Standing Wave Solutions Research Articles

A Harnack inequality for the parabolic Allen–Cahn equation

We prove a differential Harnack inequality for the solution of the parabolic Allen–Cahn equation $$ \frac{\partial f}{\partial t}=\triangle f-(f^3-f)$$ on a closed n-dimensional manifold. As a corollary, we find a classical Harnack inequality. We also formally compare the standing wave solution to a gradient estimate of Modica from the 1980s for the elliptic equation.

A new rearrangement inequality and its application for $$L^2$$ L 2 -constraint minimizing problems

In this paper, we introduce a new type rearrangement inequality based on the Steiner rearrangement. To show orbital stability of standing wave solutions with respect to a nonlinear Schrodinger equation, \(H^1\)-precompactness of minimizing sequence of \(L^2\)-constraint minimizing problem is very important. Usually, by using the concentration compactness principle and some scaling argument, \(H^1\)-precompactness is obtained. To prove \(H^1\)-precompactness without scaling arguments, we introduce the rearrangement. The Steiner rearrangement is defined as a map from \(H^1\) to \(H^1\), whereas our rearrangement \((\cdot \star \cdot )\) is defined as a map from \(H^1 \times H^1\) to \(H^1\). By using the rearrangement, we show a strict inequality \(\Vert \nabla (u \star v) \Vert _{L^2}^2 < \Vert \nabla u\Vert _{L^2}^2 + \Vert \nabla v\Vert _{L^2}^2\) under simple assumptions.

Quasi-periodic water waves

We present the result and the ideas of the recent paper (Berti and Montalto, Quasi-periodic standing wave solutions of gravity-capillary water waves, http://arxiv.org/abs/1602.02411 , 2016) concerning the existence of Cantor families of small-amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean, with infinite depth, in irrotational regime, under the action of gravity and surface tension at the free boundary. These quasi-periodic solutions are linearly stable.

Global wellposedness of the equivariant Chern–Simons–Schrödinger equation

In this article we consider the initial value problem for the m -equivariant Chern–Simons–Schrödinger model in two spatial dimensions with coupling parameter g \in \mathbb R . This is a covariant NLS type problem that is L^2 -critical. We prove that at the critical regularity, for any equivariance index m \in \mathbb Z , the initial value problem in the defocusing case ( g &lt; 1 ) is globally wellposed and the solution scatters. The problem is focusing when g \geq 1 , and in this case we prove that for equivariance indices m \in \mathbb Z , m \geq 0 , there exist constants c = c_{m, g} such that, at the critical regularity, the initial value problem is globally wellposed and the solution scatters when the initial data \phi_0 \in L^2 is m -equivariant and satisfies \| \phi_0 \|_{L^2}^2 &lt; c_{m, g} . We also show that \sqrt{c_{m, g}} is equal to the minimum L^2 norm of a nontrivial m -equivariant standing wave solution. In the self-dual g = 1 case, we have the exact numerical values c_{m, 1} = 8\pi(m + 1) .

On a Quasilinear Schrödinger Problem at Resonance

Abstract In this paper we establish the existence of standing wave solutions for quasilinear Schrödinger equations involving subcritical growth at resonance. By using a change of variables, the quasilinear equation is reduced to a semilinear one, whose associated functional is well defined in the usual Sobolev space. The “first” eigenvalue type of a nonhomogeneous operator has been studied. Using this fact and a variant of the monotone operator theorem, we show that the problem at resonance has at least one nontrivial solution.

Standing wave solutions for a class of nonhomogeneous systems in dimension two

In this article, minimax procedures and a Trudinger–Moser type inequality in weighted Sobolev spaces are employed to establish sufficient conditions for the existence of solutions for a class of nonhomogeneous elliptic systems involving nonlinear Schrödinger equations with subcritical or critical exponential growth and with potentials which are singular and/or vanishing. The solutions are obtained by suitable control of the size of the perturbations.

Standing wave solutions for the discrete nonlinear Schrödinger equations with indefinite sign subquadratic potentials

We investigate the discrete nonlinear Schrödinger equations with indefinite sign subquadratic potentials. Making use of the critical point theory, we obtain a new result concerning the existence of a standing wave solution.

Higher dimensional vortex standing waves for nonlinear Schrödinger equations

ABSTRACTWe study standing wave solutions to nonlinear Schrödinger equations, on a manifold with a rotational symmetry, which transform in a natural fashion under the group of rotations. We call these vortex solutions. They are higher dimensional versions of vortex standing waves that have been studied on the Euclidean plane. We focus on two types of vortex solutions, which we call spherical vortices and axial vortices.

On a deformation of the nonlinear Schrödinger equation

We study a deformation of the nonlinear Schrödinger (NLS) equation recently derived in the context of deformation of hierarchies of integrable systems. Although this new equation has not been shown to be completely integrable, its solitary wave solutions exhibit typical soliton behaviour, including near elastic collisions. We will first focus on standing wave solutions which can be smooth or peaked, then with the help of numerical simulations we will study solitary waves, their interactions and finally rogue waves in the modulational instability regime. Interestingly, the structure of the solution during the collision of solitary waves or during the rogue wave events is sharper and has larger amplitudes than in the classical NLS equation.

