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  • 1
    Electronic Resource
    Electronic Resource
    A global attracting set for the Kuramoto-Sivashinsky equation (1993)
    Springer
    Communications in mathematical physics 152 (1993), S. 203-214 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract New bounds are given for the L2-norm of the solution of the Kuramoto-Sivashinsky equation $$\partial _t U(x,t) = - (\partial _x^2 + \partial _x^4 )U(x,t) - U(x,t)\partial _x U(x,t)$$ , for initial data which are periodic with periodL. There is no requirement on the antisymmetry of the initial data. The result is $$\mathop {\lim \sup }\limits_{t \to \infty } \left\| {U( \cdot ,t)} \right\|_2 \leqslant const. L^{8/5} $$ .
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02097064
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  • 2
    Electronic Resource
    Electronic Resource
    The non-linear stability of front solutions for parabolic partial differential equations (1994)
    Springer
    Communications in mathematical physics 161 (1994), S. 323-334 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract For the Ginzburg-Landau equation and similar reaction-diffusion equations on the line, we show convergence ofcomplex perturbations of front solutions towards the front solutions, by exhibiting a coercive functional.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02099781
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  • 3
    Electronic Resource
    Electronic Resource
    Extensive Properties of the Complex Ginzburg–Landau Equation (1999)
    Springer
    Communications in mathematical physics 200 (1999), S. 699-722 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract: We study the set of solutions of the complex Ginzburg–Landau equation in R d , d 〈3. We consider the global attracting set (i.e., the forward map of the set of bounded initial data), and restrict it to a cube Q L of side L. We cover this set by a (minimal) number N Q L (ɛ) of balls of radius ɛ in $L infin(Q L ). We show that the Kolmogorov ɛ-entropy per unit length, exists. In particular, we bound by , which shows that the attracting set is smaller than the set of bounded analytic functions in a strip. We finally give a positive lower bound: .
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002200050546
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  • 4
    Electronic Resource
    Electronic Resource
    Geometric Stability Analysis for Periodic Solutions of the Swift-Hohenberg Equation (1997)
    Springer
    Communications in mathematical physics 190 (1997), S. 173-211 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract: In this paper we describe invariant geometrical structures in the phase space of the Swift-Hohenberg equation in a neighborhood of its periodic stationary states. We show that in spite of the fact that these states are only marginally stable (i.e., the linearized problem about these states has continuous spectrum extending all the way up to zero), there exist finite dimensional invariant manifolds in the phase space of this equation which determine the long-time behavior of solutions near these stationary solutions. In particular, using this point of view, we obtain a new demonstration of Schneider's recent proof that these states are nonlinearly stable.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002200050238
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  • 5
    Electronic Resource
    Electronic Resource
    Spectral duality for planar billiards (1995)
    Springer
    Communications in mathematical physics 170 (1995), S. 283-313 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract For a bounded open domain Ω with connected complement inR 2 and piecewise smooth boundary, we consider the Dirichlet Laplacian-Δ Ω on Ω and the S-matrix on the complementΩ c . We show that the on-shell S-matricesS k have eigenvalues converging to 1 ask↑k 0 exactly when--Δ Ω has an eigenvalue at energyk 0 2 . This includes multiplicities, and proves a weak form of “transparency” atk=k 0. We also show that stronger forms of transparency, such asS k 0 having an eigenvalue 1 are not expected to hold in general.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02108330
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  • 6
    Electronic Resource
    Electronic Resource
    Representations of the CCR in the (φ4)3 model: Independence of space cutoff (1972)
    Springer
    Communications in mathematical physics 25 (1972), S. 1-61 
    Details
    ISSN: 1432-0916
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics , Physics
    Notes: Abstract The algebra of observables for the renormalized φ4 interaction in 3-dimensional space-time is constructed. It is shown that the von Neumann algebras associated with observables in a bounded regionB are independent of the space cutoff which is used in the construction of a Hamiltonian for this interaction. This result is shown to be useful in the construction of a translation invariant φ4 theory in three dimensions. It also gives a physical criterion for the equivalence of non-Fock representations of the canonical commutation relations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01877586
