Stability of source defects in oscillatory media

Patterns are ubiquitous in nature, and understanding their formation and their dynamical behavior is always challenging and of great interest. Examples include patterns in fluids (e.g., Rayleigh-Benard convection between two flat plates, Taylor-Couette flow between rotating cylinders, surface waves in hydrothermal fluid flows,…), as well as in nonlinear optics, oscillatory chemical reactions and excitable biological media. Many of them arise from linear instabilities of an homogenous equilibrium, having space, time, or space-time periodic coherent structures such as wave trains (spatially periodic travelling waves). In presence of boundaries or defects, complex patterns form and thus break the symmetry or the periodic structures. Below, I shall briefly discuss some defect structures and my recent work on the subject.

To fix the discussion, consider the following system of reaction-diffusion equations

$\displaystyle u_t = u_{xx} + f(u), \quad (x,t)\in\mathbb{R}\times\mathbb{R}^+, \quad u\in \mathbb{R}^n, \ \ \ \ \ (1)$

with ${f(u)}$ being sufficiently smooth.

Wave trains. Typically, the system exhibits one-parameter families of spatially-periodic travelling wave solutions or wave trains that are parametrized by their spatial wavenumber ${k}$ : namely,

$\displaystyle u(x,t) = u_\mathrm{wt}(kx-\omega_\mathrm{nl}(k) t;k)$

for ${k}$ in an open nonempty interval, where the profile ${u_\mathrm{wt}(\theta;k)}$ is ${2\pi}$ -periodic in ${\theta}$ and ${\omega=\omega_\mathrm{nl}(k)}$ denotes the temporal frequency or is often referred to as the nonlinear dispersion relation associated to the wave trains. Wave trains therefore propagate with the phase velocity ${c_\mathrm{p}=\omega_\mathrm{nl}(k)/k}$ , as one can directly check that wave trains ${u_\mathrm{wt}}$ solve the wave equation ${(\partial_t^2 - c_\mathrm{p}^2 \partial_x^2) u_\mathrm{wt} =0}$ . To visualize, one may think of the simplest possible example: the traveling sinusoidal wave ${e^{i (kx - \omega_\mathrm{nl}(k) t)}}$ , for some nonzero ${k}$ .

Group velocity. An interesting quantity associated with a wave train is its group velocity ${c_\mathrm{g}}$ , which is defined by the derivative of the nonlinear dispersion relation:

$\displaystyle c_\mathrm{g} = \frac{\mathrm{d}\omega_\mathrm{nl}(k)}{\mathrm{d} k}. \ \ \ \ \ (2)$

The group velocity turns out to be the speed of propagation of small localized wave-package perturbations along the wave train as a function of time ${t}$ . More precisely, for ${k}$ near ${k_0}$ , we may write

$\displaystyle u_\mathrm{wt}(kx-\omega t;k) = u_\mathrm{wt} \Big( k_0 x - \omega_0 t + (k-k_0) \Big[ x-\frac{\omega - \omega_0}{k-k_0} t\Big];k\Big) .$

That is, the “envelope” of the wave package at ${k_0}$ formally travels with the speed ${\lim_{k\rightarrow k_0}\frac{\omega - \omega_0}{k-k_0} = c_\mathrm{g}}$ , the group velocity speed. This can also be made rigorously; for instance, see Doelman-Sandstede-Scheel-Schneider.

Defects. Defect solutions in fact arise in many biological, chemical, and physical processes: examples are planar spiral waves, flip-flops in chemical reactions, and surface waves in hydrothermal fluid flows. They are characterized as interfaces between stable wave trains with possibly different wavenumbers on each side. Precisely, defects are an exact solution to (1) of the form

$\displaystyle u(x,t) = \bar{u}(x-c_\mathrm{d}t,t),$

where the defect profile ${\bar{u}(\xi,t)}$ is assumed to be periodic in ${t}$ and, for appropriate left and right wavenumbers ${k_\pm}$ , we have that

