I have two papers: one focuses on calculating osci...

Yes, it is feasible, but only as a controlled reduction of the full lattice problem, not as a direct one-to-one transplantation of the method in 2507.13276. The method is especially suitable if your goal is: “Given post-resonance lattice data, estimate the subsequent lifetime distribution of localized oscillon-like objects under a φ–χ coupling without evolving the full 3D lattice for the entire lifetime.”

The key caveat is that 2507.13276 is primarily an individual-oscillon, spherically symmetric, flat-spacetime study. It first analyzes χ growth on a fixed inhomogeneous φ-oscillon background, then performs full two-field radial simulations with initial conditions of the form

$$\phi(0,r)=2\psi(r),\qquad \chi(0,r)=\chi_0\,\bar\chi(r),\qquad \dot\phi(0,r)=\dot\chi(0,r)=0,$$

where $\psi(r)$ is a single-field oscillon profile and $\bar\chi(r)$ is the relaxed normalized χ profile obtained on that φ background. The paper emphasizes that the χ instability depends strongly on coupling strength, oscillon shape, particle escape from the finite localized region, and whether the oscillon remains large enough to sustain resonance. (2507.13276Decay and lifetime of oscillons .pdf) (2507.13276Decay and lifetime of oscillons .pdf) (2507.13276Decay and lifetime of oscillons .pdf)

Your proposed strategy is therefore viable, but I would formulate it as:

Extract a catalog of post-resonance localized objects from the 3D lattice; classify φ–χ spatial overlap object by object; fit physically meaningful φ and χ profiles only where justified; then evolve representative or all selected objects with coupled radial equations, while validating the radial reduction against local 3D patch simulations.

The mismatch issue you anticipate must be taken into account. In fact, it should become one of the central diagnostics of the project.

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1. What is directly feasible, and what is not

The part that is feasible is the following. You can use the finite-lattice oscillon extraction philosophy of 1907.00611: identify stabilized localized objects in the 3D lattice, fit their shapes, and then use those fitted objects as initial data for cheaper single-object simulations. That paper explicitly uses 3D lattice simulations to extract oscillon energies, amplitudes, and radii after stabilization, then uses spherically symmetric simulations to follow individual oscillons until decay. (1907.00611v2(1).pdf)

The part that needs modification is the χ treatment. In 2507.13276, χ is not simply fitted arbitrarily from the lattice. For the fixed-background instability analysis, they start χ from a convenient Gaussian seed and let it relax to the fastest-growing configuration; they argue the initial χ seed affects mainly the length of the early relaxation stage, not the final growth rate. (2507.13276Decay and lifetime of oscillons .pdf) In the full two-field simulations, they then use a normalized, time-averaged χ profile $\bar\chi(r)$ obtained from that fixed-background analysis. (2507.13276Decay and lifetime of oscillons .pdf)

So, for your lattice data, the rigorous approach is not “always fit χ to a prescribed profile.” It is:

$$\text{fit φ profile} \;\Rightarrow\; \text{compute/measure χ mode appropriate to that φ profile} \;\Rightarrow\; \text{compare with actual lattice χ profile}.$$

If the actual χ profile agrees with the relaxed mode, use it. If it does not, either evolve the actual radial χ profile or treat the local χ field as radiation/noise rather than as a bound profile.

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2. Minimum data needed from your lattice

You said you have $\phi$, $\chi$, and energy-density components at each lattice site. That is excellent, but for a true continuation of the equations of motion you also need the signed canonical momenta:

$$\dot\phi(\mathbf{x}),\qquad \dot\chi(\mathbf{x}).$$

Knowing the kinetic energy density gives only

$$K_\phi=\frac12\dot\phi^2,\qquad K_\chi=\frac12\dot\chi^2,$$

which loses the sign of $\dot\phi$ and $\dot\chi$. For profile-fitting studies at a turning point, one may set $\dot\phi=\dot\chi=0$, as in 2507.13276. But for continuation from an arbitrary post-resonance snapshot, signed momenta matter.

So the first practical requirement is:

$$\{\phi,\chi,\dot\phi,\dot\chi,\rho_\phi^{K},\rho_\phi^{G},\rho_\phi^{V}, \rho_\chi^{K},\rho_\chi^{G},\rho_\chi^{V},\rho_{\rm int}\} \quad \text{at } t=t_* .$$

If you do not have $\dot\phi,\dot\chi$, then choose $t_*$ near a local extremum of the relevant object, or reconstruct the momenta from several closely spaced snapshots. Otherwise the radial simulations will represent an idealized ensemble, not a faithful continuation.

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3. Object identification on the post-resonance lattice

Start by building an object catalog from the lattice. Do not identify oscillons only from $|\phi|$ or $|\chi|$. Use energy-density clustering.

A direct generalization of 1907.00611 is useful. That paper defines oscillons as connected regions with local energy density at least about ten times the spatial average, and further requires at least one point in the object with overdensity above a stronger threshold to reject small fluctuation regions. (1907.00611v2(1).pdf)

For your two-field case, define

$$\rho_{\rm tot} = \rho_\phi^{K} +\rho_\phi^{G} +\rho_\phi^{V} +\rho_\chi^{K} +\rho_\chi^{G} +\rho_\chi^{V} +\rho_{\rm int},$$

and also the sector-resolved densities

$$\rho_\phi=\rho_\phi^{K}+\rho_\phi^{G}+\rho_\phi^{V}, \qquad \rho_\chi=\rho_\chi^{K}+\rho_\chi^{G}+\rho_\chi^{V}.$$

Then construct three catalogs:

$$\mathcal{C}_{\rm tot}: \Delta\rho_{\rm tot}>\Delta_{\rm cut},$$ $$\mathcal{C}_{\phi}: \Delta\rho_{\phi}>\Delta_{\phi,{\rm cut}},$$ $$\mathcal{C}_{\chi}: \Delta\rho_{\chi}>\Delta_{\chi,{\rm cut}}.$$

The thresholds should be varied, for example $\Delta=5,10,20$, to estimate systematic uncertainty. The high-core requirement, analogous to the $\Delta\rho>50$ condition in 1907.00611, should also be retained to avoid classifying noise or outgoing radiation as oscillons. (1907.00611v2(1).pdf)

For each connected object, compute:

$$\mathbf{x}_{\phi,c} = \frac{\int_{\mathcal{C}}\mathbf{x}\,\rho_\phi\,d^3x} {\int_{\mathcal{C}}\rho_\phi\,d^3x}, \qquad \mathbf{x}_{\chi,c} = \frac{\int_{\mathcal{C}}\mathbf{x}\,\rho_\chi\,d^3x} {\int_{\mathcal{C}}\rho_\chi\,d^3x}.$$

Then define the displacement

$$d_{\phi\chi}=|\mathbf{x}_{\phi,c}-\mathbf{x}_{\chi,c}|.$$

This displacement is one of the most important quantities in your problem.

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4. Classify localized structures before fitting

Each object should be assigned to one of four classes.

Class A: φ-dominated oscillon with negligible χ

This is a normal φ oscillon with little local χ energy:

$$\frac{E_\chi^{\rm loc}}{E_\phi^{\rm loc}}\ll 1, \qquad d_{\phi\chi}\ \text{irrelevant or poorly defined}.$$

For these objects, you should fit only the φ profile initially. χ should be initialized either from the actual lattice noise level or from a controlled seed. Then use the instability-band calculation of 2507.13276 to determine whether χ will later grow.

This class is physically important because 2507.13276 shows that χ resonance can turn off when the φ oscillon becomes too small, and that not every oscillon is destroyed by the external field. (2507.13276Decay and lifetime of oscillons .pdf)

Class B: co-located φ–χ composite object

Here both fields are localized at roughly the same center:

$$d_{\phi\chi}\lesssim \alpha R_\phi, \qquad \alpha\sim 0.3\text{--}0.5,$$

and the overlap is significant. Define an overlap integral such as

$$\mathcal{O}_{\phi\chi} = \frac{ \int W(\mathbf{x})\,\phi^2(\mathbf{x})\chi^2(\mathbf{x})\,d^3x }{ \left[\int W(\mathbf{x})\,\phi^4(\mathbf{x})\,d^3x\right]^{1/2} \left[\int W(\mathbf{x})\,\chi^4(\mathbf{x})\,d^3x\right]^{1/2} }.$$

This is the cleanest case for applying a two-field radial simulation.

