Nonscattering Solutions Research Articles

Global, Non-Scattering Solutions to the Energy Critical Wave Maps Equation

We consider the 1-equivariant energy critical wave maps problem with two-sphere target. Using a method based on matched asymptotic expansions, we construct infinite time relaxation, blow-up, and intermediate types of solutions that have topological degree one. More precisely, for a symbol class of admissible, time-dependent length scales, we construct solutions which can be decomposed as a ground state harmonic map (soliton) re-scaled by an admissible length scale, plus radiation, and small corrections which vanish (in a suitable sense) as time approaches infinity. Our class of admissible length scales includes positive and negative powers of t, with exponents sufficiently small in absolute value. In addition, we obtain solutions with soliton length scale undergoing damped or undamped oscillations in a bounded set, or undergoing unbounded oscillations, for all sufficiently large t.

On threshold solutions of the equivariant Chern–Simons–Schrödinger equation

We consider the self-dual Chern-Simons-Schr\"odinger model in two spatial dimensions. This problem is $L^2$-critical. Under equivariant setting, global wellposedness and scattering were proved in [Liu-Smith, 2016] for solution with initial charge below certain threshold given by the ground state. In this work, we show that the only non-scattering solutions with threshold charge are exactly the ground state up to scaling, phase rotation and the pseudoconformal transformation. We also obtain partial result for non-self-dual system.

Dynamics of Threshold Solutions for Energy Critical NLS with Inverse Square Potential

We consider the focusing energy critical NLS with inverse square potential in dimension $d= 3, 4, 5$ with the details given in $d=3$ and remarks on results in other dimensions. Solutions on the energy surface of the ground state are characterized. We prove that solutions with kinetic energy less than that of the ground state must scatter to zero or belong to the stable/unstable manifolds of the ground state. In the latter case they converge to the ground state exponentially in the energy space as $t\to \infty$ or $t\to -\infty$. (In 3-dim without radial assumption, this holds under the compactness assumption of non-scattering solutions on the energy surface.) When the kinetic energy is greater than that of the ground state, we show that all radial $H^1$ solutions blow up in finite time, with the only two exceptions in the case of 5-dim which belong to the stable/unstable manifold of the ground state. The proof relies on the detailed spectral analysis, local invariant manifold theory, and a global Virial analysis.

The [formula omitted] sequential convergence of a solution to the mass-critical NLS above the ground state

In this paper we generalize a weak sequential result of Fan (2021) to a non-scattering solutions in dimension d≥2. No symmetry assumptions are required for the initial data. We build on a previous result of Dodson (2020) for one dimension.

Global, non-scattering solutions to the energy critical Yang–Mills problem

We consider the Yang–Mills problem on [Formula: see text] with gauge group [Formula: see text]. In an appropriate equivariant reduction, this Yang–Mills problem reduces to a single scalar semilinear wave equation. This semilinear equation admits a one-parameter family of solitons, each of which is a re-scaling of a fixed solution. In this work, we construct a class of solutions, each of which consists of a soliton whose length scale is asymptotically constant, coupled to large radiation, plus corrections which slowly decay to zero in the energy norm. Our class of solutions includes ones for which the radiation component is only “logarithmically” better than energy class. As such, the solutions are not constructed by a priori assuming the length scale to be constant. Instead, we use an approach similar to a previous work of the author regarding wave maps. In the setup of this work, the soliton length scale asymptoting to a constant is a necessary condition for the radiation profile to have finite energy. An interesting point of our construction is that, for each radiation profile, there exist one-parameter families of solutions consisting of the radiation profile coupled to a soliton, which has any asymptotic value of the length scale.

The $L^{2}$ Sequential Convergence of a Solution to the One-Dimensional, Mass-Critical NLS above the Ground State

In this paper we generalize a weak sequential result of [C. Fan, Int. Math. Res. Not. IMRN, 2021 (2021), pp. 4864--4906] to any nonscattering solutions in one dimension. No symmetry assumptions are...

Cooperation of Helix Insertion and Lateral Pressure to Remodel Membranes.