$\displaystyle \bar{u}(\xi,t) \rightarrow u_\mathrm{wt}\left(k_\pm\xi-(\omega_\mathrm{nl}(k_\pm)-c_\mathrm{d}k_\pm)t;k_\pm\right) \mbox{ as } \xi\rightarrow\pm\infty$

uniformly in ${t}$ so that the defect converges to, possibly, different, wave trains in the far field, that is, as ${x\rightarrow\pm\infty}$ . We remark that periodicity of the defect profile in time implies that ${\omega_\mathrm{nl}(k_\pm)-c_\mathrm{d}k_\pm = \omega_\mathrm{d}}$ , with ${\frac{2\pi}{\omega_\mathrm{d}}}$ being the time periodicity of the defect, and hence the defect velocity ${c_\mathrm{d}}$ is determined by the Rankine–Hugoniot condition

$\displaystyle c_\mathrm{d} = \frac{\omega_\mathrm{nl}(k_+)-\omega_\mathrm{nl}(k_-)}{k_+-k_-} .$

Depending on the relative sign of group velocities of the asymptotic wave trains on each side of the defect, Sandstede and Scheel have systematically classified them into four distinct types:

sinks:	exist for arbitrary ${k_\pm}$	when	${c_\mathrm{g}(k_-)>c_\mathrm{d}>c_\mathrm{g}(k_+)}$
transmission defects:	exist for arbitrary ${k_+=k_-}$	when	${c_\mathrm{g}(k_\pm)>c_\mathrm{d}}$ or ${c_\mathrm{g}(k_\pm)<c_\mathrm{d}}$
contact defects:	exist for arbitrary ${k_+=k_-}$	when	${c_\mathrm{g}(k_\pm)=c_\mathrm{d}}$
sources:	exist for unique ${k_\pm}$	when	${c_\mathrm{g}(k_-)<c_\mathrm{d}<c_\mathrm{g}(k_+)}$ .

This way, sinks can be thought of as passive interfaces that accommodate two colliding wave trains. Similarly, transmission and contact defects accommodate phase-shift dislocations within wave-train solutions. On the other hand, source defects occur only for discrete wavenumbers ${k_\pm}$ and therefore select the wavenumbers of the wave trains that emerge from the defect core into the surrounding medium: hence, they may be thought of as organizing the surrounding global dynamics, rather than the reverse. Yet, among the main non-characteristic varieties (sinks, transmission defects, and sources), source defects are the only type whose stability properties have not been understood mathematically.

Only recently, the team M. Beck, B. Sandstede, K. Zumbrun, and myself are able to settle this longstanding open problem: the nonlinear stability of spectrally stable source defects, which I shall now describe. Working in the co-moving frame, we study the perturbed source solution of the form

$\displaystyle u(x,t) = \bar u(x,t) + v(x,t),\ \ \ \ \ (3)$

leading to the following system for perturbation ${v}$ :

$\displaystyle v_t - Lv = \mathcal{O}(v^2), \qquad L: = \partial_x^2 + c_\mathrm{d} \partial_x + f_u(\bar{u}(x, t)). \ \ \ \ \ (4)$

The problem is to prove that ${v}$ remains small, if initially so (which of course appears impossible, since solutions to the heat equation with quadratic nonlinearity ${v^2}$ blow up in finite time), or rather to understand the large time dynamics of ${v}$ . Assuming that source defects are spectrally stable (to the linearized problem), there are several difficulties in proving the nonlinear stability. First, sources are time periodic solutions, and hence their spectrum has to be understood in the sense of Floquet via the the ${2\pi}$ -periodic map of the corresponding linearization. In addition, there are two eigenvalues at the origin, with corresponding eigenfunctions ${\bar u_x, \bar u_t}$ (corresponding to the space-time invariants), while the continuous spectrum touches the origin, leaving no spectral gap between the continuous spectrum and the imaginary axis. These difficulties are in fact not new, and have been treated elsewhere in similar contexts; for instance, Beck-Sandstede-Zumbrun treats time-periodic viscous Lax shocks with asymptotically constant end states. The present situation is much delicate, as asymptotic “states” are non constants, or precisely, spatially periodic wave trains.