Class C: φ oscillon with displaced χ lump

This is the case where φ and χ are both localized nearby but not concentric. A single radial profile centered at either $\mathbf{x}_{\phi,c}$ or $\mathbf{x}_{\chi,c}$ is generally not reliable. The coupling term is local, so the backreaction depends strongly on overlap:

$$E_{\rm int}^{\rm loc} = \int_{\mathcal{C}} g\phi^2\chi^2\,d^3x.$$

If $d_{\phi\chi}\sim R_\phi$, a radial reduction will artificially increase or decrease the interaction. These cases require either a local 3D patch simulation or a two-center approximation.

Class D: χ-localized structure without φ

In the model of 2507.13276, χ has no nonlinear self-interaction and does not support its own oscillon when decoupled. The paper explicitly treats φ as the oscillon-forming sector and χ as an external scalar field without self-interactions. (2507.13276Decay and lifetime of oscillons .pdf) Therefore, a χ lump with no φ at the same location is usually not a χ oscillon. It is more likely to be outgoing radiation, a wave packet, or a transient resonance remnant.

Such χ-only objects should not be fitted as long-lived oscillons unless your model gives χ its own oscillon-supporting potential. They should instead be treated as part of the χ radiation bath that can later pass through φ oscillons.

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5. Fit the φ profile first

For each φ-localized candidate, construct a radial profile around $\mathbf{x}_{\phi,c}$:

$$\phi_{\rm rad}(r,t_*)= \langle \phi(\mathbf{x},t_*)\rangle_{|\mathbf{x}-\mathbf{x}_{\phi,c}|=r}.$$

But do not fit a single snapshot blindly. Oscillons have a breathing mode: 1907.00611 observes oscillations of amplitude and radius in opposite phase, and notes that the extracted profile depends on the extraction time. (1907.00611v2(1).pdf) Ideally, use several snapshots over a few oscillon periods and extract the envelope, central frequency, amplitude, and radius.

Possible φ profile families:

$$\phi(r,t)\approx \phi_{\rm vac}+A_\phi e^{-r^2/(2R_\phi^2)}\cos(\omega_\phi t+\theta_\phi),$$

or, for flatter-top objects,

$$\phi(r,t)\approx \phi_{\rm vac}+A_\phi f_{\rm flat}(r;R_\phi,\Delta R)\cos(\omega_\phi t+\theta_\phi).$$

In 2507.13276, the oscillon profile $\psi(r)$ is obtained from a radial boundary-value problem, and Gaussian profiles are good mainly in the thick-wall regime, while flatter profiles appear in the thin-wall regime. (2507.13276Decay and lifetime of oscillons .pdf) Therefore, you should not assume Gaussianity for all objects.

For each fit, record:

$$A_\phi,\quad R_\phi,\quad \omega_\phi,\quad E_\phi^{\rm loc},\quad I_\phi,\quad \text{fit residual},\quad \text{ellipticity/asphericity}.$$

Reject or separately treat objects with large asphericity.

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6. Fit or reconstruct the χ profile conditionally

For χ, the correct treatment depends on class.

For Class A, where χ is negligible near a φ oscillon, do not force a χ profile. Instead initialize χ using one of three controlled choices:

$$\chi(r,0)=0,$$ $$\chi(r,0)=\chi_{\rm noise}\,\xi(r),$$

or

$$\chi(r,0)=\chi_0\bar\chi_{\rm eig}(r),$$

where $\bar\chi_{\rm eig}$ is the fastest-growing profile obtained by solving the linearized χ equation on the extracted φ background. This follows the logic of 2507.13276, where an initial Gaussian χ seed relaxes to the relevant growing configuration on a fixed φ background. (2507.13276Decay and lifetime of oscillons .pdf)

For Class B, compare the actual lattice χ profile to the relaxed eigenmode profile:

$$\chi_{\rm rad}(r,t_*) \quad \text{vs.} \quad \bar\chi_{\rm eig}(r).$$

Use the actual lattice radial profile if it has a good fit and energy closure. Use the eigenmode profile only for idealized scans.

For Class C, do not compress the system into one radial χ profile unless $d_{\phi\chi}\ll R_\phi$. The displacement affects the coupling energy and hence the backreaction.

For Class D, treat χ as radiation unless your χ sector has its own oscillon-forming self-interaction.

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7. Perform fixed-background instability analysis before full evolution

Before evolving two nonlinear fields, compute the χ growth rate on each extracted φ profile. This is the most important bridge to 2507.13276.

For quartic coupling

$$\mathcal{L}_{\rm int}=-g\phi^2\chi^2,$$

the equations used in 2507.13276 are

$$\ddot{\phi}-\nabla^2\phi+V_\phi'(\phi)+2g\chi^2\phi=0,$$ $$\ddot{\chi}-\nabla^2\chi+m_\chi^2\chi+2g\phi^2\chi=0.$$

The paper then studies χ on an inhomogeneous oscillon background and finds distinct regimes: no growth when enhancement cannot beat escape, indirect weak growth from inhomogeneous-profile mode mixing for some attractive couplings, ordinary narrow parametric resonance, and tachyonic growth. (2507.13276Decay and lifetime of oscillons .pdf)

For each extracted φ oscillon, compute

$$\Re\mu(g,m_\chi;A_\phi,R_\phi,\omega_\phi).$$

Then determine whether this object lies in a stable region or an unstable region. This step matters because 2507.13276 finds that small oscillons may fail to sustain parametric resonance due to limited size, so χ growth may terminate before φ reaches its single-field collapse threshold. (2507.13276Decay and lifetime of oscillons .pdf)

This gives a fast first estimate:

$$\Gamma_\chi(E)\sim \Re\mu(E),$$

and

$$\tau(E_{\rm ini}) \sim \sum_i\frac{1}{\Re\mu_i} + \tau_{\rm single}\!\left[\min(E_0^{\rm osc},E_{\rm ini})\right],$$

which is the same qualitative lifetime decomposition used in 2507.13276: coupled χ-resonance stages followed by ordinary single-field oscillon decay once the object exits the χ instability band. (2507.13276Decay and lifetime of oscillons .pdf)

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8. Full nonlinear radial evolution

For objects that pass your sphericality and co-location cuts, evolve the coupled radial equations:

$$\ddot\phi-\frac{\partial^2\phi}{\partial r^2} -\frac{2}{r}\frac{\partial\phi}{\partial r} +V_\phi'(\phi) +2g\chi^2\phi =0,$$ $$\ddot\chi-\frac{\partial^2\chi}{\partial r^2} -\frac{2}{r}\frac{\partial\chi}{\partial r} +m_\chi^2\chi +2g\phi^2\chi =0.$$

If cosmic expansion is still relevant, use the FLRW form instead:

$$\ddot\phi+3H\dot\phi-\frac{1}{a^2}\nabla^2\phi +V_\phi'(\phi)+2g\chi^2\phi=0,$$ $$\ddot\chi+3H\dot\chi-\frac{1}{a^2}\nabla^2\chi +m_\chi^2\chi+2g\phi^2\chi=0.$$

Use local flat evolution only when

$$R_{\rm osc}\ll H^{-1}, \qquad T_{\rm osc}\ll H^{-1},$$

and when the lifetime segment you simulate is short compared with the Hubble time, or when expansion can be absorbed into rescaled variables.

