2.1. Linearization around the steady states

Let us now consider the linearized problem associated with the system (2.4). To this end, we adopt the variables

\begin{equation*} u=\phi+p+i q, v=\psi+r+i s \end{equation*}

Plugging this ansatz in (2.4), taking real and imaginary parts separate and ignoring all quadratic and higher-order terms leads to

(2.9)\begin{equation} \left\{ \begin{matrix} p_{tt}-2\omega q_t-p_{xx}+(4-\omega^2) p -[(18 \phi^2+12 \psi^2+12\phi\psi)p+(6 \phi^2+24 \phi\psi)r]=0 \\ q_{tt}+ 2\omega p_t-q_{xx}+(4-\omega^2) q-[(6\phi^2+12\psi^2-12\phi\psi)q+6 \phi^2 s]=0 \\ r_{tt}-6 \omega s_t -r_{xx}+(4-9 \omega^2) r -[(12\phi^2+18 \psi^2)r+(6\phi^2+24 \psi^2) p]=0 \\ s_{tt}+6 \omega r_t-s_{xx} +(4-9\omega^2) s - [6 \phi^2 q +(12\phi^2+6 \psi^2)s]=0. \end{matrix} \right. \end{equation}

This is the form of the linearized problem. We now need to convert this to a more standard, first order in time, eigenvalue problem. This is done in a number of steps.

2.1.1. The eigenvalue problem

In matrix form,

(2.10)\begin{equation} \left(\begin{matrix} p \\q \\ r\\s \end{matrix}\right)_{tt}+ \left(\begin{matrix} 0 & -2\omega & 0 & 0 \\ 2\omega &0 & 0 & 0 \\ 0&0& 0& -6\omega \\ 0 & 0 & 6\omega & 0 \end{matrix}\right) \left(\begin{matrix} p \\q \\ r\\s \end{matrix}\right)_{t} + \mathcal{H} \left(\begin{matrix} p \\q \\ r\\s \end{matrix}\right)=0, \end{equation}

where

(2.11)\begin{equation} \mathcal{H}:=\left(\begin{matrix} \mathcal{H}_1& 0 & - 24 \phi\psi -6\phi^2 & 0 \\ 0 & \mathcal{H}_2 &0 & -6 \phi^2\\ - 24 \phi\psi -6\phi^2 & 0 &\mathcal{H}_3 & 0\\ 0 & -6 \phi^2 & 0 & \mathcal{H}_4 \end{matrix}\right) \end{equation}

and

\begin{align*} \mathcal{H}_1 &= -\partial_{xx}+(4-\omega^2) - 18 \phi^2-12\psi^2- 12 \phi\psi; \ \ \mathcal{H}_2 = -\partial_{xx}+(4-\omega^2) - 6 \phi^2 - 12 \psi^2 +12 \phi\psi \\ \mathcal{H}_3&= -\partial_{xx}+(4-9 \omega^2) -12 \phi^2 - 18 \psi^2; \ \ \mathcal{H}_4 = -\partial_{xx}+(4-9 \omega^2) -12 \phi^2 - 6 \psi^2 \end{align*}

Note that the operator $\mathcal{H}$ is self-adjoint, for nice enough $\phi, \psi$, when taken with the domain $H^2(\mathbb{R})\times H^2(\mathbb{R})\times H^2(\mathbb{R})\times H^2(\mathbb{R})$, while

\begin{equation*} \mathcal{J}:=\left(\begin{matrix} 0 & -2\omega & 0 & 0 \\ 2\omega &0 & 0 & 0 \\ 0&0& 0& -6\omega \\ 0 & 0 & 6\omega & 0 \end{matrix}\right) , \end{equation*}

is skew-symmetric, i.e., $\mathcal{J}^*=-\mathcal{J}$. The corresponding eigenvalue problem, which can be obtained by the assignment

\begin{equation*} \left(\begin{matrix} p \\q \\ r\\s \end{matrix}\right)\to e^{\lambda t} \left(\begin{matrix} p \\q \\ r\\s \end{matrix}\right)=:e^{\lambda t}\vec{z} \end{equation*}

can be written in the form

\begin{equation*} \lambda^2 \vec{z} +\lambda \mathcal{J} \vec{z}+ \mathcal{H} \vec{z}=0. \end{equation*}

This eigenvalue problem is a pencil of order two, and these have been an object of intense investigations. A standard equivalence in a more classical Hamiltonian form is as followsFootnote ³

(2.12)\begin{equation} \left(\begin{matrix} 0 & -Id \\ Id & -\mathcal{J} \end{matrix}\right) \left(\begin{matrix} \mathcal{H} & 0 \\ 0 & I \end{matrix}\right) \left(\begin{matrix} \vec{z} \\ \vec{w} \end{matrix}\right)= \lambda \left(\begin{matrix} \vec{z} \\ \vec{w} \end{matrix}\right) \end{equation}

This prompts the following definition.

Definition 2.2. We say that the steady state solution $(\phi, \psi)$ of (2.6) is spectrally stable if the eigenvalue problem (2.12) does not have non-trivial solutions with $\mathrm{Re}\lambda \gt 0, \vec{z}\neq 0$.

2.2. Stability of the waves

Next, we address the important issue of symmetries and the corresponding eigenvalues at zero for the linearized problem, that arise.

