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  1. Abstract

    In the seminal work (Weinstein 1999Nonlinearity12673), Weinstein considered the question of the ground states for discrete Schrödinger equations with power law nonlinearities, posed onZd. More specifically, he constructed the so-called normalised waves, by minimising the Hamiltonian functional, for fixed powerP(i.e.l2mass). This type of variational method allows one to claim, in a straightforward manner, set stability for such waves. In this work, we revisit these questions and build upon Weinstein’s work, as well as the innovative variational methods introduced for this problem in (Laedkeet al1994Phys. Rev. Lett.731055 and Laedkeet al1996Phys. Rev.E544299) in several directions. First, for the normalised waves, we show that they are in fact spectrally stable as solutions of the corresponding discrete nonlinear Schroedinger equation (NLS) evolution equation. Next, we construct the so-called homogeneous waves, by using a different constrained optimisation problem. Importantly, this construction works for all values of the parameters, e.g.l2supercritical problems. We establish a rigorous criterion for stability, which decides the stability on the homogeneous waves, based on the classical Grillakis–Shatah–Strauss/Vakhitov–Kolokolov (GSS/VK) quantityωφωl22. In addition, we provide some symmetry results for the solitons. Finally, we complement our results with numerical computations, which showcase the full agreement between the conclusion from the GSS/VK criterion vis-á-vis with the linearised problem. In particular, one observes that it is possible for the stability of the wave to change as the spectral parameterωvaries, in contrast with the corresponding continuous NLS model.

     
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  2. ABSTRACT

    In this work, we search for negative superhumps (nSHs) in poorly studied cataclysmic variables using Transiting Exoplanet Survey Satellite data. We find three eclipsing binaries with nSH signatures: HBHA 4204−09, Gaia DR3 5931071148325476992, and SDSS J090113.51+144704.6. The last one exhibits IW And-like behaviour in archival Zwicky Transient Facility data, and appears to have shallow, grazing eclipses. In addition, we detect nSH signatures in two non-eclipsing systems: KQ Mon and Gaia DR3 4684361817175293440, by identifying the orbital period from the superorbital-dependent irradiation of the secondary. We discover nSH signatures in one more system, [PK2008] HalphaJ103959, by using an orbital period from another work. An improved mass ratio–nSH deficit relation q(ε−) is suggested by us, which agrees with independent measurements on nova-like variables. With this relation, we estimate the mass ratios of all systems in our sample, and determine the orbital inclinations for the three that are eclipsing. All systems with discovered nSHs in this work are excellent targets for follow-up spectroscopic studies.

     
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  3. Abstract In the present work we provide a characterization of the ground states of a higher-dimensional quadratic-quartic model of the nonlinear Schrödinger class with a combination of a focusing biharmonic operator with either an isotropic or an anisotropic defocusing Laplacian operator (at the linear level) and power-law nonlinearity. Examining principally the prototypical example of dimension d = 2, we find that instability arises beyond a certain threshold coefficient of the Laplacian between the cubic and quintic cases, while all solutions are stable for powers below the cubic. Above the quintic, and up to a critical nonlinearity exponent p , there exists a progressively narrowing range of stable frequencies. Finally, above the critical p all solutions are unstable. The picture is rather similar in the anisotropic case, with the difference that even before the cubic case, the numerical computations suggest an interval of unstable frequencies. Our analysis generalizes the relevant observations for arbitrary combinations of Laplacian prefactor b and nonlinearity power p . 
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  4. For generalized Korteweg–De Vries (KdV) models with polynomial nonlinearity, we establish a local smoothing property in [Formula: see text] for [Formula: see text]. Such smoothing effect persists globally, provided that the [Formula: see text] norm does not blow up in finite time. More specifically, we show that a translate of the nonlinear part of the solution gains [Formula: see text] derivatives for [Formula: see text]. Following a new simple method, which is of independent interest, we establish that, for [Formula: see text], [Formula: see text] norm of a solution grows at most by [Formula: see text] if [Formula: see text] norm is a priori controlled. 
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