Boundary treatment is crucial. 2507.13276 uses an adiabatic damping boundary for both fields in full two-field simulations. (2507.13276Decay and lifetime of oscillons .pdf) 1907.00611 uses a truncation/absorbing strategy to avoid reflected scalar waves and checks energy conservation after accounting for removed radiation. (1907.00611v2(1).pdf) (1907.00611v2(1).pdf) For your case, use a large radial box plus absorbing layer for both φ and χ, and track the absorbed energy separately.

Monitor

$$E_\phi^{\rm loc}(t),\quad E_\chi^{\rm loc}(t),\quad E_{\rm int}^{\rm loc}(t),\quad E_{\rm rad}^{\rm absorbed}(t),\quad E_{\rm total}(t).$$

The local energies should be time-averaged over several oscillation periods. 2507.13276 averages energy and charge over windows longer than the oscillation period but shorter than the adiabatic decay time. (2507.13276Decay and lifetime of oscillons .pdf)

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9. Define lifetime carefully in the coupled system

In a two-field system, “oscillon lifetime” is ambiguous unless you define it explicitly. I recommend using three related definitions.

φ-lifetime

The time until the φ localized object disappears:

$$\Delta\rho_\phi^{\rm core}(t)<\Delta_{\rm death}$$

for at least $N$ consecutive oscillation periods, or until the fitted φ amplitude/radius/frequency leaves the stable oscillon branch.

total localized-energy lifetime

The time until

$$E_{\rm loc}^{\rm tot}(t) = E_\phi^{\rm loc}+E_\chi^{\rm loc}+E_{\rm int}^{\rm loc}$$

falls below a fixed fraction of its initial value or below an overdensity threshold.

resonance lifetime

The time interval during which

$$\frac{dE_\chi^{\rm loc}}{dt}>0$$

and $\chi$ growth is exponential. This is not necessarily the same as oscillon destruction. 2507.13276 explicitly finds that χ can grow quickly, backreact on φ, terminate the resonance, propagate away, and leave behind a φ oscillon that can recover with reduced energy. (2507.13276Decay and lifetime of oscillons .pdf)

That distinction is very important for your study.

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10. The φ–χ mismatch issue must be included

Your anticipated issue is real and important.

The reason is simple: for quartic coupling, the local interaction energy and backreaction scale as

$$\rho_{\rm int}\sim g\phi^2\chi^2,$$

and the χ effective mass is

$$m_{\chi,\rm eff}^2(\mathbf{x},t) = m_\chi^2+2g\phi^2(\mathbf{x},t).$$

Therefore, χ growth, χ trapping, and χ backreaction are controlled by the spatial overlap between φ and χ. A φ oscillon without local χ can still produce χ later if it lies in an instability band, but the onset time depends on the seed amplitude. A χ lump without φ generally cannot be interpreted as a coupled oscillon in the model of 2507.13276, because χ lacks the nonlinear self-interaction needed to support an independent oscillon. (2507.13276Decay and lifetime of oscillons .pdf)

So yes, the issue must be incorporated as a formal classification criterion. I would use at least three diagnostics:

$$f_{\chi|\phi} = \frac{\int_{|\mathbf{x}-\mathbf{x}_{\phi,c}|<R_{\rm cut}} \rho_\chi\,d^3x} {\int_{\rm box}\rho_\chi\,d^3x},$$ $$\eta_{\chi/\phi} = \frac{E_\chi^{\rm loc}}{E_\phi^{\rm loc}},$$ $$\mathcal{O}_{\phi\chi} = \frac{ \int W\phi^2\chi^2\,d^3x }{ \left(\int W\phi^4\,d^3x\right)^{1/2} \left(\int W\chi^4\,d^3x\right)^{1/2} }.$$

Then decide the simulation type:

$$\mathcal{O}_{\phi\chi}\approx 1: \quad \text{radial two-field simulation is justified}.$$ $$\mathcal{O}_{\phi\chi}\ll 1,\quad E_\phi^{\rm loc}\gg E_\chi^{\rm loc}: \quad \text{φ-only oscillon plus χ seed/noise}.$$ $$\mathcal{O}_{\phi\chi}\ll 1,\quad E_\chi^{\rm loc}\gg E_\phi^{\rm loc}: \quad \text{χ radiation/wave packet, not an oscillon}.$$ $$d_{\phi\chi}\sim R_\phi: \quad \text{local 3D patch required}.$$

This classification will prevent you from artificially creating composite φ–χ oscillons by fitting χ around every φ oscillon.

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11. Validation tests

A rigorous project should include the following checks.

First, reproduce the single-field lifetime pipeline. Turn off $g$, fit φ objects from the lattice, evolve radial φ-only configurations, and check whether lifetimes and energy decay behavior are stable under changes of fitting time, lattice spacing, radial box size, and boundary layer. This mirrors the logic of 1907.00611, where the 3D lattice gives initial shapes and radial simulations give lifetimes. (1907.00611v2(1).pdf)

Second, reproduce the fixed-background χ growth rates of 2507.13276 for a benchmark potential and coupling. You should recover the qualitative regions: no growth, weak indirect growth for some attractive couplings, narrow parametric resonance, and tachyonic growth. (2507.13276Decay and lifetime of oscillons .pdf)

Third, compare three χ initializations:

$$\chi=\chi_{\rm lattice}(r), \qquad \chi=\chi_0\bar\chi_{\rm eig}(r), \qquad \chi=\chi_{\rm noise}\xi(r).$$

The measured lifetime should not depend strongly on arbitrary χ fitting choices, except through the expected logarithmic delay in the onset of exponential growth.

Fourth, run local 3D patch simulations for representative problematic cases: displaced φ–χ centers, asymmetric φ oscillons, nearby oscillon pairs, and χ-only lumps passing near φ oscillons. This is necessary because both papers note limitations of the single-object spherical approximation: 1907.00611 neglects asymmetry and oscillon–oscillon interactions in the radial stage, while 2507.13276 focuses on individual oscillons and explicitly does not include collective effects from a population. (1907.00611v2(1).pdf) (2507.13276Decay and lifetime of oscillons .pdf)

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12. Recommended final workflow

A clean research workflow would be:

1. Choose $t_*$ after the resonance stage, preferably with several nearby snapshots over multiple oscillation periods.

2. Build φ, χ, and total-energy object catalogs using overdensity clustering.

3. Compute object centers, radii, ellipticities, sector energies, and φ–χ overlap.

4. Classify objects into φ-only, co-located φ–χ, displaced φ–χ, and χ-only classes.

5. Fit φ profiles for all φ-localized objects; extract $A_\phi,R_\phi,\omega_\phi,E_\phi,I_\phi$.

6. For each φ profile, compute χ instability bands and $\Re\mu$ on the inhomogeneous φ background.

7. Fit χ only where the lattice χ profile is co-located and physically consistent with a bound or resonant χ mode.

8. Evolve coupled radial equations for spherical, isolated, co-located objects.

9. Evolve φ-only plus χ-seed radial equations for φ oscillons with no χ counterpart.

10. Treat χ-only lumps as radiation unless your χ sector supports independent oscillons.

11. Validate ambiguous/displaced/asymmetric cases with local 3D patch simulations.

12. Report not one lifetime but a lifetime distribution conditioned on object class, overlap, energy, coupling, and χ mass.

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Bottom line

Your plan is feasible, but I would slightly revise the central assumption:

Do not fit χ everywhere simply because you have a χ lattice distribution. Fit χ only after classifying its spatial relation to φ and verifying that it corresponds to a localized bound/resonant mode rather than radiation.

The mismatch between φ and χ localized structures is not a nuisance to ignore; it is a physical diagnostic. It determines whether the local object behaves like a φ oscillon with a χ decay channel, a genuine two-field composite, a displaced interacting pair, or merely a φ oscillon plus a χ radiation bath.

Yes—this is feasible, but the clean interpretation is:

You can use the end-of-resonance 3D data to construct an ensemble of effective radial $1+1$ two-field simulations for individual, isolated, nearly spherical $\phi$-oscillons.
This is not equivalent to continuing the whole 3D lattice population. It is a controlled single-object reduction, where the $1+1$ coordinate is really $(t,r)$ with the 3D spherical term $2r^{-1}\partial_r$, not a Cartesian one-dimensional theory.