Nature has developed different protein mediated mechanisms to remodel cellular membranes. One of the proteins that is implicated in these processes is α-synuclein (αS). Here we investigate if besides αS’s membrane bound amphipathic helix the disordered, solvent exposed tail of the protein contributes to membrane reshaping. We produced αS variants with elongated or truncated disordered solvent exposed domains. We observe a transformation of opaque multi lamellar vesicle solutions into nonscattering solutions containing smaller structures upon addition of all αS variants. Experimental data combined with model calculations show that the cooperation of helix insertion and lateral pressure exerted by the disordered domain makes the full length protein decidedly more efficient in membrane remodeling than the truncated version. Using disordered domains may not only be cost-efficient, it may also add a new level of control over vesicle fusion/fission by expansion or compaction of the domain.

Blow-up of a critical Sobolev norm for energy-subcritical and energy-supercritical wave equations

This work concerns the semilinear wave equation in three space dimensions with a power-like nonlinearity which is greater than cubic, and not quintic (i.e. not energy-critical). We prove that a scale-invariant Sobolev norm of any non-scattering solution goes to infinity at the maximal time of existence. This gives a refinement on known results on energy-subcritical and energy-supercritical wave equation, with a unified proof. The proof relies on the channel of energy method, as in arXiv:1204.0031, in weighted scale-invariant Sobolev spaces which were introduced in arXiv:1506.00788. These spaces are local, thus adapted to finite speed of propagation, and related to a conservation law of the linear wave equation. We also construct the adapted profile decomposition.

Refinement of Strichartz Estimates for Airy Equation in Nondiagonal Case and its Application

In this paper, we give an improvement of nondiagonal Strichartz estimates for the Airy equation by using a Morrey-type space. As its applications, we prove the small data scattering and the existence of special nonscattering solutions, which are minimal in a suitable sense, to the mass-subcritical generalized Korteweg--de Vries equation. Especially, the use of a refined nondiagonal estimate removes several technical restrictions on the previous work [S. Masaki and J. Segata, Existence of a Minimal Non-Scattering Solution to the Mass-Subcritical Generalized Korteweg-de Vries Equation, preprint, https://arxiv.org/abs/1602.05331] about the existence of the special non-scattering solution.

Existence of a minimal non-scattering solution to the mass-subcritical generalized Korteweg–de Vries equation

In this article, we prove the existence of a non-scattering solution, which is minimal in some sense, to the mass-subcritical generalized Korteweg–de Vries (gKdV) equation in the scale critical Lˆr space where Lˆr={f∈S′(R)|‖f‖Lˆr=‖fˆ‖Lr′<∞}. We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted to Lˆr-framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schrödinger equation.

On minimal nonscattering solution for focusing mass-subcritical nonlinear Schrödinger equation

ABSTRACTWe consider time global behavior of solutions to the focusing mass-subcritical NLS equation in a weighted L2 space. We prove that there exists a threshold solution such that (i) it does not scatter; (ii) with respect to a certain scale-invariant quantity, this solution attains minimum value in all nonscattering solutions. In the mass-critical case, it is known that ground states are this kind of threshold solution. However, in our case, it turns out that the above threshold solution is not a standing wave solution.

Minimal mass non-scattering solutions of the focusing &lt;i&gt;L&lt;/i&gt;&lt;sup&gt;2&lt;/sup&gt;-critical Hartree equations with radial data

We prove that for the Cauchy problem of focusing \begin{document}$L^2$\end{document} -critical Hartree equations with spherically symmetric \begin{document}$H^1$\end{document} data in dimensions \begin{document}$3$\end{document} and \begin{document}$4$\end{document} , the global non-scattering solution with ground state mass must be a solitary wave up to symmetries of the equation. The approach is a linearization analysis around the ground state combined with an in-out spherical wave decomposition technique.

A sharp scattering condition for focusing mass-subcritical nonlinear Schrödinger equation

This article is concerned with time global behavior of solutionsto focusing mass-subcritical nonlinear Schrödinger equation of power type withdata in a critical homogeneous weighted $L^2$ space. We givea sharp sufficient condition for scattering by proving existence ofa threshold solution which does not scatter at least forone time direction and of which initial data attains minimumvalue of a norm of the weighted $L^2$ space inall initial value of non-scattering solution. Unlike in the mass-criticalor -supercritical case, ground state is not a threshold. Thisis an extension of previous author's result to the casewhere the exponent of nonlinearity is below so-called Strauss number.A main new ingredient is a stability estimate in aLorenz-modified-Bezov type spacetime norm.

