[Paper Review] Remarks on solitary waves and Cauchy problem for a Half-wave-Schrödinger equations

Yakine Bahri, Slim Ibrahim+1 more|arXiv (Cornell University)|Oct 2, 2018

Advanced Mathematical Physics Problems22 references4 citations

TL;DR

This paper establishes the existence and orbital stability of ground state solitary waves for a half-wave-Schrödinger equation in two spatial dimensions, proves the existence of traveling waves that vanish as velocity approaches 1, and solves the Cauchy problem for initial data in $L^2_x H^s_y(bR^2)$ with $s > 1/2$, using variational methods and contraction mapping arguments in a critical regularity regime.

ABSTRACT

In this paper, we study the solitary wave and the Cauchy problem for Half-wave-Schrödinger equations in the plane. First, we show the existence and orbital stability of the ground states. Secondly, we prove that traveling waves exist and converge to zero as the velocity tends to $1$. Finally, we solve the Cauchy problem for initial data in $L^{2}_{x}H^{s}_{y}(\mathbb{R}^{2})$, with $s>\frac{1}{2}$.

Motivation & Objective

• To establish the existence of ground state solutions to the half-wave-Schrödinger equation via constrained minimization in a mixed Sobolev space.
• To prove the orbital stability of standing wave solutions using the concentration-compactness method and the method of Cazenave and Lions.
• To investigate the existence and asymptotic behavior of traveling wave solutions as the velocity approaches the relativistic limit of 1.
• To resolve the local well-posedness of the Cauchy problem in the critical regularity space $L^2_x H^s_y(bR^2)$ for $s > 1/2$, despite the equation's non-standard scaling and energy space.

Proposed method

• Employed a constrained minimization problem to construct ground states $Q_ ho$ as minimizers of the functional $\mathcal{S}_\omega(u)$ under the constraint $\mathcal{N}_\omega(u) = 0$, leveraging the variational structure of the elliptic equation (1.3).
• Applied the concentration-compactness principle and the method of Cazenave and P.-L. Lions to prove orbital stability of the ground state orbit in the $L^2$-subcritical regime ($1 < p < 7/3$).
• Used a scaling argument and the implicit function theorem to show the existence of traveling wave solutions $\psi(t,x,y) = e^{i\omega t}Q_\omega(x - vt, y)$ that decay to zero as $v \to 1^-$.
• Established local well-posedness in $L^2_x H^s_y(\bbR^2)$ for $s > 1/2$ via a contraction mapping argument in a suitable function space, relying on Strichartz estimates and small-time dependence on initial data norm.
• Proved that the ground state $Q_\omega$ belongs to the energy space $X = H^1_x L^2_y \cap L^2_x H^{1/2}_y(\bbR^2)$ using Fourier multiplier theory and regularity results from Lizorkin and Esfahani et al.
• Utilized Brezis-Lieb lemma and compactness tools to handle convergence of minimizing sequences and to verify the strong convergence of $|u_n|^{p+1}$ in $L^1$ under pointwise a.e. convergence.

Experimental results

Research questions

• RQ1 Do ground state solitary wave solutions exist for the half-wave-Schrödinger equation in $\bbR^2$ with $1 < p < 5$?
• RQ2 Is the orbit of the ground state solution orbitally stable under the flow of the equation in the $L^2$-subcritical regime?
• RQ3 Do traveling wave solutions exist, and what is their behavior as the velocity approaches 1?
• RQ4 Is the Cauchy problem locally well-posed for initial data in $L^2_x H^s_y(\bbR^2)$ with $s > 1/2$?
• RQ5 What is the regularity and decay behavior of the ground state solutions in the anisotropic energy space $X$?

Key findings

• For $1 < p < 5$ and $\omega > 0$, there exist non-trivial ground states $Q_\omega \in X$ that minimize the functional $\mathcal{S}_\omega(u)$ under the constraint $\mathcal{N}_\omega(u) = 0$, proving existence via variational methods.
• In the $L^2$-subcritical case ($1 < p < 7/3$), the orbit $\mathcal{O}_\omega = \{ e^{i\theta} Q_\omega(\cdot + \tau_1, \cdot + \tau_2) \}$ is orbitally stable in the space $X$.
• Traveling wave solutions exist for all velocities $v \in (0,1)$ and satisfy $\|Q_v\|_{X} \to 0$ as $v \to 1^-$, indicating a vanishing amplitude in the relativistic limit.
• The Cauchy problem for (1.1) is locally well-posed in $L^2_x H^s_y(\bbR^2)$ for $s > 1/2$, with the maximal existence time $T$ depending continuously on the initial data norm.
• The ground state $Q_\omega$ belongs to the energy space $X$, and its regularity is established via Fourier multiplier estimates and the use of Lizorkin’s multiplier theorem for $L^2$-boundedness.
• For $p > 7/3$, the function $R_1 = \partial_\omega Q_\omega|_{\omega=1}$ lies in $L^2(\bbR^2)$, which is essential for proving the regularity of the ground state under scaling.