The method has strong literature support in its core pieces. The 1907.00611 paper extracts oscillon shapes from $3+1$ lattice simulations after stabilization, fits them with spherically symmetric Gaussian profiles, and then uses those fitted shapes as initial data for radial lifetime simulations. (1907.00611v2(1).pdf) The 2507.13276 paper then extends the single-oscillon radial strategy to a coupled $\phi$-$\chi$ system with $g\phi^2\chi^2$, computes $\chi$ instability bands on an inhomogeneous oscillon background, and performs full radial two-field simulations. (2507.13276Decay and lifetime of oscillons .pdf) What I add below is the data-analysis layer needed because you already have real 3D $\phi,\chi,\dot\phi,\dot\chi$ lattice data, rather than the idealized initial profiles used in those papers.

Physical picture

A $\phi$-oscillon is the localized object. In the type of model studied in 2507.13276, $\phi$ has the nonlinear potential that supports oscillons, while $\chi$ is an external scalar without nonlinear self-interactions, so $\chi$ does not by itself form a stable oscillon in the same sense. (2507.13276Decay and lifetime of oscillons .pdf)

The $\phi$-oscillon acts as a localized, time-periodic pump for $\chi$. In a radial approximation,

$$\phi(t,r)\simeq 2\psi_\omega(r)\cos(\omega t)$$

makes the $\chi$ equation look like a parametric-resonance problem with a spatially localized driving region,

$$\ddot\chi-\chi''-\frac{2}{r}\chi' +\left[m_\chi^2+2g\phi^2(t,r)\right]\chi=0 .$$

The key point is that $\chi$ growth depends on both the coupling and the shape/size of the $\phi$-oscillon. 2507.13276 finds that finite oscillon size can prevent parametric resonance: a small oscillon may have a large central amplitude but still be too spatially narrow to amplify the relevant $\chi$ modes efficiently. (2507.13276Decay and lifetime of oscillons .pdf)

Once $\chi$ grows, it backreacts on $\phi$ through the term $2g\chi^2\phi$. But this backreaction is local in overlap: a $\chi$ lump far from a $\phi$-oscillon does not strongly affect that oscillon. This is why your anticipated mismatch—$\phi$ localized here, $\chi$ localized elsewhere—is not a technical nuisance; it is part of the physics and must be tracked.

Important caution about “fitting the $\chi$ field”

You should not automatically fit $\chi$ to an oscillon-like Gaussian at every $\phi$-oscillon site. The physically motivated options are:

1. If $\chi$ is small near a $\phi$-oscillon: treat it as a seed/noise field, not as a pre-existing localized $\chi$ profile.

2. If $\chi$ is coherently localized around the same center as the $\phi$-oscillon: fit $\chi$ as a resonant mode supported by the $\phi$-oscillon background.

3. If $\chi$ energy is present but the field average cancels angularly: do not fit the angular average $\langle\chi\rangle_\Omega$ alone; the coupling depends on $\chi^2$, so an incoherent $\chi$ bath can still backreact even when the mean field is nearly zero.

4. If $\chi$ is localized at a different position from $\phi$: that is not a co-centered two-field oscillon initial condition. Either exclude it from the single-oscillon radial sample or treat it as a separate 3D/multi-center problem.

This is also consistent with the 1907.00611 paper’s observation that field-amplitude plots alone can miss oscillons that are near the middle of their oscillation; energy density is more reliable than instantaneous field value alone. (1907.00611v2(1).pdf)

Detailed research strategy

1. Decide what your $1+1$ problem represents

Use the radial 3D equations, not a Cartesian 1D model:

$$\ddot\phi-\phi''-\frac{2}{r}\phi' +\frac{\partial V_\phi}{\partial\phi} +2g\phi\chi^2=0,$$ $$\ddot\chi-\chi''-\frac{2}{r}\chi' +m_\chi^2\chi+2g\phi^2\chi=0.$$

If your original simulation is in an expanding background, decide whether the local oscillon problem can be treated in Minkowski space. This is often reasonable when $H R_{\rm osc}\ll 1$, but you should document the approximation. If expansion is retained, include Hubble friction and physical/comoving coordinate factors consistently.

2. Identify $\phi$-oscillon candidates from energy density, not field value alone

For each lattice site you have

$$\rho_\phi = \frac12\dot\phi^2 +\frac12|\nabla\phi|^2 +V_\phi(\phi),$$ $$\rho_\chi = \frac12\dot\chi^2 +\frac12|\nabla\chi|^2 +\frac12m_\chi^2\chi^2,$$

and, if you want the total local energy,

$$\rho_{\rm int}=g\phi^2\chi^2.$$

Use connected components of $\rho_\phi$, or of the localized oscillon energy density, to identify candidate $\phi$-oscillons. The 1907.00611 paper used an energy overdensity criterion, roughly $\rho/\langle\rho\rangle\gtrsim 10$, plus a stronger peak condition to avoid mistaking fluctuations for oscillons. (1907.00611v2(1).pdf) In your two-field case, use this only as a starting point; avoid selecting candidates from total energy alone because outgoing $\chi$ radiation can also produce high-energy regions.

For each candidate, compute:

$$E_{\phi,i}=\int_{B_i} d^3x\,\rho_\phi, \qquad E_{\chi,i}=\int_{B_i} d^3x\,\rho_\chi, \qquad E_{{\rm int},i}=\int_{B_i} d^3x\,g\phi^2\chi^2 .$$

Also compute shape diagnostics: inertia tensor, principal-axis ratios, distance to nearest oscillon, and angular multipoles. Only nearly spherical, isolated objects should be put into the radial $1+1$ sample.

3. Cross-match $\phi$ and $\chi$ structures

For every $\phi$-oscillon center ${\bf x}_{\phi,i}$, check whether $\chi$ is actually co-located.

A useful overlap diagnostic is

$${\cal O}_{\phi\chi}^{(i)} = \frac{ \int_{B_i} d^3x\, \left(\rho_\phi-\bar\rho_\phi\right)_+ \left(\rho_\chi-\bar\rho_\chi\right)_+ }{ \left[ \int_{B_i} d^3x\, \left(\rho_\phi-\bar\rho_\phi\right)_+^2 \right]^{1/2} \left[ \int_{B_i} d^3x\, \left(\rho_\chi-\bar\rho_\chi\right)_+^2 \right]^{1/2} }.$$

Then classify objects:

Case	Meaning	Recommended treatment
$\phi$-lump, negligible local $\chi$	ordinary $\phi$-oscillon with small $\chi$ seed	radial two-field run with measured small $\chi$, or controlled seed/noise
co-located $\phi$ and $\chi$	true local two-field interaction	fit both fields and evolve coupled radial EOM
$\chi$-lump, no $\phi$-oscillon	likely $\chi$ radiation/bound packet, not a $\phi$-oscillon lifetime problem	do not count as oscillon decay; analyze separately
offset $\phi$ and $\chi$ peaks	non-spherical/multi-center configuration	radial co-centered simulation is invalid; use 3D or exclude

This step is essential because the coupled decay rate depends on local overlap. Transplanting a $\chi$ profile onto a $\phi$-oscillon where no $\chi$ exists would artificially enhance the decay.

4. Build radial profiles from the 3D lattice

For each accepted $\phi$-oscillon center, define spherical shells and compute radial shell averages:

$$\bar f(r_j) = \frac{1}{N_j} \sum_{{\bf x}_n\in {\rm shell}\,j} f({\bf x}_n),$$

for

$$f\in \{\phi,\dot\phi,\chi,\dot\chi,\rho_\phi,\rho_\chi,\rho_{\rm int}\}.$$

Also compute RMS profiles:

$$f_{\rm rms}(r_j) = \sqrt{ \left\langle f^2\right\rangle_{\Omega,j} },$$

and angular coherence, for example

$$C_\chi(r_j) = \frac{ |\langle \chi\rangle_{\Omega,j}| }{ \sqrt{\langle\chi^2\rangle_{\Omega,j}} }.$$

If $C_\chi\simeq 1$, $\chi$ is coherent and a radial field profile is meaningful. If $C_\chi\ll1$, the angular average of $\chi$ is not a faithful initial condition even though $\chi^2$ may be dynamically important.