2.2.1. Symmetries of the system

We can identify at least two symmetries of the system (2.6), namely a translational and (phase) modulational invariance. Specifically, for every solution $(\phi, \psi)$ of (2.6) and arbitrary real $y$, then $(\phi(\cdot-y), \psi(\cdot-y))$, $(e^{i y} \phi, e^{3i y} \psi)$ are solutions as well. The usual differentiation in the parameter $y$, and taking into account the setup of $\mathcal{H}$, provides two elements in $Ker[\mathcal{H}]$. More concretely,

\begin{equation*} \left(\begin{matrix} \phi' \\ 0 \\ \psi'\\ 0 \end{matrix} \right), \ \ \ \left(\begin{matrix} 0 \\ \phi \\ 0 \\ 3 \psi \end{matrix} \right) \in Ker[\mathcal{H}]. \end{equation*}

2.2.2. Non-degeneracy of the wave pair φ, ψ

We now introduce the notion of a non-degeneracy for the wave pair $(\phi, \psi)$.

Definition 2.3. We say that the solution $(\phi, \psi)$ is non-degenerate if there are no more elements of $Ker[\mathcal{H}]$ other than those already found. Specifically,

\begin{equation*} Ker[\mathcal{H}] = span \left[\left(\begin{matrix} \phi' \\ 0 \\ \psi'\\ 0 \end{matrix} \right), \ \ \ \left(\begin{matrix} 0 \\ \phi \\ 0 \\ 3 \psi \end{matrix} \right) \right] \end{equation*}

In other words, non-degeneracy means that the only zeros in the kernel of $\mathcal{H}$ are those generated by the symmetries of the system.

We are now ready to state the main stability result.

Theorem 2.4. Let $\omega: |\omega| \lt \frac{2}{3}$. Assume that the steady state solution $(\phi, \psi)$ of (2.4), constructedFootnote ⁴ in Theorem 2.1, is a non-degenerate, smooth function of $\omega$. That is, the mapping $\omega\to (\phi, \psi)$ is Frechet differentiable.

Then, it is spectrally stable if and only if the following holds true

\begin{equation*} \omega \partial_\omega \int_{\mathbb{R}} \left(\phi^2+9 \psi^2\right) dx + \int_{\mathbb{R}} \left(\phi^2+9 \psi^2\right) dx \lt 0. \end{equation*}

Equivalently, $\omega\to \omega \int_{\mathbb{R}} \left(\phi^2+9 \psi^2\right) dx$ is decreasing.

3.1. Decreasing rearrangements and compactness criteria

One tool that will be useful in the sequel is the notion of the decreasing rearrangement. For a function $f:\mathbb{R}\to \mathbb{R}$, there exists an even, positive, non-increasing on $(0, +\infty)$ function $f^*$, so that the corresponding level functions coincide, i.e., $ d_f(\alpha)=\{x: |f(x)| \gt \alpha\}=\{x: f^*(x) \gt \alpha\}=d_{f^*}(\alpha) $ Alternatively, a constructive definition may be given as follows

\begin{equation*} f^*(t)=\inf\left\{s: d_f(s)\leq 2|t| \right\}, \ \ t\in \mathbb{R} \end{equation*}

Either way, we conclude that $\|f\|_p=\|f^*\|_p$. Additionally, there are the standard inequalities

\begin{equation*} \int_{\mathbb{R}} f(x) g(x) dx\leq \int_{\mathbb{R}} f^*(x) g^*(x) dx, \int_{\mathbb{R}^2} f(x-y) g(y) h(x) dx dy \leq \int_{\mathbb{R}^2} f^*(x-y) g^*(y) h^*(x) dx dy \end{equation*}

The Szegö inequality guarantees that $f^*\in H^1$, whenever $f\in H^1(\mathbb{R})$ and moreover,

(3.1)\begin{equation} \|\partial_x f\|\geq \|\partial_x f^*\| \end{equation}

In case of equality in (3.1), one concludes that the function $f$ coincides with its rearrangement, i.e., $f=f^*$, and therefore it belongs to a class of functions, usually referred to as bell-shaped.

The Riesz–Relich–Kolmogorov compactness criteria (or rather a consequence thereof) is the following – a sequence $\{u_n\}_n$ is compact in $L^p(\mathbb{R}), 1 \lt p \lt \infty$, if

• • $\sup_n \|u_n\|_{H^1(\mathbb{R})} \lt \infty$

• • For every $\epsilon \gt 0$, there exists $R=R_\epsilon \gt 0$, so that

  \begin{equation*} \sup_n \int_{|x| \gt R} |u_n(x)|^p dx \lt \epsilon. \end{equation*}

3.2. Basics of the instability index theory

Consider a Hamiltonian eigenvalue problem in the form

(3.2)\begin{equation} J L u =\lambda u \end{equation}

where $J^*=-J$ and $L^*=L$, with appropriate domains for the composition operator $J L$. Assume in addition that both $J, L$ act on spaces of functions, and they do preserve real-valued functions. The stability of (3.2) is a fundamental problem in modern dynamics. Specifically, as in Definition 2.2, we say that the problem (3.2) is stable if there are no non-trivial solutions of (3.2) with $\mathrm{Re}\lambda \gt 0$. In order to introduce some relevant notions, we assume that $L$ is semi-bounded from below, that is, there exists a constant $C$, so that $L\geq C$ in the sense of quadratic forms. In fact, we assume that its negative subspace $X_-$ is finite dimensional and $P_{X_-} L P_{X_-}$ has only point spectrum, each with finite multiplicity. Denote its Morse index by

\begin{equation*} n(L)=dim[X_-]=\#\{\sigma: \sigma \lt 0: \sigma \in \sigma_{p.p.}(L)\}. \end{equation*}

