On the one-dimensional cubic nonlinear Schrodinger equation below L^2 [PDF]
, 2010 In this paper, we review several recent results concerning well-posedness of the one-dimensional, cubic Nonlinear Schrodinger equation (NLS) on the real line R and on the circle T for solutions below the L^2-threshold. We point out common results for NLS
Oh, Tadahiro, Sulem, Catherine
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The Painleve Test and Reducibility to the Canonical Forms for Higher-Dimensional Soliton Equations with Variable-Coefficients [PDF]
, 2006 The general KdV equation (gKdV) derived by T. Chou is one of the famous (1+1) dimensional soliton equations with variable coefficients. It is well-known that the gKdV equation is integrable.
Kobayashi, Tadashi, Toda, Kouichi
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On a Whitham-Type Equation [PDF]
, 2009 The Hunter-Saxton equation and the Gurevich-Zybin system are considered as two mutually non-equivalent representations of one and the same Whitham-type equation, and all their common solutions are obtained ...
Sakovich, Sergei
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Deep quench approximation and optimal control of general Cahn-Hilliard systems with fractional operators and double obstacle potentials [PDF]
, 2018 The paper arXiv:1804.11290 contains well-posedness and regularity results for a system of evolutionary operator equations having the structure of a Cahn-Hilliard system.
Colli, Pierluigi +2 more
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A remark on normal forms and the "upside-down" I-method for periodic NLS: growth of higher Sobolev norms [PDF]
, 2011 We study growth of higher Sobolev norms of solutions to the one-dimensional periodic nonlinear Schrodinger equation (NLS). By a combination of the normal form reduction and the upside-down I-method, we establish \|u(t)\|_{H^s} \lesssim (1+|t|)^{\alpha (s-
Colliander, James +2 more
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Complex plane representations and stationary states in cubic and quintic resonant systems
, 2019 Weakly nonlinear energy transfer between normal modes of strongly resonant PDEs is captured by the corresponding effective resonant systems. In a previous article, we have constructed a large class of such resonant systems (with specific representatives ...
Biasi, Anxo, Bizon, Piotr, Evnin, Oleg
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Solvable cubic resonant systems
, 2019 Weakly nonlinear analysis of resonant PDEs in recent literature has generated a number of resonant systems for slow evolution of the normal mode amplitudes that possess remarkable properties.
Biasi, Anxo, Bizon, Piotr, Evnin, Oleg
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, 2012 Two unusual time-integral conditional regularity results are presented for the three-dimensional Navier-Stokes equations. The ideas are based on $L^{2m}$-norms of the vorticity, denoted by $\Omega_{m}(t)$, and particularly on $D_{m} = \Omega_{m}^{\alpha_{
Constantin P. +9 more
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On the longtime behavior of a viscous Cahn-Hilliard system with convection and dynamic boundary conditions [PDF]
, 2018 In this paper, we study the longtime asymptotic behavior of a phase separation process occurring in a three-dimensional domain containing a fluid flow of given velocity. This process is modeled by a viscous convective Cahn-Hilliard system, which consists
Colli, Pierluigi +2 more
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Conformal flow on $S^3$ and weak field integrability in AdS$_4$
, 2017 We consider the conformally invariant cubic wave equation on the Einstein cylinder $\mathbb{R} \times \mathbb{S}^3$ for small rotationally symmetric initial data.
Bizoń, Piotr +5 more
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