5. Fit the $\phi$ profile first

There are two levels of rigor.

The simpler 1907.00611-style fit is Gaussian:

$$\phi(r,t_e) = \phi_{\rm vac} + A_\phi \exp\left[-\frac{r^2}{2R_\phi^2}\right],$$

with a separate fit for $\dot\phi(r,t_e)$ if the snapshot is not at a turning point. The 1907.00611 paper explicitly used Gaussian fitted amplitudes and radii from 3D simulations as initial conditions for radial lifetime simulations. (1907.00611v2(1).pdf)

The more physical fit, closer to 2507.13276, is to solve the oscillon profile equation for your potential and fit

$$\phi(r,t_e) = \phi_{\rm vac} + 2\psi_\omega(r)\cos\theta_\phi,$$ $$\dot\phi(r,t_e) = -2\omega\psi_\omega(r)\sin\theta_\phi .$$

Here $\omega$ fixes the oscillon energy and shape. 2507.13276 uses numerically solved single-field profiles $\psi_\omega(r)$ and then evolves the coupled system from those profiles. (2507.13276Decay and lifetime of oscillons .pdf)

Since you have momenta, do not set $\dot\phi=0$ unless your snapshot is actually at a local extremum of the oscillon oscillation. The $\dot\phi=0$ initialization is justified in the papers only when the fitted time is chosen at a local minimum/maximum of the central field. (1907.00611v2(1).pdf)

6. Fit $\chi$: recommended hierarchy

Best option: project $\chi$ onto the dominant resonant mode of the fitted $\phi$-oscillon

For each fitted $\phi$-oscillon, freeze

$$\phi_{\rm bg}(t,r)=2\psi_\omega(r)\cos(\omega t),$$

then solve the linearized $\chi$ equation,

$$\ddot\chi-\chi''-\frac{2}{r}\chi' +\left[m_\chi^2+2g\phi_{\rm bg}^2(t,r)\right]\chi=0.$$

This gives you the dominant Floquet/growing mode $u_\chi(r,t)$ and growth rate ${\rm Re}\,\mu$. 2507.13276 does exactly this type of fixed-background calculation and shows that the inhomogeneous profile changes the instability bands. (2507.13276Decay and lifetime of oscillons .pdf)

Then fit your actual lattice $\chi,\dot\chi$ data by phase-space projection:

$$\chi_{\rm lat}(r,t_e) \simeq a\,u_1(r,t_e)+b\,u_2(r,t_e)+\chi_{\rm res}(r),$$ $$\dot\chi_{\rm lat}(r,t_e) \simeq a\,\dot u_1(r,t_e)+b\,\dot u_2(r,t_e)+\dot\chi_{\rm res}(r),$$

where $(u_1,\dot u_1)$ and $(u_2,\dot u_2)$ are two independent real solutions spanning the relevant Floquet phase. The fit should use the radial 3D inner product,

$$\langle f,g\rangle = 4\pi\int_0^{R_{\rm fit}} dr\,r^2 f(r)g(r).$$

This is the most faithful way to use your $\chi$ field and momentum data.

Practical option: use the relaxed $\bar\chi(r)$ profile of 2507.13276

2507.13276 initializes the full two-field simulation as

$$\phi(0,r)=2\psi(r), \qquad \chi(0,r)=\chi_0\bar\chi(r), \qquad \dot\phi(0,r)=\dot\chi(0,r)=0,$$

where $\bar\chi(r)$ is the normalized time-averaged $\chi$ profile relaxed on a fixed oscillon background. (2507.13276Decay and lifetime of oscillons .pdf) They also show that starting from a Gaussian $\chi$ seed, most non-growing components dissipate and the field relaxes to the mode that grows fastest; the initial Gaussian details mainly affect the relaxation time. (2507.13276Decay and lifetime of oscillons .pdf)

For your data, improve this by setting $\chi_0$ from projection onto the measured $\chi$ field:

$$\chi_0 = \frac{ 4\pi\int dr\,r^2 \bar\chi(r)\, \bar\chi_{\rm lat}(r) }{ 4\pi\int dr\,r^2 \bar\chi^2(r) }.$$

Use this only when the local $\chi$ configuration is coherent and approximately co-centered.

Least rigorous option: empirical Gaussian fit

A co-centered Gaussian,

$$\chi(r,t_e) = \chi_{\infty} + A_\chi \exp\left[-\frac{r^2}{2R_\chi^2}\right],$$

is acceptable as a diagnostic or robustness test, but it is not the best physical model unless the actual $\chi$ profile is visibly Gaussian and coherent. Unlike $\phi$, $\chi$ is not generally a self-supported oscillon profile in the model of 2507.13276.

7. Initialize and evolve the radial two-field simulation

For each accepted object, initialize

$$\phi(0,r)=\phi_{\rm fit}(r),\qquad \dot\phi(0,r)=\pi_{\phi,{\rm fit}}(r),$$ $$\chi(0,r)=\chi_{\rm fit}(r),\qquad \dot\chi(0,r)=\pi_{\chi,{\rm fit}}(r).$$

Use a large radial box and an absorbing/damping boundary. 2507.13276 imposes an adiabatic damping boundary on both fields in its radial two-field simulations. (2507.13276Decay and lifetime of oscillons .pdf) 1907.00611 also emphasizes boundary treatment/truncation because reflected waves can contaminate long lifetime simulations. (1907.00611v2(1).pdf)

Monitor

$$E_\phi(t)=4\pi\int_0^{r_{\max}}dr\,r^2\rho_\phi,$$ $$E_\chi(t)=4\pi\int_0^{r_{\max}}dr\,r^2\rho_\chi,$$ $$E_{\rm int}(t)=4\pi\int_0^{r_{\max}}dr\,r^2g\phi^2\chi^2.$$

2507.13276 computes the total energy including the interaction contribution and notes that the monitored energy should remain constant until outgoing waves leave the monitored region. (2507.13276Decay and lifetime of oscillons .pdf)

8. Define lifetime consistently

Use more than one lifetime diagnostic:

$$t_{\rm dec}^{(E)}: \quad E_\phi^{\rm loc}(t) \text{ falls below a critical value or threshold},$$ $$t_{\rm dec}^{(A)}: \quad \text{central oscillation amplitude disappears},$$ $$t_{\rm dec}^{({\rm profile})}: \quad \text{no localized Gaussian/oscillon profile can be fitted}.$$

The 2507.13276 picture is that $\chi$ resonance can dominate while the oscillon energy is above a critical instability-band boundary $E_0^{\rm osc}$; once the oscillon becomes small enough, $\chi$ growth can stop and the remaining object may return to single-field-like slow decay. (2507.13276Decay and lifetime of oscillons .pdf) Their order-of-magnitude lifetime estimate is

$$\tau(E_{\rm ini}) \sim \sum_i \frac{1}{{\rm Re}\,\mu_i} + \tau_{\rm single} \!\left( \min(E_0^{\rm osc},E_{\rm ini}) \right),$$

with the first term representing stages of $\chi$ resonance and the second the remaining single-field lifetime. (2507.13276Decay and lifetime of oscillons .pdf)

For your project, report a distribution of lifetimes over many extracted oscillons, not just one representative object.

Does the $\phi$-$\chi$ mismatch need to be taken into account?

Yes, absolutely.

The mismatch determines whether the radial two-field initial condition is physically meaningful. The coupling is $g\phi^2\chi^2$, so what matters is not just whether both fields have peaks somewhere in the box, but whether their localized energies overlap in space.