Next, let $k_r$ be the number of positive eigenvalues of (3.2), $k_c$ be the number of quadruplets of eigenvalues with non-zero real and imaginary parts, and $k_i^-$, the number of pairs of purely imaginary eigenvalues with negative Krein-signature. For a simple pair of imaginary eigenvalues $\pm i \mu, \mu\neq 0$, and the corresponding eigenvector $\vec{h} = \left(\begin{array}{c} h_1 \\ h_2 \end{array}\right) $, the Krein signature, either $\pm 1$ is the following quantity $ sgn(\langle \mathcal{L} \vec{h}, \vec{h} \rangle). $ Consider the generalized kernel of $J L$

\begin{equation*} gKer(J L)=span[(Ker(J L)^l, l=1, 2, \ldots] . \end{equation*}

Assume that $dim(gKer(J L)) \lt \infty$, although this is, strictly speaking, not required in [Reference Lin and Zeng19]. Select a basis in

\begin{equation*} gKer(J L)\ominus Ker(L)=span[\eta_j, j=1, \ldots, N]. \end{equation*}

Introduce $\mathcal{D}\in \mathcal{M}_{N\times N}$,

\begin{equation*} \mathcal{D}:=\{\mathcal{D}_{i j}\}_{i,j=1}^N: \mathcal{D}_{i j}=\langle \mathcal{L} \eta_i, \eta_j \rangle . \end{equation*}

Then, following [Reference Lin and Zeng19], we have the following formula, relating the number of ‘instabilities’ or Hamiltonian index of the eigenvalue problem (3.2) and the Morse indices of the operators $L$ and $\mathcal{D}$,

(3.3)\begin{equation} k_{Ham}:=k_r+2 k_c+2k_i^-=n(L)-n(\mathcal{D}). \end{equation}

Specifically, if $n(L)=1$, it follows from (3.3) that $k_c=k_i^-=0$ and

(3.4)\begin{equation} k_r=1-n(\mathcal{D}). \end{equation}

Note that in the case $n(L)=1$, instability occurs exactly when $n(\mathcal{D})=0$, while stability occurs whenever $n(\mathcal{D})=1$. We formulate this in a corollary as follows.

6.1. Numerical oscillons

In our efforts to identify solutions to Eqs. (2.6), we found three branches of possible solutions. We thus hereafter explore these solution branches that can be discerned in the diagrams of the left panel of Figure 1. This is complemented by an example of the profile of each of the oscillon branches that we have identified in the right panel of the figure. The former represents the amplitude of the $\phi(x)$ component, providing a useful bifurcation diagnostic accordingly. For completeness, we also show the bifurcation diagram in terms of the Hamiltonian defined in Eq. (2.5), although the latter does not carry any stability information. The first branch of solutions, depicted in blue, corresponds to $(0,+)$ oscillons, i.e., $\phi(x)=0$, $\psi(x) \gt 0$, with only the third harmonic being present. These waveforms have been found to exist for any $\omega\in(0,2/3)$, which is in line with earlier dynamical observations of the original PDE of Eq. (2.1) [Reference Barashenkov and Alexeeva7]. In the latter the frequency of these waves $3 \omega \lt 2 \Rightarrow \omega \lt 2/3$.Footnote ⁶ On the other hand, the red line in Figure 1 corresponds to solutions of the $(-,+)$ form, i.e., $\phi(x) \lt 0$, $\psi(x) \gt 0$, in the interval $\omega\in(0,\omega_0)$ with $\omega_0=0.5242$; as one can see, the amplitude of the $\phi$ component tends to zero when $\omega\rightarrow\omega_0$ and becomes zero just at $\omega=\omega_0$. For $\omega \gt \omega_0$, $\phi(x) \gt 0$ and $\psi(x) \gt 0$, meaning that the solution, in our notation, becomes of the $(+,+)$ form, until the solution disappears for only slightly larger frequency, i.e., at $\omega=\omega_+=0.5323$ through a saddle-centre bifurcation with a third branch, also of the $(+,+)$ form and depicted in black. In particular, the branch depicted with a black line has a larger (positive) amplitude within its $\phi$ component, while the red one has a lower amplitude, and the two collide and disappear hand-in-hand at $\omega=\omega_+=0.5323$. However, this saddle-centre event is not the only bifurcation occurring. As the $(-,+)$ branch turns to a $(+,+)$ one with $\phi(0)$ crossing through 0, this red branch ‘collides’ in a transcritical-like bifurcation with symmetry, as we will see below, with the blue branch of the form $(0,+)$. We will explore the implications of these bifurcations for the branches involved in what follows.

Figure 1.

(Left panels) Bifurcation diagrams of the solutions to Eq. (2.6) that were identified in our analysis. The values of the first component at $x=0$ ( $\phi(0)$) and the Hamiltonian defined in Eq. (2.5) are used as a bifurcation diagnostic. The black branch is a $(+,+)$ branch with both components featuring a positive (sign-definite) waveform. The blue solid line represents a $(0,+)$ branch with only the third harmonic being ‘populated’. Finally, the red branch is a $(-,+)$ branch with the two components starting with opposite values, but the first component crosses $\phi(x)=0$ when $\omega\rightarrow\omega_0=0.5242$, leading to both components becoming positive before this branch collides in a saddle-centre bifurcation with the other $(+,+)$ branch at $\omega=\omega_+=0.5323$. Dashed lines represent unstable solutions. (Right panels) Profiles of the oscillons for the different branches (from top to bottom: $(+,+)$, $(0,+)$ and $(-,+)$ branches) at $\omega=0.5$. Green (purple) curves correspond to the $\phi(x)$ ( $\psi(x)$) component.