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  • 7
    Electronic Resource
    Electronic Resource
    A complete proof of the Feigenbaum conjectures (1987)
    Springer
    Journal of statistical physics 46 (1987), S. 455-475 
    Details
    ISSN: 1572-9613
    Keywords: Nonlinear functional equation ; renormalization group ; Feigenbaum phenomenon ; computer-assisted proof ; rigorous bounds on critical indices
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract The Feigenbaum phenomenon is studied by analyzing an extended renormalization group map ℳ. This map acts on functionsΦ that are jointly analytic in a “position variable” (t) and in the parameter (μ) that controls the period doubling phenomenon. A fixed pointΦ * for this map is found. The usual renormalization group doubling operatorN acts on this functionΦ * simply by multiplication ofμ with the universal Feigenbaum ratioδ *= 4.669201..., i.e., (N Φ *(μ,t)=Φ *(δ * μ,t). Therefore, the one-parameter family of functions,Ψ μ * ,Ψ μ * (t)=(Φ *(μ,t), is invariant underN. In particular, the functionΨ 0 * is the Feigenbaum fixed point ofN, whileΨ μ * represents the unstable manifold ofN. It is proven that this unstable manifold crosses the manifold of functions with superstable period two transversally.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01013368
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  • 8
    Electronic Resource
    Electronic Resource
    Liapunov spectra for infinite chains of nonlinear oscillators (1988)
    Springer
    Journal of statistical physics 50 (1988), S. 853-878 
    Details
    ISSN: 1572-9613
    Keywords: Liapunov exponents ; random matrices ; coupled oscillators
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We argue that the spectrum of Liapunov exponents for long chains of nonlinear oscillators, at large energy per mode, may be well approximated by the Liapunov exponents of products of independent random matrices. If, in addition, statistical mechanics applies to the system, the elements of these random matrices have a distribution which may be calculated from the potential and the energy alone. Under a certain isotropy hypothesis (which is not always satisfied), we argue that the Liapunov exponents of these random matrix products can be obtained from the density of states of a typical random matrix. This construction uses an integral equation first derived by Newman. We then derive and discuss a method to compute the spectrum of a typical random matrix. Putting the pieces together, we see that the Liapunov spectrum can be computed from the potential between the oscillators.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01019144
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  • 9
    Electronic Resource
    Electronic Resource
    Hydrodynamic Lyapunov Modes in Translation-Invariant Systems (2000)
    Springer
    Journal of statistical physics 98 (2000), S. 775-798 
    Details
    ISSN: 1572-9613
    Keywords: nonlinear dynamics ; Hamiltonian dynamics ; extended systems ; random matrices ; Lyapunov spectrum ; hydrodynamic modes
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We study the implications of translation invariance on the tangent dynamics of extended dynamical systems, within a random matrix approximation. In a model system, we show the existence of hydrodynamic modes in the slowly growing part of the Lyapunov spectrum, which are analogous to the hydrodynamic modes discovered numerically by Dellago, Posch, and Hoover. The hydrodynamic Lyapunov vectors lose the typical random structure and exhibit instead the structure of weakly perturbed coherent long-wavelength waves. We show further that the amplitude of the perturbations vanishes in the thermodynamic limit, and that the associated Lyapunov exponents are universal.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018679609870
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  • 10
    Electronic Resource
    Electronic Resource
    Entropy Production in Nonlinear, Thermally Driven Hamiltonian Systems (1999)
    Springer
    Journal of statistical physics 95 (1999), S. 305-331 
    Details
    ISSN: 1572-9613
    Keywords: open systems ; nonequilibrium steady states ; control theory ; entropy production
    Source: Springer Online Journal Archives 1860-2000
    Topics: Physics
    Notes: Abstract We consider a finite chain of nonlinear oscillators coupled at its ends to two infinite heat baths which are at different temperatures. Using our earlier results about the existence of a stationary state, we show rigorously that for arbitrary temperature differences and arbitrary couplings, such a system has a unique stationary state. (This extends our earlier results for small temperature differences.) In all these cases, any initial state will converge (at an unknown rate) to the stationary state. We show that this stationary state continually produces entropy. The rate of entropy production is strictly negative when the temperatures are unequal and is proportional to the mean energy flux through the system
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1004537730090
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