Liouville Integrability of Conservative Peakons for a Modified CH equation

The modified Camassa-Holm (also called FORQ) equation is one of numerous cousins of the Camassa-Holm equation possessing non-smoth solitons (peakons) as special solutions. The peakon sector of solutions is not uniquely defined: in one peakon sector (dissipative^a) the Sobolev H¹ norm is not preserved, in the other sector (conservative), introduced in [2], the time evolution of peakons leaves the H¹ norm invariant. In this Letter, it is shown that the conservative peakon equations of the modified Camassa-Holm can be given an appropriate Poisson structure relative to which the equations are Hamiltonian and, in fact, Liouville integrable. The latter is proved directly by exploiting the inverse spectral techniques, especially asymptotic analysis of solutions, developed elsewhere [3].     

PT Symmetry in Quantum Field Theory

The Hamiltonian H specifies the energy levels and the time evolution of a quantum theory. It is an axiom of quantum mechanics that H be Hermitian. The Hermiticity of H guarantees that the energy spectrum is real and that the time evolution is unitary (probability preserving). In this talk we investigate an alternative formulation of quantum mechanics in which the mathematical requirement of Hermiticity is replaced by the more physically transparent condition of space-time reflection (PT) symmetry. We show that if the PT symmetry of a Hamiltonian H is not broken, then the spectrum of H is real. Examples of PT-symmetric non-Hermitian Hamiltonians are H=p ²+ix ³ and H=p ²-x ⁴. The crucial question is whether PT-symmetric Hamiltonians specify physically acceptable quantum theories in which the norms of states are positive and the time evolution is unitary. The answer is that a Hamiltonian that has an unbroken PT symmetry also possesses a physical symmetry that we call C. Using C, we show how to construct an inner product whose associated norm is positive definite. The result is a new class of fully consistent complex quantum theories. Observables exhibit CPT symmetry, probabilities are positive, and the dynamics is governed by unitary time evolution.     

On the Spectral Dynamics of the Deformation Tensor and New A Priori Estimates for the 3D Euler Equations

In this paper we study the dynamics of eigenvalues of the deformation tensor for solutions of the 3D incompressible Euler equations. Using the evolution equation of the L² norm of spectra, we deduce new a priori estimates of the L² norm of vorticity. As an immediate corollary of the estimate we obtain a new sufficient condition of L² norm control of vorticity. We also obtain decay in time estimates of the ratios of the eigenvalues. In the remarks we discuss what these estimates suggest in the study of searching initial data leading to possible finite time singularities. We find that the dynamical behaviors of L² norm of vorticity are controlled completely by the second largest eigenvalue of the deformation tensor. Part of this work was done while the author was visiting CSCAMM, University of Maryland, USA. The author would like to thank to Professor E. Tadmor for his hospitality     

Algebraic Rate of Decay for the Excess Free Energy and Stability of Fronts for a Nonlocal Phase Kinetics Equation with a Conservation Law. I

This is the first of two papers devoted to the study of a nonlocal evolution equation that describes the evolution of the local magnetization in a continuum limit of an Ising spin system with Kawasaki dynamics and Kac potentials. We consider subcritical temperatures, for which there are two local equilibria, and begin the proof of a local nonlinear stability result for the minimum free energy profiles for the magnetization at the interface between regions of these two different local equilibria; i.e., the fronts. We shall show in the second paper that an initial perturbation v ₀ of a front that is sufficiently small in L ² norm, and sufficiently localized that x ² v ₀(x)² dx<, yields a solution that relaxes to another front, selected by a conservation law, in the L ¹ norm at an algebraic rate that we explicitly estimate. There we also obtain rates for the relaxation in the L ² norm and the rate of decrease of the excess free energy. Here we prove a number of estimates essential for this result. Moreover, the estimates proved here suffice to establish the main result in an important special case.on leave from     

Some Stability and Instability Criteria for Ideal Plane Flows

We investigate stability and instability of steady ideal plane flows for an arbitrary bounded domain. First, we obtain some general criteria for linear and nonlinear stability. Second, we find a sufficient condition for the existence of a growing mode to the linearized equation. Third, we construct a steady flow which is nonlinearly and linearly stable in the L² norm of vorticity but linearly unstable in the L² norm of velocity.     