1 Long description

The image consists of two bifurcation diagrams on the left and three oscillon profiles on the right. The top left diagram plots phi subscript 0 left parenthesis 0 right parenthesis against omega, showing three branches: black, blue and red. The black branch curves upward, the blue branch is horizontal and the red branch curves downward. The bottom left diagram plots H against omega, with dashed lines representing unstable solutions for the same branches. The right side shows three oscillon profiles labeled as black, blue and red branches. Each profile plots phi left parenthesis x right parenthesis and psi left parenthesis x right parenthesis against x, with distinct curves for each branch. The black branch profile shows a higher peak for phi left parenthesis x right parenthesis, the blue branch has a moderate peak and the red branch shows a lower peak for phi left parenthesis x right parenthesis.

In Figure 2, we show the dependence on $\omega$ of the VK-like quantity discussed earlier, namely, $\omega \int_{\mathbb{R}} \left(\phi^2+9 \psi^2\right) dx$. We observe a maximum of the $(0,+)$ oscillon of the blue branch at $\omega=0.4719$ and another one for the $(-,+)$ oscillon of the red branch at $\omega=0.5200$. Interestingly, for the blue branch, we will see that the change of monotonicity of the relevant quantity is associated with a real eigenvalue changing to imaginary, as the relevant quantity changes from increasing to decreasing. On the other hand, for the black $(+,+)$ branch, the ‘wrong’ monotonicity (i.e., its increasing nature) implies that this branch will always feature real eigenvalue pairs, as we will see to be the case in what follows. Interestingly, the situation for the red branch will be seen in our eigenvalue analysis to be more complicated.

Figure 2.

In this figure, we demonstrate the $\omega \int_{\mathbb{R}} \left(\phi^2+9 \psi^2\right) dx$ quantity that we associated in Theorem 2.4 with stability. The red $(-,+)$ branch changes monotonicity (and hence stability) at $\omega=0.5200$; see also the zoom-in of the right panel. The black $(+,+)$ branch is always increasing and hence always bears real eigenvalues. The blue $(0,+)$ branch has a change of monotonicity at $\omega=0.4719$ analysed further below. Dashed lines represent unstable solutions.

Figure 2 Long description

The left graph shows three curves representing the relationship between omega and the integral of phi squared plus nine psi squared. The x-axis is labeled omega, ranging from 0 to 0.8 and the y-axis is labeled omega times the integral of phi squared plus nine psi squared, ranging from 0 to 4. The blue curve peaks around 0.6, the red curve peaks slightly lower and the black curve is the lowest. The right graph zooms in on the range from 0.51 to 0.53 on the x-axis and from 3.87 to 3.9 on the y-axis. It highlights the red and blue curves, showing a detailed view of their behavior in this specific range. The red curve peaks around 0.52, while the blue curve continues to rise sharply.

Another quantity that was deemed central to our theoretical analysis of Section 4, when constructing the waves, was the functional $J/I^2$, where $J$ and $I$ are defined in Proposition 4.1. Accordingly, in Figure 3, we illustrate the quantity $J/I^2$ versus $\omega$. One can see that the largest value of the relevant quantity pertains to the black $(+,+)$ branch for $\omega \lt 0.5316$, when the blue branch acquires the largest $J/I^2$ value and becomes the global maximizer until the branch ceases to exist at $\omega=2/3$. The relevant global maximizer is unstable throughout its existence (irrespectively of whether it pertains to the black or blue branch), as we will confirm in our eigenvalue analysis.

Figure 3.

The quantity $J/I^2$, where $J$ and $I$ are defined in Proposition 4.1, is given on the left panel for each of the previously discussed branches. The right panel is again a zoom-in where a crossing of this quantity for the black and blue branch takes place at $\omega=0.5316$.

Figure 3 Long description

The left graph shows the relationship between omega (x-axis) and J over I squared (y-axis) with three curves: black, red and blue. The x-axis ranges from 0 to 0.5 and the y-axis ranges from 0 to 0.4. The black curve is the highest, followed by the red and blue curves. The right graph is a zoomed-in view of the left graph, focusing on the range of omega from 0.53162 to 0.53172 and J over I squared from 0.3211 to 0.3215. The black and blue curves intersect within this range, while the red curve remains below them.

We now turn to an analysis of the eigenvalues of the different branches to help elucidate their stability and connect them to the VK-criterion discussed in Theorem 2.4. As discussed already above, and in line with the VK criterion of the Theorem, the $(+,+)$ branch is always unstable because of a real eigenvalue pair. Figure 4 shows the relevant real part of the (maximal) eigenvalue of the linearization around such solutions. A typical example of the relevant solution and its spectral plane is shown in Figure 5.

Figure 4.

Dependence of the real part of the eigenvalues of the $(+,+)$ family with respect to the frequency. The black line indicates that such a real eigenvalue exists for all the frequencies for which the branch exists, thus rendering it spectrally unstable.

Figure 4 Long description

A graph showing the real part of eigenvalues on the y-axis labeled as 'Re left parenthesis lambda right parenthesis' and frequency on the x-axis labeled as 'omega'. The curve starts at approximately 3.5 on the y-axis and decreases as it moves right, ending near 0.5 on the x-axis and 1 on the y-axis.

Figure 5.

Oscillon in the $(+,+)$ branch with $\omega=0.5$ (left panel) and the corresponding spectral plane featuring a real (unstable) eigenvalue. The green (magenta) curve in the left panel corresponds to the $\phi(x)$ ( $\psi(x)$) component.

