Journal ArticleDOI
Nonelliptic Schrödinger equations
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In this article, a brief survey of the main nonelliptic Schrodinger equations known by the authors, with emphasis on water waves, is made, and a theory is developed for the Cauchy problem for selected examples, based on linear estimates which are strongly related to the dispersion relation of the problem.Abstract:
Nonelliptic Schrodinger equations are defined as multidimensional nonlinear dispersive wave equations whose linear part in the space variables is not an elliptic equation. These equations arise in a natural fashion in several contexts in physics and fluid mechanics. The aim of this paper is twofold. First, a brief survey is made of the main nonelliptic Schrodinger equations known by the authors, with emphasis on water waves. Second, a theory is developed for the Cauchy problem for selected examples. The method is based on linear estimates which are strongly related to the dispersion relation of the problem.read more
Citations
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Journal ArticleDOI
A Relaxation Scheme for the Nonlinear Schrödinger Equation
TL;DR: A relaxation-type scheme that avoids solving for nonlinear systems and preserves density and energy is presented and convergence results for the semidiscretized version of the scheme are given.
Journal ArticleDOI
Long-Time Asymptotics for the Focusing Nonlinear Schrödinger Equation with Nonzero Boundary Conditions at Infinity and Asymptotic Stage of Modulational Instability
TL;DR: In this paper, the authors characterize the long-time asymptotic behavior of the focusing nonlinear Schrodinger (NLS) equation on the line with symmetric, nonzero boundary conditions at infinity.
Journal ArticleDOI
The nonlinear Schrödinger equation on the half-line
TL;DR: In this article, the initial bounding value problem for the cubic nonlinear Schrödinger (NLS) equation on the half-line with data in Sobolev spaces is analyzed via the formula obtained through the unified transform method, and a contraction mapping approach.
Journal ArticleDOI
The Fourier restriction norm method for the Zakharov-Kuznetsov equation
Axel Grünrock,Sebastian Herr +1 more
TL;DR: In this paper, the Cauchy problem for the Zakharov-Kuznetsov equation was shown to be locally well-posed in O(H^s(mathbb{R}^2)$ for all $s>\frac{1}{2}$ by using the Fourier restriction norm method and bilinear refinements of Strichartz type inequalities.
Journal ArticleDOI
The Fourier restriction norm method for the Zakharov-Kuznetsov equation
Axel Grünrock,Sebastian Herr +1 more
TL;DR: In this article, the Cauchy problem for the Zakharov-Kuznetsov equation was shown to be locally well-posed in H^s(R^2) for all s>1/2 by using the Fourier restriction norm method and bilinear refinements of Strichartz type inequalities.
References
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Journal ArticleDOI
H p spaces of several variables
Journal ArticleDOI
Stability of periodic waves of finite amplitude on the surface of a deep fluid
TL;DR: In this article, the stability of steady nonlinear waves on the surface of an infinitely deep fluid with a free surface was studied. And the authors considered the problem of stability of surface waves as part of the more general problem of nonlinear wave in media with dispersion.
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Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations
TL;DR: In this paper, the authors give a complete solution when S is a quadratic surface given by the duality argument for the special case S {(x, y) yZ xz I} and give the interpretation of the answer as a space-time decay for solutions of the Klein-Gordon equation with finite relativistic invariant norm.
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Water waves, nonlinear Schrödinger equations and their solutions
TL;DR: In this article, a number of ases in which these equations reduce to a one dimensional nonlinear Schrodinger (NLS) equation are enumerated, and several analytical solutions of NLS equations are presented, with discussion of their implications for describing the propagation of water waves.