Search

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]

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]

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]

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]