Figure 5 Long description

The left graph displays two functions, phi left parenthesis x right parenthesis and psi left parenthesis x right parenthesis, plotted over the x-axis ranging from negative 20 to positive 20. The y-axis ranges from 0 to 0.6. The purple curve represents phi left parenthesis x right parenthesis and the green curve represents psi left parenthesis x right parenthesis. Both curves peak at x equals 0. The right graph shows eigenvalues with the real part on the x-axis ranging from negative 2 to positive 2 and the imaginary part on the y-axis ranging from negative 1 to positive 1. The eigenvalues are represented by circles, with clusters around the real part equals 0 and imaginary parts at approximately positive and negative 0.5.

From the representation of the real and imaginary parts of the spectrum for the $(0,+)$ solutions in Figure 6, one can observe that such a solution is unstable for $\omega \lt 0.5121$, as at such a value of the frequency parameter, there is a Hopf bifurcation where a quartet of complex eigenvalues becomes imaginary (note that the VK-like quantity does not change monotonicity at this point), while at $\omega=0.4719$, a real eigenvalue pair becomes imaginary as $\omega$ is increased (consistent with the VK-like quantity reaching to a maximum at this point); in addition, at $\omega=\omega_0 \equiv 0.5242$ there is an eigenvalue that becomes zero although it is imaginary past this value. This is fairly remarkable as what happens here is that the positive and negative imaginary parts of the eigenvalue ‘exchange sides’ (i.e., the positive becomes negative and vice versa) without the state changing its stability at the transcritical-like point $\omega_0$. We will use here the term ‘transcritical-like’, since in a regular transcritical bifurcation, the two branches exist both before and after their collision (where they become identical), yet they exchange stability at the critical point. Here, the branches involved collide (meaning that they exist before and after the critical point and coincide at that point), yet due to each of them involving a pair of eigenvalues, their stability is not exchanged; rather, the relevant eigenvalue pairs switch sides (positives become negatives and vice versa, for real eigenvalues in one of the branches and imaginary ones in the other). This behaviour is not predicted by the VK-like quantity.

Figure 6.

Dependence of the real (left panel) and imaginary (right panel) parts of the eigenvalues of the $(0,+)$ family with respect to the frequency $\omega$. A black line indicates that an eigenvalue is real, and red lines are associated with eigenvalues with non-zero imaginary part, i.e., complex ones. Vertical lines correspond to the values of $\omega$ for which the spectral plane is represented in Figure 7. Notice that the solution is fully spectrally stable for $\omega \gt 0.5121$.

Figure 6 Long description

The left graph displays the real parts of eigenvalues plotted against frequency, labeled as omega. The y-axis is labeled Re lambda, ranging from 0 to 1.2. The graph features black and red lines, indicating different eigenvalue behaviors. Vertical dashed lines are present at various frequency values. The right graph illustrates the imaginary parts of eigenvalues, with the y-axis labeled Im lambda, ranging from 0 to 1. Blue lines represent the imaginary components, intersecting and overlapping at different points. Vertical dashed lines are also visible, corresponding to specific frequency values on the x-axis.

Importantly, it is relevant to note that these results are semi-quantitatively in line with the findings of [Reference Barashenkov and Alexeeva7] at the level of the full original PDE of Eq. (2.1), as the latter findings suggest that the single frequency waveform is dynamically robust for $\sqrt{2}/3 \approx0.4714 \lt \omega \lt 2/3$ in our notation. We indeed identify an eigenvalue transition at $0.4719$ (from real to imaginary) very close to the above threshold, while the complete stabilization of this steady state (when it becomes devoid of Hamiltonian Hopf bifurcations) occurs at $\omega=0.5121$. Some typical examples of the corresponding the $(0,+)$ solution branch spectral planes are shown in Figure 7.

Figure 7.

Oscillons in the $(0,+)$ branch whose frequency corresponds to the dashed vertical lines in Figure 6. Each value of the frequency is displayed at the corresponding panel.

Figure 7 Long description

The image consists of eight spectral plane graphs, each displaying eigenvalues for different omega values. The graphs are arranged in two columns and four rows. Each graph plots imaginary lambda on the y-axis against real lambda on the x-axis, with both axes ranging from negative one to positive one. The omega values are labeled at the top left of each graph: 0.470, 0.480, 0.510, 0.520, 0.523, 0.526, 0.540 and 0.560. The eigenvalues are represented by circles distributed along the vertical axis, indicating their real and imaginary components. The distribution of eigenvalues varies slightly across the different omega values, with some graphs showing more clustered values along the vertical axis than others.

Finally, as can be deduced from Figure 8, the $(-,+)$ solutions are unstable for $\omega \lt 0.5183$, as at such a value, the oscillon becomes stable via a Hopf bifurcation; i.e., a quartet turns into two imaginary pairs at this frequency. As with the $(0,+)$ case, this bifurcation is not predicted by the VK-like quantity. Importantly, in the case of this branch, the conditions of Theorem 2.1 are not satisfied (as the solutions are not bell-shaped) and hence Theorem 2.4 does not apply. Indeed, this branch with $n({\cal L})=2$ can be observed in Figure 2 (right panel) to have a stable segment even though the VK-quantity is increasing. The relevant branch is stable for a very narrow frequency interval until $\omega=0.5200$, where an eigenvalue pair turns from imaginary to real, leading to the subsequent destabilization of the branch. Notice that we have checked the latter to be consonant with the Hamiltonian–Krein index theory, given that for this branch $n({\cal H})=2$, while upon change of monotonicity $n({\cal D})=1$. Hence, per the relevant index theory [Reference Kapitula and Promislow15], there should be $n({\cal H})-n({\cal D})=1$ positive real eigenvalue(s), which is what we observe.

Figure 8.

Dependence of the real (left panel) and imaginary (right panel) parts of the eigenvalues of the $(-,+)$ family on the frequency. The black line indicates a real eigenvalue, while the red lines correspond to eigenvalues with a non-zero imaginary part.

