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In this paper, we study long time behavior of solution to the quantum Zakharov system in two dimensions. We construct a small global solution to the quantum Zakharov system which scatters to a given free solution by using space-time resonance method developed by Gustafson-Nakanishi-Tsai [Commun. Contemp. Math. 11 (2009), pp. 657–707] and Germain-Masmoudi-Shatah [Int. Math. Res. Not. IMRN 3 (2009), 414–432; J. Math. Pures Appl. (9) 97 (2012), pp. 505–543] etc. [ABSTRACT FROM AUTHOR]

We consider the initial value problem of a linearized hyperbolic-parabolic coupled systems of thermoviscoelasticity around the constant state with arbitrary spatial dimensions (n ≥ 1) and study the decay property of the systems. In the first place, we construct the fundamental solutions in Fourier space. Then we prove the decay property for the linearized systems provided that the initial data are in L 2 ∩ L 1 . [ABSTRACT FROM AUTHOR]

We consider the one-dimensional nonlinear Schrödinger equation with an attractive delta potential and mass-supercritical nonlinearity. This equation admits a one-parameter family of solitary wave solutions in both the focusing and defocusing cases. We establish asymptotic stability for all solitary waves satisfying a suitable spectral condition, namely, that the linearized operator around the solitary wave has a two-dimensional generalized kernel and no other eigenvalues or resonances. In particular, we extend our previous result [35] beyond the regime of small solitary waves and extend the results of [19,29] from orbital to asymptotic stability for a suitable family of solitary waves. [ABSTRACT FROM AUTHOR]

In this paper, we consider the large time asymptotic behavior of solutions to systems of two cubic nonlinear Klein-Gordon equations in one space dimension. We classify the systems by studying the quotient set of a suitable subset of systems by the equivalence relation naturally induced by the linear transformation of the unknowns. It is revealed that the equivalence relation is well described by an identification with a matrix. In particular, we characterize some known systems in terms of the matrix and specify all systems equivalent to them. An explicit reduction procedure from a given system in the suitable subset to a model system, i.e., to a representative, is also established. The classification also draws our attention to some model systems which admit solutions with a new kind of asymptotic behavior. Especially, we find new systems which admit a solution of which decay rate is worse than that of a solution to the linear Klein-Gordon equation by logarithmic order. [ABSTRACT FROM AUTHOR]

In this paper, we study large time behavior of complex-valued solutions to nonlinear Klein-Gordon equation with a gauge invariant quadratic nonlinearity in two spatial dimensions. To find a possible asymptotic behavior, we consider the final value problem. It turns out that one possible behavior is a linear solution with a logarithmic phase correction as in the real-valued case. However, the shape of the logarithmic correction term has one more parameter which is also given by the final data. In the real case the parameter is constant so one cannot see its effect. However, in the complex case it varies in general. The one dimensional case is also discussed. [ABSTRACT FROM AUTHOR]

We consider the initial-value problem for the one-dimensional cubic nonlinear Schrödinger equation with a repulsive delta potential. We prove that small initial data in a weighted Sobolev space lead to global solutions that decay in |$L^{\infty }$| and exhibit modified scattering. [ABSTRACT FROM AUTHOR]

We consider the asymptotic behavior in time of solutions to the nonlinear Schrödinger equation with fourth order anisotropic dispersion (4NLS) which describes the propagation of ultrashort laser pulses in a medium with anomalous time-dispersion in the presence of fourth-order time-dispersion. We prove existence of a solution to (4NLS) which scatters to a solution of the linearized equation of (4NLS) as t → ∞. [ABSTRACT FROM AUTHOR]

In this paper, we consider the final state problem for the nonlinear Klein-Gordon equation (NLKG) with a critical nonlinearity in three space dimensions: (⇔ + 1)u = &#955|u|2/3u, t ∈ R, x ∈ R³, where ⇔ = ∂² t - Δ is d'Alembertian. We prove that for a given asymptotic profile uap, there exists a solution u to (NLKG) which converges to uap as t → 1. Here the asymptotic profile uap is given by the leading term of the solution to the linear Klein-Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on the combination of Fourier series expansion for the nonlinearity used in our previous paper [23] and smooth modification of phase correction by Ginibre and Ozawa [6]. [ABSTRACT FROM AUTHOR]

We consider the asymptotic behavior in time of solutions to the cubic nonlinear Schrödinger equation with repulsive delta potential (δ-NLS). We shall prove that for a given small asymptotic profileuap, there exists a solutionuto (δ-NLS) which converges touapinL2(ℝ) ast → ∞. To show this result we exploit the distorted Fourier transform associated to the Schrödinger equation with delta potential. [ABSTRACT FROM PUBLISHER]

This paper is concerned with the orbital stability of a two parameter family of solitary waves for the fourth order nonlinear Schrödinger type equation (4NLS). By using variational approach, we prove that those solitary waves are orbitally stable under the time evolution by 4NLS. [ABSTRACT FROM AUTHOR]

We give an asymptotic formula in time of solutions to the linear fourth-order Schrödinger type equation in one space dimension without the mean zero condition on the initial data. Then we apply this formula to study the long time behavior of solution to the fourth-order Schrödinger type equation with the cubic non-linearity. [ABSTRACT FROM AUTHOR]

This paper is a continuation of our previous study [13] on the long time behavior of solution to the nonlinear Schrödinger equation with higher order anisotropic dispersion (4NLS). We prove the long range scattering for (4NLS) with the quadratic nonlinearity in two dimensions. More precisely, for a given asymptotic profile u + , we construct a solution to (4NLS) which converges to u + as t → ∞ , where u + is given by the leading term of the solution to the linearized equation of (4NLS) with a logarithmic phase correction. [ABSTRACT FROM AUTHOR]