What greatly troubles us is that localized perturbations can lead to a non-localized response, precisely due to the fact the group velocities on each side of the defect have opposite sign and point outward away from the defect core, thus forming a plateau-like structure in the phase dynamics. In particular, the ${L^p}$ norm of perturbations can in fact grow in time, for ${p<\infty}$ (only ${L^\infty}$ norm remains bounded). I will now detail this point.

First of all, due to the spacetime translation invariants, ${\bar u_x, \bar u_t}$ are in the kernel of ${L}$ . As a consequence, we need to take care of projections of ${v}$ onto the kernel of ${L}$ ; namely, we write perturbation ${v = \psi \bar u_x + \phi \bar u_t + w}$ , and aim to bound unknowns ${\psi, \phi, w}$ . As a matter of fact, it turns out to be more convenient to look for solutions of the form, in place of (3),

$\displaystyle u(x,t) = \bar u(x+ \psi(x,t),t+\phi(x,t)) + v(x,t),\ \ \ \ \ (5)$

for unknowns ${\psi, \phi, v}$ . A direct computation then yields, in place of (4),

$\displaystyle (\partial_t - L) (v - \psi \bar u_x - \phi \bar u_t)= \mathcal{O}(v^2, \psi_x^2, \phi_x^2). \ \ \ \ \ (6)$

We stress that the nonlinearity appears precisely in derivatives of the spacetime shift functions ${\phi, \psi}$ , which is crucial in the nonlinear iteration. Recall now that ${L}$ is the linearization around a time-periodic defect solution ${\bar u(x,t)}$ , which at the far fields are the asymptotic wave trains whose group velocities have opposite sign and point outward away from the defect core (i.e., ${x=0}$ ).

Loosely speaking, let us consider the following toymodel

$\displaystyle \phi_t + c\tanh\left(\frac{cx}{2}\right) \phi_x = \phi_{xx} + \phi_x^2, \qquad c>0 , \ \ \ \ \ (7)$

which is the simplest possible model that incorporates both the dynamical effect of the core and the correct far-field dynamics of the spacetime translation functions. Notice that the group velocities at far fields have opposite sign and point away from each other: ${-c < 0 < c}$ . The problem is reduced to study the large-time stability of the zero solution of (7). Here, we wish to obtain the precise large time dynamics of the phase solution ${\phi}$ . The Green function of the linearized problem (7) (around zero) is in fact explicit:

$\displaystyle G(x,y,t) = \frac{c}{4}\left[ \mathrm{errfn}\left(\frac{y-x+ct}{\sqrt{4t}}\right) - \mathrm{errfn}\left(\frac{y-x-ct}{\sqrt{4t}}\right) \right] e^{-c |y|}+ \mathrm{Gaussian},$

where ${\mathrm{errfn}(\cdot)}$ denotes the error function, and the Gaussian term behaves like ${t^{-1/2} e^{-(x-y\pm ct)/4t}}$ . That is, the two error functions precisely produce a plateau of constant height ${\frac{c}{4}}$ and of an expanding width of order ${\sqrt t}$ that spreads outward; this term arises because of the zero eigenvalue of the corresponding linearized problem. In particular, the ${L^p}$ norm of this plateau grows in time of order ${t^{1/2p}}$ . In the nonlinear analysis, we write ${\phi(x,t) = \phi^*(x + p(t),t) + \tilde \phi(x,t)}$ , where ${\phi^*(x+p(t),t)}$ denotes the leading plateau due to the error functions in the Green function, with an unknown function ${p(t)}$ carefully introduced to collect the height of the plateau, formally leaving the remainder ${\tilde \phi(x,t)}$ to solve

$\displaystyle \tilde \phi(x,t) = \mathrm{Gaussian} \star v_0 + \int_0^t \mathrm{Gaussian} \star \tilde \phi_x^2 \; ds.$

The nonlinear iteration follows similarly to the case of heat equations with nonlinearity ${\tilde \phi_x^2}$ .