A $\phi$-oscillon with no local $\chi$ profile is still a valid oscillon. Its subsequent coupled decay should be studied with the actual small local $\chi$ fluctuation as the seed. By contrast, a $\chi$-localized structure with no corresponding $\phi$-oscillon is not a $\phi$-oscillon lifetime problem; in the spectator-field setup it will usually disperse unless it is trapped by a nearby $\phi$ background. If the two peaks are offset, a co-centered radial simulation is not reliable.

This issue is especially important because 2507.13276 finds that $\chi$ can grow, backreact, propagate away, and then leave a surviving $\phi$-oscillon that resumes single-field-like evolution. (2507.13276Decay and lifetime of oscillons .pdf) So the spatial relationship between $\phi$ and $\chi$ at the extraction time affects not only the instantaneous decay rate, but also the late-time remnant energy.

What is literature-supported and what is my proposed extension?

The following parts are directly supported by the two papers:

The extraction of individual oscillons from $3+1$ lattice data, fitting them with spherical profiles, and using those fits as initial conditions for radial lifetime simulations follows 1907.00611. (1907.00611v2(1).pdf)

The coupled two-field radial equations, the $g\phi^2\chi^2$ interaction, the instability-band analysis of $\chi$ on an inhomogeneous oscillon background, and the full radial two-field evolution follow 2507.13276. (2507.13276Decay and lifetime of oscillons .pdf)

The use of a relaxed normalized $\bar\chi(r)$ mode as the initial $\chi$ profile follows 2507.13276. (2507.13276Decay and lifetime of oscillons .pdf)

The following parts are my recommended extension for your specific dataset:

Projecting the measured $\chi,\dot\chi$ lattice fields onto the dominant $\chi$ Floquet mode of each fitted $\phi$-oscillon.

Classifying $\phi$-only, $\chi$-only, co-located, and offset objects before doing radial simulations.

Using angular coherence and RMS diagnostics to avoid losing incoherent $\chi^2$ backreaction when $\langle\chi\rangle_\Omega$ is small.

These extensions are not arbitrary; they are the natural way to combine the two papers’ methods with the richer information you have from the two-field 3D lattice snapshot.

Yes. You can study the late decay of individual oscillons by reducing the 3D lattice data to a radial two-field problem, but only after you have verified that the object is approximately isolated, approximately spherical, and that the relevant parts of $\phi$ and $\chi$ are spatially overlapping. Strictly speaking, this is not an ordinary planar $1+1$ dimensional system; it is a spherically symmetric $3+1$ problem written as a $t+r$ simulation, so the radial Laplacian contains $2\partial_r/r$, and energies are integrated with $4\pi r^2dr$.

The key point is that in the model of 2507.13276 the $\chi$ field is not an independent oscillon-forming field; it is an external scalar with no nonlinear self-interaction, coupled through $g\phi^2\chi^2$. In that paper, $\phi$ supports oscillons, while $\chi$ is amplified or localized only because it sees the oscillating, spatially localized $\phi$ background as a time-dependent effective mass. The equations used there are the two-field equations with coupling terms $2g\chi^2\phi$ and $2g\phi^2\chi$. (2507.13276Decay and lifetime of oscillons .pdf)

1. Physical picture of fitting $\chi$

Think of a $\phi$-oscillon as a localized, periodically oscillating pump. The $\chi$ field sees

$$m_{\chi,\mathrm{eff}}^2(t,\mathbf{x}) = m_\chi^2+2g\phi^2(t,\mathbf{x}).$$

If $\chi$ overlaps the oscillon core, this effective mass is both localized and periodic. Depending on $g$, $m_\chi$, the oscillon frequency, and the oscillon radius, $\chi$ may have an unstable mode that grows by parametric or tachyonic resonance. If $\chi$ is far from the $\phi$ oscillon, the local overlap $g\phi^2\chi^2$ is small, so that particular $\phi$ oscillon is not efficiently decaying into that $\chi$ lump.

This is why the position issue is not a technical detail; it is physically central. A co-centered $\phi$-$\chi$ radial simulation assumes that the interaction energy density

$$\rho_{\rm int}(\mathbf{x}) = g\phi^2(\mathbf{x})\chi^2(\mathbf{x})$$

is localized in the same region as the $\phi$ oscillon. If your 3D data show a $\phi$ lump at one site and a $\chi$ lump elsewhere, forcing them into the same radial profile would artificially overestimate the coupling and the decay rate.

2. What “fit $\chi$ on a fixed $\phi$ background” means

This phrase does not mean that you first take the 3D $\chi$ field and fit it while keeping the actual lattice $\phi$ field fixed. In 2507.13276, the authors first construct an idealized single $\phi$-oscillon background,

$$\phi_{\rm bg}(t,r)=2\psi(r)\cos(\omega t),$$

and then solve the linear $\chi$ equation on this prescribed background:

$$\ddot{\chi} - \chi'' - \frac{2}{r}\chi' + \Big[m_\chi^2+2g\phi_{\rm bg}^2(t,r)\Big]\chi = 0.$$

Here $\phi_{\rm bg}$ is fixed by hand at every time step, so there is no backreaction from $\chi$ onto $\phi$ during this stage. The purpose is to find the unstable $\chi$ mode, if one exists.

In the paper, the initial $\chi$ seed is chosen as a simple Gaussian,

$$\chi(0,r)=\chi_0 e^{-r^2/R_\chi^2}, \qquad \dot{\chi}(0,r)=0.$$

They explain that most non-growing components dissipate, while the unstable component relaxes to the fastest-growing configuration on the given $\phi$ background. The precise seed parameters affect mainly the duration of the initial relaxation stage, not the final growth rate. (2507.13276Decay and lifetime of oscillons .pdf)

So the correct interpretation is:

$$\boxed{ \text{fixed-$\phi$ calculation} = \text{a linear instability/eigenmode calculation for $\chi$, not a direct fit of a random $\chi$ snapshot.} }$$

After this fixed-background step, one extracts a normalized relaxed profile $\bar{\chi}(r)$. Then the full two-field simulation is initialized with

$$\phi(0,r)=2\psi(r), \qquad \chi(0,r)=\chi_0\bar{\chi}(r), \qquad \dot{\phi}(0,r)=\dot{\chi}(0,r)=0,$$

or with the corresponding phase-corrected profiles if you want to use the actual lattice phase. This is exactly the structure used in the two-field simulation stage of 2507.13276. (2507.13276Decay and lifetime of oscillons .pdf)

3. How to define the centers of $\phi$ and $\chi$

Use energy density, not field amplitude alone, to locate oscillons. A field amplitude snapshot can miss an oscillon if the field happens to be crossing the vacuum value, while its kinetic energy is large. The 1907.00611 paper explicitly identifies oscillons by energy overdensity rather than only by field amplitude; they define oscillon candidates as connected regions whose local energy density is much larger than the box average and additionally require a stronger peak to reject random fluctuations. (1907.00611v2(1).pdf)

For your data, define

$$\rho_\phi = \frac12\dot{\phi}^2 + \frac12|\nabla\phi|^2 + V_\phi(\phi),$$ $$\rho_\chi = \frac12\dot{\chi}^2 + \frac12|\nabla\chi|^2 + V_\chi(\chi),$$

and, if available,

$$\rho_{\rm int}=g\phi^2\chi^2.$$

For locating a $\phi$-oscillon, use $\rho_\phi$ or a smoothed version of it. For locating a possible composite $\phi+\chi$ object, also inspect $\rho_{\rm tot}=\rho_\phi+\rho_\chi+\rho_{\rm int}$.

A practical definition is:

$$\mathbf{x}_\phi = \frac{\sum_{i\in C_\phi} w_{\phi,i}\mathbf{x}_i} {\sum_{i\in C_\phi} w_{\phi,i}}, \qquad w_{\phi,i}= \max(\rho_{\phi,i}-\rho_{\phi,\rm bg},0),$$

where $C_\phi$ is the connected high-energy region around the $\phi$ oscillon. For periodic boxes, compute this barycenter using circular coordinates in each direction to avoid errors across the boundary. Define $\mathbf{x}_\chi$ analogously using $w_{\chi,i}$.