<Superscript>6</Superscript>He + <Superscript>6</Superscript>He Clustering of <Superscript>12</Superscript>Be in a Microscopic Algebraic Approach

The norm kernel of the A=12 system composed of two ⁶He clusters, and the L=0 basis functions (in the SU(3) and angular momentum-coupled schemes) are analytically obtained in the Fock-Bargmann space. The norm kernel has a diagonal form in the former basis, but the asymptotic conditions are naturally defined in the latter one. The system is a good illustration for the method of projection of the norm kernel to the basis functions in the presence of SU(3) degeneracy that was proposed by the authors. The coupled-channel problem is considered in the algebraic version of the resonating-group method, with the multiple decay thresholds being properly accounted for. The structure of the ground state of ¹²Be obtained in the approximation of zero-range nuclear force is compared with the shell-model predictions. In the continuum part of the spectrum, the S-matrix is constructed, the asymptotic normalization coefficients are deduced and their energy dependence is analyzed.     

Remarks on Singularities, Dimension and Energy Dissipation for Ideal Hydrodynamics and MHD

For weak solutions of the incompressible Euler equations, there is energy conservation if the velocity is in the Besov space B ³ _s with s greater than 1/3. B ³ _s consists of functions that are Lip(s) (i.e., H?lder continuous with exponent s) measured in the L ^p norm. Here this result is applied to a velocity field that is Lip(α₀) except on a set of co-dimension on which it is Lip($agr;₁), with uniformity that will be made precise below. We show that the Frisch-Parisi multifractal formalism is valid (at least in one direction) for such a function, and that there is energy conservation if . Analogous conservation results are derived for the equations of incompressible ideal MHD (i.e., zero viscosity and resistivity) for both energy and helicity . In addition, a necessary condition is derived for singularity development in ideal MHD generalizing the Beale-Kato-Majda condition for ideal hydrodynamics. Received: 21 March 1995 / Accepted: 6 August 1996     

Global Regularity of Wave Maps¶II. Small Energy in Two Dimensions

We show that wave maps from Minkowski space ℝ^{1+ n} to a sphere S ^{m −1} are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space , in all dimensions n≥ 5. This generalizes the results in the prequel [40] of this paper, which addressed the high-dimensional case n≥ 5. In particular, in two dimensions we have global regularity whenever the energy is small, and global regularity for large data is thus reduced to demonstrating non-concentration of energy. Received: 14 December 2000 / Accepted: 18 June 2001     

Banach spaces of weights on quasimanuals

Section 1 is a brief introduction. Section 2 contains the basic definitions of quasimanuals, weights, and operational logics. The linear spaceW of all weights on a quasimanualA is introduced and given a norm.W with this norm is seen to be a Banach space. The subspaceV ofW generated by the positive cone ofW is given the base norm and is also shown to be an Archimedian ordered Banach space with an additive norm. In Section 3 normal linear functionals onV ^* are defined in analogy with normal linear functionals onw ^* algebras. The spaceV is shown to be the set of normal functionals onV ^* and we showV to be the unique partially ordered Banach space with a closed generating cone which is predual toV ^*. Next, weakly compact subsets ofW are characterized in terms of eventwise convergence. This is the Hahn-Vitali-Saks theorem of classical measure theory in this noncommutative setting; several weak compactness results are drawn from this and compared with their classical counterparts. Section 4 introduces the ultraweak topology forV ^* in analogy with the same for the trace class operators on Hubert space. Here the condition for a compact base for the cone ofV is examined and shown to be a poor and unnecessary hypothesis in many circumstances. Many connections with the existent literature are made and throughout the paper there are many examples and open questions.     

Banach Synaptic Algebras

Using a representation theorem of Erik Alfsen, Frederic Schultz, and Erling Størmer for special JB-algebras, we prove that a synaptic algebra is norm complete (i.e., Banach) if and only if it is isomorphic to the self-adjoint part of a Rickart C^?-algebra. Also, we give conditions on a Banach synaptic algebra that are equivalent to the condition that it is isomorphic to the self-adjoint part of an AW^?-algebra. Moreover, we study some relationships between synaptic algebras and so-called generalized Hermitian algebras.     

Removable singularities in Yang-Mills fields

We show that a field satisfying the Yang-Mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite. We obtain the result that every Yang-Mills field overR ⁴ with bounded functional (L ² norm) may be obtained from a field onS ⁴=R ⁴{}. Hodge (or Coulomb) gauges are constructed for general small fields in arbitrary dimensions including 4.     

A confinement model calculation of h1(x)

The transverse polarization distribution of quarks h₁(x) is computed in a confinement model, the chiral chromodielectric model. The flavor structure of h₁, its Q² evolution and Soffer's inequality are studied. The Drell-Yan double transverse asymmetry A_TT is evaluated and found to be one order of magnitude smaller than some previous estimates which equated it to the double longitudinal asymmetry.     