Going back to the case of source defects, it’s crucial in our analysis to get the “right” Green function behavior for the linearized problem about a source defect; namely,

$\displaystyle G(x,t;y,s) = \bar{u}_x(x,t) E_1(x,t; y,s) + \bar{u}_t(x,t) E_2(x,t; y,s) + G_R(x,t; y,s) \ \ \ \ \ (8)$

with ${E_j}$ collecting the leading behavior (error functions, plus a Gaussian) and ${G_R}$ being the remainder which behaves as a derivative of a Gaussian. Recall that ${\bar u_x, \bar u_t}$ are two eigenfunctions associated with the two eigenvalues at the origin. Here, it is very crucial that the remainder ${G_R}$ is of order of a derivative of a Gaussian, since otherwise (i.e., behaving as a Gaussian) the solution would blowup as in the case of heat equations with nonlinearity ${v^2}$ ; see (6). The leading Green kernels ${E_1, E_2}$ correspond to the spacetime translations.

In view of (6), we write Duhamel’s principle

$\displaystyle \begin{aligned} v(x,t) &= \bar{u}_x \psi (x,t)+ \bar{u}_t \phi (x,t)+ \mbox{I.C.} + \int_0^t \int G(x,t;y,s) \mathcal{O}(v^2, \psi_x^2, \phi_x^2)(y,s) \; \mathrm{d} y\mathrm{d} s \end{aligned}$

in which I.C. denotes the contribution from initial data. Thanks to the Green function decomposition (8), the unknown spacetime shifts ${\psi, \phi}$ are constructed via

$\displaystyle \begin{aligned} \psi(x,t) : &= \mbox{I.C.} - \int_0^t \int E_1(x,t;y,s)\mathcal{O}(v^2, \psi_x^2, \phi_x^2)(y,s) \mathrm{d} y\mathrm{d} s \\ \phi(x,t) : &= \mbox{I.C.} - \int_0^t \int E_2(x,t;y,s) \mathcal{O}(v^2, \psi_x^2, \phi_x^2)(y,s) \mathrm{d} y\mathrm{d} s, \end{aligned}\ \ \ \ \ (9)$

which are corresponding projections of solutions onto the kernel of ${L}$ , leaving the remainder satisfying

$\displaystyle \begin{aligned} v(x,t) &= \mbox{I.C.} + \int_0^t \int G_R(x,t;y,s) \mathcal{O}(v^2, \psi_x^2, \phi_x^2)(y,s) \; \mathrm{d} y\mathrm{d} s. \end{aligned}$

Now, the standard nonlinear iteration follows. To conclude, our main result is roughly as follows:

Theorem 1 [Beck, Toan, Sandstede, Zumbrun, preprint 2018] Assume that source defects are spectrally stable. Then, for initial data ${u(x,0)}$ that are ${C^2}$ sufficiently close to the defect, the solution ${u(x,t)}$ of (1) exists globally in time, and there are functions ${\phi(x,t)}$ and ${\psi(x,t)}$ such that

$\displaystyle |u(x,t) - \bar{u}(x+ \psi(x,t),t+ \phi(x,t))| \lesssim \mathrm{Gaussian}(x,t).$

That is, perturbed solutions converge to a shifted defect solutions. In addition, the spacetime shifts are approximately an expanding plateau: namely, there are smooth functions ${\delta_\phi(t),\delta_\psi(t)}$ , which are exponentially close to constants, such that

$\displaystyle \begin{aligned} |\phi(x,t) - e(x,t) \delta_\phi(t)| + |\psi(x,t) - e(x,t)\delta_\psi(t)| &\lesssim (1+t)^{\frac12} \mathrm{Gaussian}(x,t) \end{aligned}$

for all ${t\ge 0}$ . Here, ${e(x,t)}$ is the expanding plateau:

$\displaystyle e(x,t) := \mathrm{errfn}\left(\frac{x-c_+ t}{\sqrt{4\mathrm{d_+}t}}\right) - \mathrm{errfn}\left(\frac{x - c_- t}{\sqrt{4\mathrm{d_-}t}}\right), \ \ \ \ \ (10)$

with group velocities ${c_\pm := c_\mathrm{g}(k_\pm)-c_\mathrm{d}.}$ This gives a rather complete picture of the dynamics near stable source defects.

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Stability of source defects in oscillatory media

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