You should also record the peak positions,

$$\mathbf{x}^{\rm peak}_\phi = \arg\max_{\mathbf{x}}\rho_\phi(\mathbf{x}), \qquad \mathbf{x}^{\rm peak}_\chi = \arg\max_{\mathbf{x}}\rho_\chi(\mathbf{x}),$$

because the difference between peak and barycenter is a useful diagnostic of asymmetry.

4. How to decide whether $\phi$ and $\chi$ are at the same position

For every $\phi$-oscillon candidate, compute the periodic distance

$$d_{\phi\chi} = |\mathbf{x}_\phi-\mathbf{x}_\chi|_{\rm periodic}.$$

Then compare it with the fitted radii $R_\phi$ and $R_\chi$. A strict co-centered criterion could be

$$d_{\phi\chi} < \max(2\Delta x,\;0.3\min(R_\phi,R_\chi)),$$

where $\Delta x$ is the lattice spacing. A looser “overlap but not exactly co-centered” criterion is

$$d_{\phi\chi} < \sqrt{R_\phi^2+R_\chi^2}.$$

But distance alone is not enough. Also compute an overlap integral, for example

$$\mathcal{O}_{\phi\chi} = \frac{ \sum_i w_{\phi,i}w_{\chi,i} }{ \sqrt{ \left(\sum_i w_{\phi,i}^2\right) \left(\sum_i w_{\chi,i}^2\right) } }.$$

Then classify the objects:

$$\boxed{ \begin{array}{ll} \text{Type A:} & \phi\text{-lump with no local }\chi\text{ overlap},\\[2mm] \text{Type B:} & \chi\text{-lump with no local }\phi\text{ overlap},\\[2mm] \text{Type C:} & \phi\text{ and }\chi\text{ co-localized or strongly overlapping}. \end{array} }$$

Only Type C objects are good candidates for a direct two-field radial simulation. Type A objects should be treated as $\phi$-oscillons with a small $\chi$ seed or with the local residual $\chi$ fluctuation. Type B objects should not be interpreted as $\chi$-oscillons unless your $\chi$ sector has its own nonlinear potential supporting localized solutions. In the 2507.13276 setup, $\chi$ does not independently support oscillon solutions. (2507.13276Decay and lifetime of oscillons .pdf)

If $d_{\phi\chi}$ is comparable to $R_\phi$, a one-center radial simulation is not valid. The $\phi^2\chi$ and $\chi^2\phi$ coupling terms are then angle-dependent, and the system cannot be represented by a single radial coordinate $r$. You would need a local 3D simulation, or at least a higher-dimensional reduced simulation.

5. Fitting the $\phi$ profile

First identify the $\phi$ oscillon using $\rho_\phi$. Then choose a fitting time. Ideally, fit when the $\phi$ field at the center is close to a turning point,

$$\dot{\phi}(t,\mathbf{x}_\phi)\simeq 0,$$

because then the field profile is largest and easiest to fit. The 1907.00611 paper fits oscillon shapes at times when the field at the center is near an extremum and then fits a spherically symmetric Gaussian profile. (1907.00611v2(1).pdf)

A simple Gaussian fit is

$$\phi(t_0,r) = \phi_{\rm vac} + A_\phi \exp\!\left[-\frac{r^2}{2R_\phi^2}\right],$$

or, depending on convention,

$$\phi(t_0,r) = \phi_{\rm vac} + A_\phi \exp\!\left[-\frac{r^2}{R_\phi^2}\right].$$

Use one convention consistently. The 1907.00611 paper uses a Gaussian amplitude-radius parametrization and fits points within a few radii of the oscillon center. (1907.00611v2(1).pdf)

For potentials like the sextic example in 2507.13276, a better fit may be a numerical oscillon profile $\psi_\omega(r)$ labeled by the oscillon frequency $\omega$. That paper characterizes profiles by central amplitude $\psi(0)$ and a radius $R$ defined by $\psi(R)=\psi(0)/e$, and notes that profiles can be flat-top or Gaussian-like depending on $\omega$. (2507.13276Decay and lifetime of oscillons .pdf)

If your snapshot is not at a turning point, use both field and momentum. Fit

$$\phi(t_0,r)-\phi_{\rm vac} = A_\phi f_\phi(r)\cos\theta_\phi,$$ $$\dot{\phi}(t_0,r) = -\omega_\phi A_\phi f_\phi(r)\sin\theta_\phi.$$

Equivalently, fit two coefficients,

$$\phi(t_0,r)-\phi_{\rm vac}=C_\phi f_\phi(r), \qquad \dot{\phi}(t_0,r)=D_\phi f_\phi(r),$$

and then reconstruct

$$A_\phi=\sqrt{C_\phi^2+\left(\frac{D_\phi}{\omega_\phi}\right)^2}, \qquad \tan\theta_\phi = -\frac{D_\phi}{\omega_\phi C_\phi}.$$

This is more faithful than forcing $\dot{\phi}=0$ at an arbitrary lattice time.

6. Direct fitting of the $\chi$ field from your 3D data

If $\chi$ is co-localized with the $\phi$ oscillon, you can fit $\chi$ around the same center. I recommend using the $\phi$ center $\mathbf{x}_\phi$ as the radial origin, because $\phi$ is the object sourcing the localized effective mass for $\chi$.

First subtract any local background:

$$\delta\chi_i=\chi_i-\chi_{\rm bg}, \qquad \delta\dot{\chi}_i=\dot{\chi}_i-\dot{\chi}_{\rm bg}.$$

Then define $r_i=|\mathbf{x}_i-\mathbf{x}_\phi|$, bin the data radially, and fit

$$\chi(t_0,r) = C_\chi f_\chi(r), \qquad \dot{\chi}(t_0,r) = D_\chi f_\chi(r).$$

Possible choices for $f_\chi(r)$ are:

$$f_\chi(r)=\exp\!\left[-\frac{r^2}{2R_\chi^2}\right],$$

or, better,

$$f_\chi(r)=\bar{\chi}(r),$$

where $\bar{\chi}(r)$ is the relaxed $\chi$ profile obtained from the fixed-$\phi$ background calculation described below.

A robust least-squares objective is

$$\mathcal{L}_\chi = \sum_i W_i \left[ \chi_i-C_\chi f_\chi(r_i) \right]^2 + \alpha \sum_i W_i \left[ \dot{\chi}_i-D_\chi f_\chi(r_i) \right]^2,$$

where $W_i$ can be chosen from the localized energy weight, for example

$$W_i=\max(\rho_{\chi,i}-\rho_{\chi,\rm bg},0),$$

or from the total localized energy if you want the fit to emphasize the interacting core. The constant $\alpha$ simply balances field and momentum units; a natural choice is $\alpha\sim 1/\Omega_\chi^2$, where $\Omega_\chi$ is the dominant $\chi$ oscillation frequency.

After fitting, check the residual:

$$\epsilon_\chi = \frac{ \sum_i W_i\left[\chi_i-C_\chi f_\chi(r_i)\right]^2 }{ \sum_i W_i\chi_i^2 }.$$

If $\epsilon_\chi$ is large, the local $\chi$ distribution is not well described by a single spherical mode. In that case, do not use a radial two-field simulation as your main result.

7. The fixed-$\phi$ way to obtain $\bar{\chi}(r)$

This is the most physically motivated way to fit $\chi$ if your goal is to study decay through the coupling.