Figure 8 Long description

The left graph shows the real part of eigenvalues, labeled as Re left parenthesis lambda right parenthesis, plotted against frequency, labeled as omega. The x-axis ranges from 0.51 to 0.53 and the y-axis ranges from 0 to 0.8. A black curve starts at approximately 0.52 and rises sharply, while a red curve descends from 0.51 to 0.52. The right graph displays the imaginary part of eigenvalues, labeled as Im left parenthesis lambda right parenthesis, against the same frequency range. The x-axis ranges from 0.51 to 0.53 and the y-axis ranges from 0 to 0.5. Multiple blue curves are visible, with one curve showing a distinct dip around 0.52 before rising again. The graphs illustrate the dependence of eigenvalues on frequency, with distinct real and imaginary components.

As $\omega$ increases further, again very proximally in frequency, i.e., at $\omega=\omega_0=0.5242$, the collision of the $(-,+)$ and the $(0,+)$ branch discussed above takes place. However, importantly, this is a rather special example of a transcritical-like bifurcation with symmetry. That is to say, as we pointed out above, for the $(-,+)$ branch, the eigenvalues of the real pair exchange sides while both remain real. Accordingly, the $(-,+)$ branch before and after the bifurcation remains unstable, just like the $(0,+)$ branch with which it collides preserves its stability before and after the bifurcation. At the critical point for the $(-,+)$, the $\phi(x)$ component crosses through zero (from negative to positive, as the frequency increases).

Typical examples of relevant solutions in the $(-,+)$ branch and their spectral planes are shown in Figure 9.

Figure 9.

Oscillons in the $(-,+)$ branch with $\omega=0.5$ (top panels), $\omega=0.521$ (middle panels) and $\omega=0.53$ (bottom panels). The left panels show the solution profile, while the right ones show the corresponding spectral plane associated with the branch’s stability. Green (magenta) curves in the left panels correspond to the $\phi(x)$ ( $\psi(x)$) component.

Figure 9 Long description

The image consists of three pairs of graphs. Each pair includes a solution profile on the left and a spectral plane on the right. The left graphs display functions phi(x) and psi(x) plotted over the x-axis, with phi(x) in purple and psi(x) in green. The x-axis ranges from negative 20 to positive 20 and the y-axis ranges from negative 0.2 to positive 0.8. The right graphs illustrate the imaginary part (Im(lambda)) versus the real part (Re(lambda)) of lambda. The x-axis for these graphs varies, with the first graph ranging from negative 1 to positive 1, the second from negative 0.5 to positive 0.5 and the third from negative 0.4 to positive 0.4. The y-axis for all spectral planes ranges from negative 1 to positive 1. Each spectral plane graph contains multiple blue circles representing data points distributed along the vertical axis, indicating the stability of the branch associated with the solution profile. The arrangement of the graphs is in three rows, with each row containing one solution profile and one spectral plane graph side by side.

6.2. Numerical breathers

The oscillons obtained through the solution of Eqs. (2.6) can be used as seeds for fixed point methods that allow to solve the original KG PDE of Eq. (2.1) in the form of periodic orbits (breathers) of the form

(6.1)\begin{equation} z(t,x)=2\sum_{k=1}^\infty y_k(x)\cos\left[(2k-1)\omega_b t\right] \end{equation}

When this Fourier series expansion is introduced into Eq. (2.1), the PDE is transformed into a system of coupled ODEs, which upon finite difference discretization, turn into a system of algebraic equations. After the choice of a suitable seed and boundary conditions (we have taken periodic ones), it can be solved by means of fixed-point methods like Newton–Raphson (see [Reference Marín and Aubry21] for more details). The numerical domain extends from $[-L,L]$, where the spatial length parameter here has been chosen as $L=25$.

The main drawback of working with breathers is that they do not exist in a genuinely infinite domain because of resonances of $k \gt 1$ harmonics with the linear modes band. However, in a finite domain, there can appear gaps in the spectrum that can be controlled by the size of the domain, whose inverse controls the discretization in wavenumbers (and accordingly in frequencies). Then, breathers whose harmonics lie in the resulting frequency gaps have a short interval of stability that is dependent on the domain size and the discretization parameter. In addition, the only oscillon family which gives rise to breathers with features similar to that of the original oscillon is that of the $(0,+)$ solutions above. Because of this, we have only considered breathers coming from the $(0,+)$ branch; in that case, the seed for the fixed point algorithm is taken as $y_1(x)=\psi(x)$ and $y_k(x)=0$ for $k\geq2$, and $\omega_b=3\omega$. In this way, one can find breathers that can potentially be stable as we now discuss.

A remark as concerns the $(-,+)$ and $(+,+)$ modes and their non-existence in the original PDE of Eq. (2.1) is relevant at this point. What we find here is intriguing in that, indeed, there are branches such as $(-,+)$ and $(+,+)$ that may bear complex mathematical structure and associated bifurcations in the reduced model, yet they do not appear – at least, to the best of our ability – in our numerical computations to be featured in the original infinite-dimensional dynamical system. While we maintain that it is of interest [from a mathematical and computational perspective] to explore the stability and bifurcation features of these branches, we believe that it is appropriate to caution physically and computationally minded practitioners about the potential existence of features of the reduced model that ‘may not make it’ to the original PDE model (and vice versa).

