Total positivity and spectral properties of linearized operators

1. Problem Statement

The paper addresses the stability analysis of traveling-wave solutions ($u(x,t) = \phi(x-ct)$) for a class of semilinear dispersive wave equations of the form:
$$u_t - (Mu)_x + (f(u))_x = 0$$
where $M$ is a linear operator and $f$ is a nonlinear term. The stability of these waves is governed by the spectral properties of the linearized operator $H$ associated with the wave profile $\phi$:
$$H = L - f'(\phi)$$
where $L = M + c$.

The Core Challenge:
Standard stability theories (e.g., Grillakis-Shatah-Strauss) rely on three specific spectral hypotheses regarding $H$:

• (H1) $H$ has exactly one simple negative eigenvalue.
• (H2) The kernel of $H$ is spanned by the translation mode $\phi'$.
• (H3) The rest of the spectrum is positive and bounded away from zero.

While (H3) is often guaranteed by the structure of the operator, verifying (H1) and (H2) is notoriously difficult. Traditionally, this requires explicit formulas for the solution $\phi$ to analyze the operator directly. However, for many higher-order dispersive equations (like the Kawahara equation or higher-order KdV), explicit formulas for $\phi$ are unknown, making standard verification impossible.

2. Methodology

The authors develop a new set of sufficient conditions to verify (H1) and (H2) without requiring an explicit formula for the solution $\phi$. Their approach relies on the theory of Total Positivity and Polya Frequency (PF) functions.

Key Steps in the Methodology:

1. Operator Transformation: They define an auxiliary operator $H_0 = L^{-1}H = I - L^{-1}f'(\phi)$. They prove that if $H_0$ satisfies (H1) and (H2), then $H$ does as well.
2. Integral Operator Formulation: The operator $H_0$ is viewed as a perturbation of the identity by an integral operator $T_1 = L^{-1}f'(\phi)$. The kernel of this integral operator is $K(x-y)$, where $K = \mathcal{F}^{-1}(1/\alpha(\xi))$ and $\alpha(\xi)$ is the symbol of $L$.
3. Total Positivity (TP): The authors utilize the concept of Strongly TP(2) (Totally Positive of order 2) functions. A kernel $K$ is strongly TP(2) if it is positive, even, and its second-order determinants are strictly positive.
4. Spectral Theory of Positive Kernels: They invoke a theorem (Proposition 2.2) stating that integral operators with continuous, non-negative, symmetric kernels possess a unique simple positive eigenvalue with a strictly positive eigenfunction.
5. Parity and Nodal Analysis: By assuming $\phi$ and $f'(\phi)$ are positive and even, and that the kernel $K$ is Strongly PF(2), they prove that the second eigenvalue of the associated operator corresponds to the translation mode $\phi'$ (which is odd and vanishes only at the origin). This establishes the minimality of the negative eigenspace and the kernel.

3. Key Contributions

A. New Sufficient Conditions (Theorem 1.9)

The paper's primary contribution is Theorem 1.9, which provides verifiable criteria for (H1) and (H2) that depend only on the symbol of the linear operator and the symmetry/positivity of the solution, not the solution's explicit form.

• Conditions:
  1. The symbol $\alpha(\xi)$ is even.
  2. The solution $\phi$ and $f'(\phi)$ are positive and even.
  3. The kernel $K(x) = \mathcal{F}^{-1}(1/\alpha(\xi))$ is Strongly PF(2).

B. Characterization of Valid Symbols (Section 3)

The authors identify specific classes of operators $L$ for which the kernel $K$ is Strongly PF(2):

• Factorizable Symbols: If $\alpha(\xi) = \prod_{i=1}^m (\xi^2 + a_i^2)$ with $a_i \neq 0$, then $K$ is a convolution of exponentials $e^{-a_i|x|}$, which is known to be Strongly PF(2).
• Counter-examples: They demonstrate that if the symbol $\alpha(\xi)$ has complex roots that do not lie on the negative real axis (in the polynomial $p(\xi^2)$), the kernel $K$ may oscillate and fail the PF(2) condition (e.g., Lemma 3.6).

C. Existence of Symmetric Solutions (Theorem 3.8)

They prove that for operators with symbols factoring as $\prod (\xi^2 + a_i^2)$, positive even solutions to the traveling wave equation exist and are unique up to translation. This ensures the symmetry assumptions required for the main theorem are not vacuous.

4. Results and Applications

The paper applies these theoretical results to two specific high-order dispersive equations:

1. 5th Order KdV Equation:
$$u_t - (u_{xxxx} - b u_{xx})_x + (f(u))_x = 0$$

• Analysis: The symbol is $\alpha(\xi) = \xi^4 + b\xi^2 + c$.
• Result: The kernel $K$ is Strongly PF(2) if and only if $b \ge 2\sqrt{c}$. In this parameter regime, the spectral hypotheses (H1) and (H2) are guaranteed.
• Stability: The stability of the wave is determined by the sign of $\frac{d}{dc} \|\phi\|_{L^2}^2$. The authors provide numerical plots showing the regions in the $(b, c)$ plane where the theory applies versus where it remains an open question.

2. 7th Order KdV Equation:
$$u_t - (-u_{xxxxxx} + a u_{xxxx} - b u_{xx})_x + (f(u))_x = 0$$

• Analysis: The symbol is $\alpha(\xi) = \xi^6 + a\xi^4 + b\xi^2 + c$.
• Result: The authors derive precise conditions on parameters $(a, b, c)$ such that the polynomial factors into three quadratic terms with imaginary roots (ensuring $K$ is Strongly PF(2)).
• Numerical Verification: For specific parameters ($a=6, b=9$), they compute the stability curve $\frac{d}{dc} \|\phi\|_{L^2}^2$ for various nonlinearities ($p=2, 4, 6, 8$). They identify critical speeds where stability transitions occur (e.g., for $p=6$, waves are unstable for low speeds and stable for higher speeds).

5. Significance

• Bridging Theory and Computation: The paper provides a rigorous theoretical framework that allows researchers to assert spectral properties for equations where no closed-form solution exists. This is crucial for higher-order dispersive models used in fluid dynamics and plasma physics.
• Generalization of Stability Theory: It extends the applicability of the Grillakis-Shatah-Strauss stability framework beyond the classical KdV and Nonlinear Schrödinger equations to a broader class of nonlocal and higher-order operators.
• Connection to Total Positivity: The work highlights a deep and previously underutilized connection between the spectral theory of linearized operators and the theory of totally positive functions (Polya frequency functions), offering a powerful new tool for spectral analysis.
• Practical Utility: By reducing the verification of complex spectral hypotheses to checking the factorization of a polynomial symbol (the symbol $\alpha$), the paper offers a practical "checklist" for determining the stability of traveling waves in new physical models.