Geometric Property of the Faddeev Model

Two functions u and v are used in expressing the solutions of the Faddeev model. The geometric property of the surface S determined by u and v is discussed and the shape of the surface is demonstrated as an example. The Gaussian curvature of the surface S is negative.     

Dynamically Stable Solitons in Optical Fibers

The existence of a dynamically stable soliton in optical fibers is established by virtue of the multiple-scale perturbation technique applied in a new way to the perturbed nonlinear Schrodinger's equation. We show that, by introducing a proper definition of the phase of the soliton, one can obtain the corrections to the pulse where the standard soliton perturbation approach fails.     

Parameter Region for Existence of Non-classical Solitons

We study a pure nonlinear model of a chain formed by particles that are linked each other by an enharmonic type. This lattice nonlinear Klein–Gordon model is subsequently studied in its continuum version. We use the dynamical systems approach for analyzing the properties of the non-classical structures that support the model. Several non-classical structures like peakons, kink compactons and crodwon or bubble compactons are generated along the chain for the specific region of the parameter space. It is shown that the phase space trajectories are nonclassical curves and show unexpected behaviors. The first type of phase transition in the parametric space occurs when the number of centers and saddles changes while the main phase state parameter becomes critical.     

Pipe Shape Solution of Faddeev Model

Utilizing the results that the Faddeev model is equivalent to the mesonic sector of the SU（2） Skyrme model, where the baryon number current vanishes everywhere, some exact solutions including the vortex solutions of the Faddeev model axe discussed. The solutions are classified by the 2, the new multisoliton solutions are obtained and the pipe shape found. first Chern number. When the Chern number equals distribution of the energy density of the solutions are     

Knotted Solitons in an Interacting Mixture of a Charged and a Neutral Superfluid for Neutron Stars

By making use of the decomposition of U（1） gauge potential theory and the C-mapping method we discuss a mixture of interacting neutral and charged Bose condensates, which is supposed being realized in the interior of neutron stars in the form of a coexistent neutron superfluid and protonic superconductor. We propose that this system possesses vortex lines and two classes of knotted solitons. The topological charge of the vortex lines are characterized by the Hopf indices and the Brower degrees of φ-mapping, and the knotted solitons are described by nontrivial Hopf invariant and the BF action respectively.     

The Existence and Stability of Noncommutative Scalar Solitons

 We establish existence and stability results for solitons in noncommutative scalar field theories in even space dimension 2d. In particular, for any finite rank spectral projection P of the number operator 𝒩 of the d-dimensional harmonic oscillator and sufficiently large noncommutativity parameter θ we prove the existence of a rotationally invariant soliton which depends smoothly on θ and converges to a multiple of P as θ→∞. In the two-dimensional case we prove that these solitons are stable at large θ, if P=P _N , where P _N projects onto the space spanned by the N+1 lowest eigenstates of 𝒩, and otherwise they are unstable. We also discuss the generalisation of the stability results to higher dimensions. In particular, we prove stability of the soliton corresponding to P=P ₀ for all θ in its domain of existence. Finally, for arbitrary d and small values of θ, we prove without assuming rotational invariance that there do not exist any solitons depending smoothly on θ. Received: 13 July 2001 / Accepted: 9 July 2002 Published online: 10 January 2003     

Asymptotic Stability of Lattice Solitons in the Energy Space

Orbital and asymptotic stability for 1-soliton solutions of the Toda lattice equations as well as for small solitary waves of the FPU lattice equations are established in the energy space. Unlike analogous Hamiltonian PDEs, the lattice equations do not conserve the adjoint momentum. In fact, the Toda lattice equation is a bidirectional model that does not fit in with the existing theory for the Hamiltonian systems by Grillakis, Shatah and Strauss. To prove stability of 1-soliton solutions, we split a solution around a 1-soliton into a small solution that moves more slowly than the main solitary wave and an exponentially localized part. We apply a decay estimate for solutions to a linearized Toda equation which has been recently proved by Mizumachi and Pego to estimate the localized part. We improve the asymptotic stability results for FPU lattices in a weighted space obtained by Friesecke and Pego.     

Regularity of Local Minimizers of the Interaction Energy Via Obstacle Problems

The repulsion strength at the origin for repulsive/attractive potentials determines the regularity of local minimizers of the interaction energy. In this paper, we show that if this repulsion is like Newtonian or more singular than Newtonian (but still locally integrable), then the local minimizers must be locally bounded densities (and even continuous for more singular than Newtonian repulsion). We prove this (and some other regularity results) by first showing that the potential function associated to a local minimizer solves an obstacle problem and then by using classical regularity results for such problems.     

Local Minimizers of the Ginzburg-Landau Energy with Magnetic Field in Three Dimensions

We establish the existence of locally minimizing vortex solutions to the full Ginzburg-Landau energy in three dimensional simply-connected domains with or without the presence of an applied magnetic field. The approach is based upon the theory of weak Jacobians and applies to nonconvex sample geometries for which there exists a configuration of locally shortest line segments with endpoints on the boundary.Research partially supported by NSERC grant number 261955     

Solitons in the Two-Dimensional Antiferromagnetic Heisenberg Model

An effective Hamiltonian of the two-dimensional antiferromagnetic Heisenberg model is derived by using the Holstein-Primakoff transformation. Three nonlinear coupled partial equations of motion are obtained. In the long-wavelength approximation, these equations, are reduced to the envelope function equations by the method of multiple scales. The amplitude functions satisfy the nonlinear Schrodinger equation. Introducing an inverse scatteriag transformation, the single-, two- and multi-soliton solutions in the two-dimensional antiferromagnetic Heisenberg model are investigated.     