6.3. Simulations of the full PDE

Returning to the full model breather solutions, we have considered the dynamics of the KG PDE by taking as an initial condition $z(0,x)$ a breather with frequency $3\omega$, i.e., $z(0,x)=2\sum_k y_k(x)$ and comparing it with that stemming from an oscillon, i.e., $z(0,x)=2(\phi(x)+\psi(x))$. It is worthwhile to note that in all the relevant examples where the KG PDE is considered, the velocity initial condition $z_t(0,x)$ is set to zero; cf. also Eq. (6.1). Notice that this is also consistent with $z_t(0,x)$ as obtained from Eq. (2.3) for real initial condition fields $\phi$ and $\psi$. When considering the $(0,+)$ solution, naturally, $\phi(x)=0$. Figure 10 considers the dynamics when taking as initial condition an oscillon with $\omega=0.45$ and comparing it with the evolution of a breather with frequency $3\omega$; to this aim, a spatio-temporal diagram is shown together with the spectrum of the Floquet operator of the breather, from which one can see that the dominant instability is of an exponential nature. The evolution of the central site and its Fourier spectrum shown also complements the picture that can be summarized as the emergence of a quasi-periodic breather-like structure that in the case of the initial breather is created after a transient. From the Fourier spectra, one can see that the dominant frequency for the dynamics emerging from the oscillon (breather) is $1.031$ ( $1.717$), and a secondary peak of the spectrum is observed around $0.999$ ( $1.665$). Although the dynamics draws a number of parallels between the two cases, there are also some notable differences. In the approximate oscillon case generated by the steady state solution of Eqs. (2.6), we observe a continuously, yet very slowly decaying structure. On the other hand, the obtained numerically exact solution is indeed a periodic orbit, up to numerical tolerance error, yet this error suffices to eventually destabilize the originally genuinely periodic evolution for sufficiently long times. The resulting structures are similar in nature, albeit with different frequencies involved in the long-term dynamics. Notice also that the frequency peaks at $\omega_p$ in the spectrum of the breather can be associated with the argument $\theta$ of the Floquet exponents related to Hopf bifurcations by $\theta=2\pi(2\omega_p/\omega_b\ \mathrm{mod}\ 1)$. In that case, the dominant peak at $\omega_p=1.717$ corresponds to $\theta=3.4162$, which is close to the argument of the dominant unstable multiplier $\Lambda=-1.3490 \pm 0.4991\mathrm{i}=1.4384\exp(\pm3.4959\mathrm{i})$.

Figure 10.

Spatio-temporal evolution of the energy density for the KG equation using as initial condition a $(0,+)$ oscillon with frequency $\omega=0.45$ (a) and a (genuinely time-periodic) breather with frequency $3\omega$ (b). Panel (c) shows the time evolution of the central site and also its Fourier spectrum (d) when the oscillon is taken as the initial condition, whereas panels (e) and (f) do the same for the breather; notice that in this case, the Fourier spectrum has been computed from the solution at $x=0$ in the time interval $t\in[400,1000]$. Finally, panel (g) displays the Floquet spectrum of the breather.

Figure 10 Long description

The image contains seven panels labeled (a) to (g). Panel (a) shows a spatio-temporal diagram with axes labeled x and t, depicting energy density evolution with a color scale from 1 to 5. Panel (b) is similar to (a), showing another spatio-temporal diagram with the same axes and color scale. Panel (c) presents a graph of z(t,0) over time t from 0 to 1000, showing oscillatory behavior. Panel (d) displays a Fourier spectrum with axes labeled omega and vertical bar S left parenthesis omega right parenthesis vertical bar, showing a peak around omega equals 1. Panel (e) shows another graph of z(t,0) over time t from 0 to 1000, with a different oscillatory pattern. Panel (f) presents a Fourier spectrum similar to (d), with a peak around omega equals 2. Panel (g) illustrates a Floquet spectrum with axes labeled Re left parenthesis Lambda right parenthesis and Im left parenthesis Lambda right parenthesis, showing points distributed around a circle. Each panel provides insights into the dynamics and spectral characteristics of the system under study.

Figure 11 shows the outcome for the oscillon with $\omega=0.5$ and the breather counterpart with frequency $3\omega$. In this case, the instabilities of the breather are very weak, and its shape remains almost invariant for the long-time evolutions of the order of 1000 time units reported. It is nevertheless relevant to point out that in this case too, the genuine breather solution is expected to be destabilized for sufficiently long times. Finally, in Figure 12, one can observe that for $\omega=0.61$, both the oscillon and the breather are stable, and the oscillon dynamics is quite similar to that of the breather. In addition, we observe the stability of the breather from its Floquet spectrum; in fact, in the bottom right panel of Figure 12, one can see the dependence of the energy versus the breather frequency, and observe the existence of the typical ladders (‘spikes’) caused by the resonance with phonons, i.e., with linear continuous spectrum modes. It is relevant to note here that in the context of oscillons, such a ladder structure has also been reported in the realm of a three-dimensional radial oscillon in a ball in the study of [Reference Alexeeva, Barashenkov, Bogolubskaya and Zemlyanaya3]. Interestingly, for $\omega_b \gt rsim1.82$, the resonances are so small that they cannot be discerned, and the breather is stable for that branch. Prior to the resonance at $\omega_b=1.8199$, the breather solution is stable in the decreasing portion of the branch, while it is unstable in the (narrow) increasing portion evident in Figure 12. While the presence of the continuous spectrum resonances renders the continuation of the relevant breather branch for all frequencies tricky, our expectation is that for $3 \omega \gt \sqrt{2}$, the breather branch will have a character similar to what is reported above with a typical marginally stable Floquet spectrum (in the exception of small finite-size induced complex [Reference Aubry6] multipliers). Nevertheless, near the resonances with each relevant phonon mode, there will be a portion of increasing energy-frequency dependence, corresponding to breather instability, in line with the corresponding stability criterion of [Reference Kevrekidis, Cuevas-Maraver and Pelinovsky16].