Lax Integrability of the Modified Camassa-Holm Equation and the Concept of Peakons

In this Letter we propose that for Lax integrable nonlinear partial differential equations the natural concept of weak solutions is implied by the compatibility condition for the respective distributional Lax pairs. We illustrate our proposal by comparing two concepts of weak solutions of the modified Camassa-Holm equation pointing out that in the peakon sector (a family of non-smooth solitons) only one of them, namely the one obtained from the distributional compatibility condition, supports the time invariance of the Sobolev H¹ norm.     

A Relation Between Completely Bounded Norms and Conjugate Channels

We show a relation between a quantum channel Φ and its conjugate Φ ^C , which implies that the p → p Schatten norm of the channel is the same as the 1 → p completely bounded norm of the conjugate. This relation is used to give an alternative proof of the multiplicativity of both norms.     

Trial Equation Method Based on Symmetry and Applications to Nonlinear Equations Arising in Mathematical Physics

To find exact traveling wave solutions to nonlinear evolution equations, we propose a method combining symmetry properties with trial polynomial solution to nonlinear ordinary differential equations. By the method, we obtain some exact traveling wave solutions to the Burgers-KdV equations and a kind of reaction-diffusion equations with high order nonlinear terms. As a result, we prove that the Burgers-KdV equation does not have the real solution in the form a ₀+a ₁tan ξ+a ₂tan ² ξ, which indicates that some types of the solutions to the Burgers-KdV equation are very limited, that is, there exists no new solution to the Burgers-KdV equation if the degree of the corresponding polynomial increases. For the second equation, we obtain some new solutions. In particular, some interesting structures in those solutions maybe imply some physical meanings. Finally, we discuss some classifications of the reaction-diffusion equations which can be solved by trial equation method.     

Global Existence and Full Regularity of the Boltzmann Equation Without Angular Cutoff

We prove the global existence and uniqueness of classical solutions around an equilibrium to the Boltzmann equation without angular cutoff in some Sobolev spaces. In addition, the solutions thus obtained are shown to be non-negative and C ^∞ in all variables for any positive time. In this paper, we study the Maxwellian molecule type collision operator with mild singularity. One of the key observations is the introduction of a new important norm related to the singular behavior of the cross section in the collision operator. This norm captures the essential properties of the singularity and yields precisely the dissipation of the linearized collision operator through the celebrated H-theorem.     

Ghost busting: Making sense of non-Hermitian Hamiltonians

The Lee model is an elementary quantum field theory in which mass, wave-function, and charge renormalization can be performed exactly. In early studies of this model in the 1950's it was found that there is a critical value of g ², the square of the renormalized coupling constant, above which g ₀ ² , the square of the unrenormalized coupling constant, is negative. For g ² larger than this critical value, the Hamiltonian of the Lee model becomes non-Hermitian. In this non-Hermitian regime a new state appears whose norm is negative. This state is called a ghost. It has always been thought that in this ghost regime the Lee model is an unacceptable quantum theory because unitarity appears to be violated. However, in this regime while the Hamiltonian is not Hermitian, it does possess symmetry. It has recently been discovered that a non-Hermitian Hamiltonian having symmetry may define a quantum theory that is unitary. The proof of unitarity requires the construction of a time-independent operator called C. In terms of C one can define a new inner product with respect to which the norms of the states in the Hilbert space are positive. Furthermore, it has been shown that time evolution in such a theory is unitary. In this talk the C operator for the Lee model in the ghost regime is constructed in the V/Nθ sector. It is then shown that the ghost state has a positive norm and that the Lee model is an acceptable unitary quantum field theory for all values of g ². Presented at the 3rd International Workshop “Pseudo-Hermitian Hamiltonians in Quantum Physics”, Istanbul, Turkey, June 20–22, 2005.     

The reconstruction of supersymmetric theories at high energy scales

The reconstruction of fundamental parameters in supersymmetric theories requires the evolution to high scales, where the characteristic regularities in mechanisms of supersymmetry breaking become manifest. We have studied a set of representative examples in this context: minimal supergravity and a left-right symmetric extension; gauge mediated supersymmetry breaking; and superstring effective field theories. Through the evolution of the parameters from the electroweak scale the regularities in different scenarios at the high scales can be unravelled if precision analyses of the supersymmetric particle sector at e ⁺ e ^- linear colliders are combined with analyses at the LHC. Received: 25 November 2002 / Published online: 31 January 2003