Standing waves for a class of quasilinear Schrödinger equations

We study the existence of standing wave solutions for a class of quasilinear Schrödinger equations with subcritical or critical growthwhere is a given potential, is a parameter. By variational methods, we investigate the range of parameters where the existence of non-trivial solutions can be guaranteed. Especially, by combing Resonance and Hahn–Banach theorems, we improve some previous results on .

Instability of standing waves for inhomogeneous Hartree equations

We are concerned with the instability of standing-wave solutions eiωtφω(x) to the Hartree equation iut+Δu+gu(|⋅|−γ⁎(g|u|2))=0 for some suitable g(x), where φω is a ground state solution. The instability of standing-wave solutions reveals a balance between ω and γ.

Standing wave solutions of a quasilinear degenerate Schrödinger equation with unbounded potential

Standing wave solutions of a quasilinear degenerate Schrödinger equation with unbounded potential

Existence and Stability of Standing Waves for Nonlinear Fractional Schrödinger Equations with Hartree Type and Power Type Nonlinearities

We consider the standing wave solutions for nonlinear fractional Schrödinger equations with focusing Hartree type and power type nonlinearities. We first establish the constrained minimization problem via applying variational method. Under certain conditions, we then show the existence of standing waves. Finally, we prove that the set of minimizers for the initial value problem of this minimization problem is stable.

THE DELTA STANDING WAVE SOLUTION FOR THE LINEAR SCALAR CONSERVATION LAW WITH DISCONTINUOUS COEFFICIENTS USING A SELF-SIMILAR VISCOUS REGULARIZATION

This paper is mainly concerned with the formation of delta standing wave for the scalar conservation law with a linear flux function involving discontinuous coefficients by using the self-similar viscosity vanishing method. More precisely, we use the self-similar viscosity to smooth out the discontinuous coefficient such that the existence of approximate viscous solutions to the delta standing wave for the Riemann problem is established and then the convergence to the delta standing wave solution is also obtained when the viscosity parameter tends to zero. In addition, the Riemann problem is also solved with the standard method and the instability of Riemann solutions with respect to the specific small perturbation of initial data is pointed out in some particular situations.

Standing wave solutions for generalized quasilinear Schrödinger equations with critical growth

We study the existence of standing wave solutions for generalized quasilinear Schrödinger equations involving critical Sobolev exponents. We use the variational methods to obtain a nonexistence theorem and two existence results.

Standing waves of a weakly coupled Schrödinger system with distinct potential functions

The standing wave solutions of a weakly coupled nonlinear Schrödinger system with distinct trapping potential functions in RN (1≤N≤3) are considered. This type of system arises from models in Bose–Einstein condensates theory and nonlinear optics. The existence of a positive ground state solution is shown when the coupling constant is larger than a sharp threshold value, which is explicitly defined in terms of potential functions and system parameters. It is also shown that such solutions concentrate near the minimum points of potential functions, and multiple positive concentration solutions exist when the topological structure of the set of minimum points satisfies certain condition. Variational approach is used for the existence and concentration of positive solutions.

A model for Rayleigh-Bénard magnetoconvection

A model for three-dimensional Rayleigh-B\'{e}nard convection in low-Prandtl-number fluids near onset with rigid horizontal boundaries in the presence of a uniform vertical magnetic field is constructed and analyzed in detail. The kinetic energy $K$, the convective entropy $\Phi$ and the convective heat flux ($Nu-1$) show scaling behaviour with $\epsilon = r-1$ near onset of convection, where $r$ is the reduced Rayleigh number. The model is also used to investigate various magneto-convective structures close to the onset. Straight rolls, which appear at the primary instability, become unstable with increase in $r$ and bifurcate to three-dimensional structures. The straight rolls become periodically varying wavy rolls or quasiperiodically varying structures in time with increase in $r$ depending on the values of Prandtl number $Pr$. They become irregular in time, with increase in $r$. These standing wave solutions bifurcate first to periodic and then quasiperiodic traveling wave solutions, as $r$ is raised further. The variations of the critical Rayleigh number $Ra_{os}$ and the frequency $\omega_{os}$ at the onset of the secondary instability with $Pr$ are also studied for different values of Chandrasekhar's number $Q$.

On the spectral stability of periodic waves of the coupled Schrödinger equations

In this present work we consider the periodic standing wave solutions for the coupled nonlinear Schrödinger equations. We restrict our problem to two cases and construct the solutions explicitly. Then we make the stability analysis for each case. We show that the relevant stationary solutions are spectrally stable for certain parameters. For those parameters, we also present our numerical results in the spectral analysis.

Dual variational methods and nonvanishing for the nonlinear Helmholtz equation

We set up a dual variational framework to detect real standing wave solutions of the nonlinear Helmholtz equation−Δu−k2u=Q(x)|u|p−2u,u∈W2,p(RN) with N≥3, 2(N+1)(N−1)<p<2NN−2 and nonnegative Q∈L∞(RN). We prove the existence of nontrivial solutions for periodic Q as well as in the case where Q(x)→0 as |x|→∞. In the periodic case, a key ingredient of the approach is a new nonvanishing theorem related to an associated integral equation. The solutions we study are superpositions of outgoing and incoming waves and are characterized by a nonlinear far field relation.

Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases

We study the stability of the standing wave solutions of a Gross-Pitaevskii equation describing Bose-Einstein condensation of dipolar quantum gases and characterize their orbit. As an intermediate step, we consider the corresponding constrained minimization problem and establish existence, symmetry and uniqueness of the ground state solutions.