On the rigidity of solitary waves for the focusing mass-critical NLS in dimensions d⩾2

For the focusing mass-critical NLS \(iu_t + \Delta u = - \left| u \right|^{\tfrac{4} {d}} u\), it is conjectured that the only global non-scattering solution with ground state mass must be a solitary wave up to symmetries of the equation. In this paper, we settle the conjecture for Hx1 initial data in dimensions d = 2, 3 with spherical symmetry and d ⩾ 4 with certain splitting-spherically symmetric initial data.

On the focusing mass critical problem in six dimensions with splitting spherically symmetric initial data

In this paper, we consider the six-dimensional focusing mass criti- cal NLS: iut+�u = | u| 2 3 u with splitting-spherical initial data u0(x1,··· x6) = u0( q x 2 + x 2 + x 2, q x 2 + x 2 + x 2). We prove that any finite mass solution which is almost periodic modulo scaling in both time directions must have Sobolev regularity H 1+ . Moreover, the kinetic energy of the solution is local- ized around the spatial origin uniformly in time. As important applications of the results, we prove the scattering conjecture for solutions with mass smaller than that of the ground state. We also prove that any two-way non-scattering solution must be global and coincides with the solitary wave up to symmetries. Here the ground state is the unique positive, radial solution of the nonlinear elliptic equationQ Q + Q 5 3 = 0. To prove the smoothness of the solution, we use a new local iteration scheme which first appears in (19).

Characterization of Minimal-Mass Blowup Solutions to the Focusing Mass-Critical NLS

Let $d\geq4$ and let u be a global solution to the focusing mass-critical nonlinear Schrodinger equation $iu_t+\Delta u=-|u|^{\frac{4}{d}}u$ with spherically symmetric $H_x^1$ initial data and mass equal to that of the ground state Q. We prove that if u does not scatter, then, up to phase rotation and scaling, u is the solitary wave $e^{it}Q$. Combining this result with that of Merle [Duke Math. J., 69 (1993), pp. 427–453], we obtain that in dimensions $d\geq4$, the only spherically symmetric minimal-mass nonscattering solutions are, up to phase rotation and scaling, the pseudoconformal ground state and the ground state solitary wave.

Fluorescence Lifetime Spectroscopy in Multiply Scattering Media with Dyes Exhibiting Multiexponential Decay Kinetics

To investigate fluorescence lifetime spectroscopy in tissue-like scattering, measurements of phase modulation as a function of modulation frequency were made using two fluorescent dyes exhibiting single exponential decay kinetics in a 2% intralipid solution. To experimentally simulate fluorescence multiexponential decay kinetics, we varied the concentration ratios of the two dyes, 3,3-diethylthiatricarbocyanine iodide and indocynanine green (ICG), which exhibit distinctly different lifetimes of 1.33 and 0.57 ns, respectively. The experimental results were then compared with values predicted using the optical diffusion equation incorporating 1) biexponential decay, 2) average of the biexponential decay, as well as 3) stretched exponential decay kinetic models to describe kinetics owing to independent and quenched relaxation of the two dyes. Our results show that while all kinetic models could describe phase-modulation data in nonscattering solution, when incorporated into the diffusion equation, the kinetic parameters failed to likewise predict phase-modulation data in scattering solutions. We attribute the results to the insensitivity of phase-modulation measurements in nonscattering solutions and the inaccuracy of the derived kinetic parameters. Our results suggest the high sensitivity of phase-modulation measurements in scattering solutions may provide greater opportunities for fluorescence lifetime spectroscopy.

Frequency‐domain photon migration measurements for quantitative assessment of powder absorbance: A novel sensor of blend homogeneity

The measurement and analysis of frequency‐domain photon migration (FDPM) measurements of powder absorbance in pharmaceutical powders is described in the context of other optical techniques. FDPM consists of launching intensity‐modulated light into a powder and detecting the phase delay and amplitude modulation of the re‐emitted light as a function of the modulation frequency. From analysis of the data using the diffusion approximation to the radiative transport equation, the absorption coefficient can be obtained. Absorption coefficient measurements of riboflavin in lactose mixtures are presented at concentrations of 0.1 to 1% (w/w) at near‐infrared wavelengths where solution absorption cross sections are difficult to accurately measure using traditional transmission measurements in nonscattering solutions. FDPM measurements in powders enabled determinations of absorption coefficients that increase linearly with concentration (w/w) according to Beer–Lambert relationship. The extension of FDPM for monitoring absorbance of low‐dose and ultralow‐dose powder blending operations is presented.