[Paper Review] On the existence and analycity of solitary waves solutions to a two-dimesional Benjamin-Ono equation

Germán Preciado López, Félix H. Soriano Méndez|arXiv (Cornell University)|Mar 14, 2015

Advanced Mathematical Physics Problems16 references3 citations

TL;DR

This paper establishes the existence, regularity, and real analyticity of solitary wave solutions for a two-dimensional generalization of the Benjamin-Ono equation involving Hilbert transforms in both spatial directions. Using minimax methods for existence and Lizorkin's theorem with interpolation spaces for analyticity, the authors prove that solitary waves are real-analytic functions in the plane under general conditions on the parameters λ and μ.

ABSTRACT

We show the existence, regularity and analyticity of solitary waves associated to the following equation \begin{eqnarray*} (u_t+u^{p}u_x+ \mathcal H\partial_x^2u+ λ\mathcal H\partial_y^2u)_x +μu_{yy}=0, \end{eqnarray*} where $\mathcal H$ is the Hilbert transform with respect to $x$ and $λ$ and $μ$ are nonnegative real numbers, not simultaneously zero.

Motivation & Objective

• To establish the existence of solitary wave solutions for a two-dimensional generalization of the Benjamin-Ono equation with mixed nonlocal dispersion.
• To prove the regularity and real analyticity of these solitary wave solutions in the full two-dimensional space.
• To resolve gaps in prior literature, particularly the incomplete proof of Lemma 3.4 in earlier work on the same equation.
• To extend the analyticity theory of solitary waves to a multidimensional setting with nonlocal operators.
• To provide a rigorous functional analytic framework using interpolation spaces and weighted Sobolev-type norms.

Proposed method

• Employ minimax theory in the Sobolev space framework $H^s \cap X^{1/2}$ to prove existence of solitary wave solutions.
• Use interpolation space techniques to control the regularity of solutions, particularly involving the Hilbert transform and fractional derivatives.
• Apply Lizorkin's theorem on multipliers in $L^p$ spaces to establish analyticity of solutions via $L^p$-boundedness of Fourier multipliers.
• Use induction on multi-indices $\alpha$ to bound the $H^2$-norms of higher-order derivatives $\partial^\alpha \phi$ by $C |\alpha|! (R/2)^{|\alpha|}$, ensuring real analyticity.
• Leverage the Banach algebra property of $H^2$ and Faà di Bruno-type formulas for derivatives of nonlinear terms.
• Construct weighted estimates using the structure of the equation $\mathcal{H}\partial_x^3\phi + \mathcal{H}\partial_x\partial_y^2\phi - \partial_y^2\phi = -\partial_x^2(\phi^2/2 - c\phi)$ to control nonlinearities.

Experimental results

Research questions

• RQ1 Do solitary wave solutions exist for the two-dimensional Benjamin-Ono equation with mixed nonlocal dispersion terms?
• RQ2 Are these solitary wave solutions real-analytic in the spatial variables, even in the presence of mixed Hilbert transform and second-order derivative terms?
• RQ3 Can the analyticity of solutions be established using multiplier theory and weighted norm estimates in $L^p$ spaces?
• RQ4 What is the role of the parameter $\lambda$ and $\mu$ in determining the regularity and analyticity of solutions?
• RQ5 How do interpolation spaces and weighted Sobolev norms contribute to the proof of existence and analyticity?

Key findings

• The existence of solitary wave solutions is rigorously proven using minimax methods in the space $H^s \cap X^{1/2}$ for $s > 2$.
• The solitary wave solutions are shown to be real-analytic functions in $\mathbb{R}^2$, satisfying $\|\partial^\alpha \phi\|_{H^2} \leq C |\alpha|! (R/2)^{|\alpha|}$ for some $R > 0$.
• The proof resolves a gap in earlier work by providing a complete and rigorous argument for the analyticity of the solitary wave, particularly addressing the missing justification in Lemma 3.4 of prior literature.
• The analyticity is established via the boundedness of specific Fourier multipliers derived from the equation’s linear part, using Lizorkin’s theorem.
• The method applies uniformly to both cases $\lambda > 0, \mu = 0$ and $\lambda = 0, \mu > 0$, with appropriate modifications in the interpolation estimates.
• The nonlinear term $\phi^2/2 - c\phi$ is handled via Faà di Bruno-type expansions and $H^2$-algebra properties, ensuring control of derivatives in the induction step.