The Interaction Energy of Well-Separated Skyrme Solitons

We prove that the asymptotic field of a Skyrme soliton of any degree has a non-trivial multipole expansion. It follows that every Skyrme soliton has a well-defined leading multipole moment. We derive an expression for the linear interaction energy of well-separated Skyrme solitons in terms of their leading multipole moments. This expression can always be made negative by suitable rotations of one of the Skyrme solitons in space and iso-space. We show that the linear interaction energy dominates for large separation if the orders of the Skyrme solitons multipole moments differ by at most two. In that case there are therefore always attractive forces between the Skyrme solitons.     

On the Existence of Hypersonic Electrostatic Solitons (Estimation of Limiting Mach Numbers of Ion-Sound Solitons in a Warm Plasma)

Journal of Experimental and Theoretical Physics - We develop a nonlinear theory of ion-sound waves in a collisionless warm electron–ion plasma. The theory is based on analysis of the Sagdeev...     

Structure of Energy Fluxes in Topological Three-Dimensional Dissipative Solitons

The internal structure of dissipative topological solitons has been revealed by example of three-dimensional dissipative optical solitons with one open and one closed dislocation lines of a wavefront. This structure is manifested in the existence of critical points, lines, and surfaces in the field of electromagnetic energy fluxes (Poynting vector). The conservation of the topological characteristics of such solitons, which can be formed in a homogeneous laser medium with saturated amplification and absorption or in lasers with quite large longitudinal and transverse dimensions, provides additional capabilities for information applications.     

Froth-like Minimizers of a Non-Local Free Energy Functional with Competing Interactions

We investigate the ground and low energy states of a one dimensional non-local free energy functional describing at a mean field level a spin system with both ferromagnetic and antiferromagnetic interactions. In particular, the antiferromagnetic interaction is assumed to have a range much larger than the ferromagnetic one. The competition between these two effects is expected to lead to the spontaneous emergence of a regular alternation of long intervals on which the spin profile is magnetized either up or down, with an oscillation scale intermediate between the range of the ferromagnetic and that of the antiferromagnetic interaction. In this sense, the optimal or quasi-optimal profiles are “froth-like”: if seen on the scale of the antiferromagnetic potential they look neutral, but if seen at the microscope they actually consist of big bubbles of two different phases alternating among each other. In this paper we prove the validity of this picture, we compute the oscillation scale of the quasi-optimal profiles and we quantify their distance in norm from a reference periodic profile. The proof consists of two main steps: we first coarse grain the system on a scale intermediate between the range of the ferromagnetic potential and the expected optimal oscillation scale; in this way we reduce the original functional to an effective “sharp interface” one. Next, we study the latter by reflection positivity methods, which require as a key ingredient the exact locality of the short range term. Our proof has the conceptual interest of combining coarse graining with reflection positivity methods, an idea that is presumably useful in much more general contexts than the one studied here.     

Existence of a Stable Polarized Vacuum in the Bogoliubov-Dirac-Fock Approximation

According to Diracs ideas, the vacuum consists of infinitely many virtual electrons which completely fill up the negative part of the spectrum of the free Dirac operator D⁰. In the presence of an external field, these virtual particles react and the vacuum becomes polarized. In this paper, following Chaix and Iracane (J. Phys. B 22, 3791–3814 (1989)), we consider the Bogoliubov-Dirac-Fock model, which is derived from no-photon QED. The corresponding BDF-energy takes the polarization of the vacuum into account and is bounded from below. A BDF-stable vacuum is defined to be a minimizer of this energy. If it exists, such a minimizer is the solution of a self-consistent equation. We show the existence of a unique minimizer of the BDF-energy in the presence of an external electrostatic field, by means of a fixed-point approach. This minimizer is interpreted as the polarized vacuum.     

Energy Splitting,Substantial Inequality,and Minimization for the Faddeev and Skyrme Models

In this paper, we prove that the Faddeev energy E ₁ at the unit Hopf charge is attainable. The proof is based on utilizing an important inequality called the substantial inequality in our previous paper which describes how the Faddeev energy splits into its sublevels in terms of energy and topology when compactness fails. With the help of an optimal Sobolev estimate of the Faddeev energy lower bound and an upper bound of E ₁, we show that E ₁ is attainable. For the two-dimensional Skyrme model, we prove that the substantial inequality is also valid, which allows us to greatly improve the range of the coupling parameters for the existence of unit-charge solitons previously guaranteed in a smaller range of the coupling parameters by the validity of the concentration-compactness method.     

Striped Periodic Minimizers of a Two-Dimensional Model for Martensitic Phase Transitions

In this paper we consider a simplified two-dimensional scalar model for the formation of mesoscopic domain patterns in martensitic shape-memory alloys at the interface between a region occupied by the parent (austenite) phase and a region occupied by the product (martensite) phase, which can occur in two variants (twins). The model, first proposed by Kohn and Müller (Philos Mag A 66(5):697–715, 1992), is defined by the following functional: