. We determine the minimum wave speed, c*, and investigate its relation to the spectral stability of the travelling wave solutions.
The new global version of the Cauchy-Kovalevskaia theorem presented here is a strengthening and extension of the regularity of similar global solutions obtained earlier by the author. Recently the space-time foam differential algebras of generalized functions with dense singularities were introduced. A main motivation for these algebras comes from the so called space-time foam structures in General Relativity, where the set of singularities can be dense. A variety of applications of these algebras have been presented elsewhere, including in de Rham cohomology, Abstract Differential Geometry, Q…
In this paper we prove the existence of the global attractor for the wave equation with nonlocal weak damping, nonlocal anti-damping and critical nonlinearity.
This paper is concerned with the nonlinear Schrodinger lattice with nonlinear hopping. Via variation approach and the Nehari manifold argument, we obtain two types of solution: periodic ground state and localized ground state. Moreover, we consider the convergence of periodic solutions to the solitary wave.
