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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]

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]

In this paper, we consider the long time behavior of the solution to the quadratic nonlinear Klein-Gordon equation (NLKG) in two space dimensions: ( + 1)u = λ|u|u, t ∈ R, x ∈ R2, where = ∂t² - Δ is d'Alembertian. For a given asymptotic profile uap, we construct a solution u to (NLKG) which converges to uap as t → ∞. 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 Fourier series expansion of the nonlinearity. [ABSTRACT FROM AUTHOR]

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 L ˆ r space where L ˆ r = { f ∈ S ′ ( R ) | ‖ f ‖ L ˆ r = ‖ f ˆ ‖ L r ′ < ∞ } . We construct this solution by a concentration compactness argument. Then, key ingredients are a linear profile decomposition result adopted to L ˆ r -framework and approximation of solutions to the gKdV equation which involves rapid linear oscillation by means of solutions to the nonlinear Schrödinger equation. [ABSTRACT FROM AUTHOR]

In this work, we study the quantum Zakharov system which describes the nonlinear interaction between high-frequency quantum Langmuir waves and low-frequency quantum ion-acoustic waves. We show the existence and the stability of the standing waves of quantum Zakharov system for the spacial dimensions d = 1 , 2 , 3 . [ABSTRACT FROM AUTHOR]

This paper considers the initial value problem for a class of fifth order dispersive models containing the fifth order KdV equation The main results show that regularity or polynomial decay of the data on the positive half-line yields regularity in the solution for positive times. [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]

Abstract: We consider the behavior of solutions to the water wave interaction equations in the limit . To justify the semiclassical approximation, we reduce the water wave interaction equation into some hyperbolic-dispersive system by using a modified Madelung transform. The reduced system causes loss of derivatives which prevents us to apply the classical energy method to prove the existence of solution. To overcome this difficulty we introduce a modified energy method and construct the solution to the reduced system. [Copyright &y& Elsevier]

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]

Abstract: We consider the time local and global well-posedness for the fourth order nonlinear Schrödinger type equation (4NLS) on the torus. The nonlinear term of (4NLS) contains the derivatives of unknown function and this prevents us to apply the classical energy method. To overcome this difficulty, we introduce the modified energy and derive an a priori estimate for the solution to (4NLS). [Copyright &y& Elsevier]

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]

Abstract: We study the global existence and asymptotic behavior in time of solutions to the Korteweg–de Vries type equation called as “Hirota” equation. This equation is a mixture of cubic nonlinear Schrödinger equation and modified Korteweg–de Vries equation. We show the unique existence of the solution for this equation which tends to the given “modified” free profile by using the two asymptotic formulae for some oscillatory integrals. [Copyright &y& Elsevier]

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]

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 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]

In this paper, we consider the initial value problem of a specific system of cubic nonlinear Schrödinger equations. Our aim of this research is to specify the asymptotic profile of the solution in L ∞ as t → ∞. It is then revealed that the solution decays slower than a linear solution does. Further, the difference of the decay rate is a polynomial order. This deceleration of the decay is due to an amplification effect by the nonlinearity. This nonlinear amplification phenomena was previously known for several specific systems, however the deceleration of the decay in these results was by a logarithmic order. As far as we know, the system studied in this paper is the first model in that the deceleration in a polynomial order is justified. [ABSTRACT FROM AUTHOR]