Figure 11.

Spatio-temporal evolution of the energy density for the KG equation using as initial condition (a) a $(0,+)$ oscillon with frequency $\omega=0.5$ and (b) a breather with frequency $3\omega$. (c) displays the Floquet spectrum of the breather.

Figure 11 Long description

The image consists of three plots. The first plot (a) shows a heatmap of energy density evolution over time, with the x-axis labeled 'x' ranging from negative 10 to 10 and the y-axis labeled 't' ranging from 0 to 1000. The color scale on the right indicates energy density values from 0.5 to 4.5. The second plot (b) is similar to the first, with the same axes and color scale, showing a slightly different pattern. The third plot (c) is a circular plot of the Floquet spectrum, with the x-axis labeled 'Re left parenthesis Lambda right parenthesis' and the y-axis labeled 'Im left parenthesis Lambda right parenthesis', both ranging from negative 1 to 1. The plot shows a circular distribution of points, indicating the stability of the system being analyzed.

Figure 12.

Spatio-temporal evolution of the energy density for the KG equation using as initial condition (a) a $(0,+)$ oscillon with frequency $\omega=0.61$ and (b) a breather with frequency $\omega_b=3\omega=1.83$. (c) displays the Floquet spectrum of the breather and (d) shows the dependence of the energy versus the breather frequency for values of $\omega_b$ near $1.83$.

Figure 12 Long description

The image A shows a graph with x-axis labeled x and y-axis labeled t, depicting energy density evolution with a color scale from 0.2 to 1.6. The image B is similar to A, showing energy density evolution with the same axes and color scale. The image C displays a circular plot with x-axis labeled Re left parenthesis capital lambda right parenthesis and y-axis labeled Im left parenthesis capital lambda right parenthesis, illustrating the Floquet spectrum. The image D shows a graph with x-axis labeled omega subscript b and y-axis labeled H subscript KG, depicting energy versus frequency with a curve showing values from approximately 1.8 to 1.9 on the x-axis and 3 to 4 on the y-axis.

Finally, we have also considered the dynamics of $(-,+)$ and $(+,+)$ oscillons. As mentioned above, we have been unable to systematically identify breathers pertaining to such steady state solutions; hence, instead, in what follows, we have focused on the initialization of the original KG PDE of Eq. (2.1) with the solution of the steady state problem of Eqs. (2.6). In the case of $(-,+)$ (see Figure 13), we have observed that the oscillon generically disperses, although it still oscillates, as one can see from the panels showing $z(t,0)$; when the frequency increases, the breather dispersion decreases and, potentially, a robust quasi-periodic breather-like excitation is generated. As an example, one can observe that for the $(-,+)$ oscillon with $\omega=0.515$, resulting in a quasi-periodic breather with dominant frequency $1.684$ and secondary frequency $1.370$. For the $(+,+)$ oscillons, the dynamics typically lead to blow up, as one can see in the left panel of Figure 14 (cf. with the right panel of the figure stemming from the $(-,+)$ branch, which leads again to quasi-periodic dynamics).

Figure 13.

(Left panels) Spatio-temporal evolution of the energy density for the KG equation using as initial condition a $(-,+)$ oscillon with frequency $\omega=0.4$ (panels (a) and (b)), $\omega=0.47$ (panels (c) and (d)), and $\omega=0.515$ (panels (e) and (f)). The right panels show the evolution for $z(t,0)$, i.e., at $x=0$, for the cases in the left panels.

Figure 13 Long description

The image consists of three pairs of graphs. The first pair (a) and (b) shows the spatio-temporal evolution and oscillation pattern for a specific frequency. Graph (a) is a heatmap with x-axis labeled 'x' ranging from -10 to 10 and y-axis labeled 't' ranging from 0 to 40, displaying energy density with a color scale from 0.2 to 1.8. Graph (b) is a line graph with x-axis labeled 't' ranging from 0 to 40 and y-axis labeled 'z(t,0)' ranging from -1 to 1, showing oscillations over time. The second pair (c) and (d) shows similar graphs for another frequency. Graph (c) is a heatmap with x-axis labeled 'x' ranging from -10 to 10 and y-axis labeled 't' ranging from 0 to 40, displaying energy density with a color scale from 0.5 to 2.5. Graph (d) is a line graph with x-axis labeled 't' ranging from 0 to 40 and y-axis labeled 'z(t,0)' ranging from -1 to 1.5, showing oscillations over time. The third pair (e) and (f) shows graphs for a third frequency. Graph (e) is a heatmap with x-axis labeled 'x' ranging from -10 to 10 and y-axis labeled 't' ranging from 0 to 100, displaying energy density with a color scale from 0.5 to 3.5. Graph (f) is a line graph with x-axis labeled 't' ranging from 0 to 100 and y-axis labeled 'z(t,0)' ranging from -1.5 to 1.5, showing oscillations over time.

Figure 14.

Spatio-temporal evolution of the energy density for the KG equation using as initial condition a $(+,+)$ and a $(-,+)$ oscillon with frequency $\omega=0.53$. Left (right) panel corresponds to the case when the initial condition is taken from the black (red) curve of Figure 1.

Figure 14 Long description

The image A shows a heat map with axes labeled x and t, ranging from negative 10 to positive 10 for x and 0 to 3.5 for t. The color scale on the right indicates values from negative 15 to positive 10. The image B shows another heat map with axes labeled x and t, ranging from negative 10 to positive 10 for x and 0 to 100 for t. The color scale on the right indicates values from 0.5 to 4.5.