Traveling modulating pulse solutions with small tails for a nonlinear wave equation in periodic media

Abstract

Traveling modulating pulse solutions consist of a small amplitude pulse-like envelope moving with a constant speed and modulating a harmonic carrier wave. Such solutions can be approximated by solitons of an effective nonlinear Schrodinger equation arising as the envelope equation. We are interested in a rigorous existence proof of such solutions for a nonlinear wave equation with spatially periodic coefficients. Such solutions are quasi-periodic in a reference frame co-moving with the envelope. We use spatial dynamics, invariant manifolds, and near-identity transformations to construct such solutions on large domains in time and space. Although the spectrum of the linearized equations in the spatial dynamics formulation contains infinitely many eigenvalues on the imaginary axis or in the worst case the complete imaginary axis, a small denominator problem is avoided when the solutions are localized on a finite spatial domain with small tails in far fields.

Figures (5)

• Figure 1: The curves of eigenvalues $\{ \pm \omega_n(l) \}_{n \in \mathbb{N}}$ plotted as functions of the Bloch wave numbers $l \in \mathbb{B}$ in a typical situation.
• Figure 2: Eigenvalues of the spatial dynamics formulation, see \ref{['spat-dyn']} below, are dense on the imaginary axis. However, due to the convolution structure w.r.t. the $z$-variable, see Theorem \ref{['thm1']}, for a certain power of $\varepsilon$ only a part of the linear operator has to be taken into account. For controlling the order $\mathcal{O}(\varepsilon)$ of the solution only the part $A_1(\omega,c)$ has to be considered. The central spectrum of $A_1(\omega,c)$ is sketched in the left panel. In the middle panel the central spectrum of $A_1(\omega,c)$ and $A_3(\omega,c)$ is sketched. It plays a role for controlling the order $\mathcal{O}(\varepsilon^3)$. The right panel shows a sketch of the central spectrum of $A_1(\omega,c)$, $A_3(\omega,c)$ and $A_5(\omega,c)$ which plays a role for controlling the order $\mathcal{O}(\varepsilon^5)$. In all cases there is a spectral gap between zero and the rest of the spectrum.
• Figure 3: A generalized modulating pulse solution as constructed in Theorem \ref{['thm1']} with $\mathcal{O} (\varepsilon^{2N})$ tails existing for $x$ in an interval of length $\mathcal{O}(\varepsilon^{-(2N+1)})$ with an envelope advancing with group velocity $c_g = \omega'_{n_0}(l_0)$, modulating a carrier wave advancing with phase velocity $c_p = \omega_{n_0}(l_0) /l_0$, and leaves behind the standing periodic Bloch wave. The wavelength of the carrier wave and the period of the coefficients $\rho$, $r$ are of a comparable order.
• Figure 4: Purely imaginary eigenvalues $\lambda = \mathrm{i} (l-ml_0), l\in {\mathbb R},$ as roots of the nonlinear equation (\ref{['speclin0']}) can be obtained graphically as intersections of the curves $l \mapsto \omega_n(l)$ and $l \mapsto m\omega_0 + c_g (l-ml_0)$ for $n\in \mathbb{N}$ and $l \in \mathbb{B}$. For $\rho\equiv 1$ we have $\omega_n(l)=\sqrt{1+(n+l)^2}$ (not ordered by magnitude) and recall that $\omega_n$ is 1-periodic. Due to the symmetry about the $l$-axis we plot only the upper part. We choose $l_0=0.35$ and $\omega_0=\omega_{1}(l_0)\approx 1.06.$
• Figure 5: Transversal intersection of the homoclinic orbit of system (\ref{['truncated-system']}) with the fixed space of the reversibility operator.

Theorems & Definitions (48)

• Remark 1.1
• Remark 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Theorem 1.7
• Remark 1.8
• Remark 1.9
• Remark 1.10