For each fitted $\phi$ oscillon:

1. Build a smooth radial $\phi$ background:

$$\phi_{\rm bg}(t,r) = \phi_{\rm vac}+A_\phi f_\phi(r)\cos(\omega_\phi t),$$

or, in the convention of 2507.13276,

$$\phi_{\rm bg}(t,r)=2\psi(r)\cos(\omega t).$$

2. Solve the linear radial $\chi$ equation:

$$\ddot{\chi} - \chi'' - \frac{2}{r}\chi' + \left[ m_\chi^2+2g\phi_{\rm bg}^2(t,r) \right]\chi = 0.$$

3. Start from a small seed, for example

$$\chi(0,r)=\chi_{\rm seed}\exp\!\left[-\frac{r^2}{R_{\rm seed}^2}\right], \qquad \dot{\chi}(0,r)=0.$$

This seed is not meant to be the physical fitted $\chi$ profile. It is just a probe. In 2507.13276, the Gaussian seed relaxes: nonresonant components dissipate and the growing mode remains. (2507.13276Decay and lifetime of oscillons .pdf)

4. During the evolution, compute a norm such as

$$N_\chi(t) = \left[ 4\pi\int_0^{r_{\rm core}}r^2\chi^2(t,r)\,dr \right]^{1/2}.$$

Fit the growth rate by

$$N_\chi(t)\propto e^{\mu_\chi t}.$$

5. After the initial transient, define the normalized radial shape. To avoid cancellation from oscillations, use an RMS profile:

$$\bar{\chi}(r) = \frac{ \left\langle \chi^2(t,r)/N_\chi^2(t)\right\rangle_t^{1/2} }{ \left[ 4\pi\int r^2 \left\langle \chi^2(t,r)/N_\chi^2(t)\right\rangle_t dr \right]^{1/2} }.$$

This gives

$$4\pi\int r^2\bar{\chi}^2(r)\,dr=1.$$

You can then use $\bar{\chi}(r)$ directly in the full two-field simulation. There is no need to force it into a Gaussian unless you want a compact parametrization.

The important diagnostic is whether the fixed-background calculation actually gives $\mu_\chi>0$. If $\mu_\chi\le 0$, that $\phi$ oscillon does not efficiently amplify a localized $\chi$ mode for your chosen $g$ and $m_\chi$. 2507.13276 emphasizes that the resonance depends strongly on the oscillon shape and may fail for small oscillons even if the central amplitude looks large, because the effective resonance region can be too small. (2507.13276Decay and lifetime of oscillons .pdf)

8. What to do if $\phi$ and $\chi$ are at different positions

You should explicitly include this issue in the analysis.

Case 1: $\phi$ localized, $\chi$ absent near the same center

For that oscillon, use a $\phi$-only fit and then either:

$$\chi(0,r)=0,$$

or initialize $\chi$ with a very small seed/noise level corresponding to the local $\chi$ fluctuation in the $\phi$ core. Then ask whether the fitted $\phi$ oscillon would later amplify $\chi$ by the fixed-background instability method.

Do not import a distant $\chi$ lump into this radial simulation.

Case 2: $\chi$ localized, $\phi$ absent

In the 2507.13276 model, this is not a two-field oscillon unless $\chi$ has its own nonlinear potential. It is more likely a radiative packet, a remnant of resonance, or a localized fluctuation. You can catalog it, but it should not be used as the $\chi$ component of a $\phi$-oscillon decay simulation.

Case 3: $\phi$ and $\chi$ centers close

If

$$d_{\phi\chi}\ll R_\phi,R_\chi,$$

you may use a common center. I would choose either $\mathbf{x}_\phi$ or the total-energy center

$$\mathbf{x}_{\rm tot} = \frac{ E_\phi\mathbf{x}_\phi+E_\chi\mathbf{x}_\chi }{ E_\phi+E_\chi }.$$

Then radially bin both fields around that common center and fit both profiles.

Case 4: moderate offset

If

$$0.3R_\phi \lesssim d_{\phi\chi}\lesssim R_\phi,$$

the radial approximation is questionable. In this regime, decompose $\chi$ around the $\phi$ center into spherical harmonics and check whether the $l=0$ component dominates. If the monopole fraction is not large, a radial simulation will erase the dipole/asymmetric structure and produce misleading backreaction.

A simple diagnostic is

$$f_{l=0}^{(\chi)} = \frac{ \int dr\, r^2 |\chi_{l=0}(r)|^2 }{ \int d^3x\,|\chi(\mathbf{x})|^2 }.$$

Use the radial reduction only if $f_{l=0}^{(\chi)}$ is large, for example $\gtrsim 0.8$.

9. Solving the coupled equations once both profiles are fitted

Once you have a co-centered pair, solve the radial two-field equations. In physical units,

$$\ddot{\phi} - \phi'' - \frac{2}{r}\phi' + \frac{\partial V_\phi}{\partial\phi} + 2g\phi\chi^2 = 0,$$ $$\ddot{\chi} - \chi'' - \frac{2}{r}\chi' + m_\chi^2\chi + 2g\phi^2\chi = 0.$$

For the dimensionless sextic model used in 2507.13276, the simulated equations are

$$\ddot{\tilde{\phi}} - \nabla^2\tilde{\phi} + \tilde{\phi} - 4\tilde{\phi}^3 + 6\tilde{g}_6\tilde{\phi}^5 + 2\tilde{g}\tilde{\chi}^2\tilde{\phi} = 0,$$ $$\ddot{\tilde{\chi}} - \nabla^2\tilde{\chi} + \tilde{m}_\chi^2\tilde{\chi} + 2\tilde{g}\tilde{\phi}^2\tilde{\chi} = 0,$$

with the radial $\nabla^2=\partial_r^2+2\partial_r/r$. (2507.13276Decay and lifetime of oscillons .pdf)

The initial conditions should be chosen in one of two ways.

Option A: phase-corrected projection from the actual 3D lattice

This is the most faithful to your data:

$$\phi(0,r)=\langle \phi(t_0,\mathbf{x})\rangle_{|\mathbf{x}-\mathbf{x}_c|=r},$$ $$\dot{\phi}(0,r)=\langle \dot{\phi}(t_0,\mathbf{x})\rangle_{|\mathbf{x}-\mathbf{x}_c|=r},$$ $$\chi(0,r)=\langle \chi(t_0,\mathbf{x})\rangle_{|\mathbf{x}-\mathbf{x}_c|=r},$$ $$\dot{\chi}(0,r)=\langle \dot{\chi}(t_0,\mathbf{x})\rangle_{|\mathbf{x}-\mathbf{x}_c|=r}.$$

Use this if the 3D object is clean, co-centered, and roughly spherical.

Option B: idealized fitted profiles

Use

$$\phi(0,r)=\phi_{\rm vac}+A_\phi f_\phi(r)\cos\theta_\phi,$$ $$\dot{\phi}(0,r)=-\omega_\phi A_\phi f_\phi(r)\sin\theta_\phi,$$ $$\chi(0,r)=A_\chi f_\chi(r)\cos\theta_\chi,$$ $$\dot{\chi}(0,r)=-\Omega_\chi A_\chi f_\chi(r)\sin\theta_\chi.$$

If you want to reproduce the 2507.13276 strategy, choose a turning-point initialization,

$$\dot{\phi}(0,r)=\dot{\chi}(0,r)=0,$$

with

$$\phi(0,r)=2\psi(r), \qquad \chi(0,r)=\chi_0\bar{\chi}(r).$$

In that paper, the amplitude $\chi_0$ is taken small; its precise value mainly shifts the time at which nonlinear backreaction becomes important, while the exponential growth rate is controlled by the instability. (2507.13276Decay and lifetime of oscillons .pdf)

10. Boundary conditions and numerical checks

Use regularity at the origin:

$$\phi'(t,0)=0, \qquad \chi'(t,0)=0.$$

At the outer boundary, use an absorbing or damping layer. 2507.13276 uses an adiabatic damping boundary for both fields, adding a damping term outside the measurement region, and checks that this agrees with absorbing boundaries or sufficiently large boxes. (2507.13276Decay and lifetime of oscillons .pdf)

Monitor

$$E_\phi = 4\pi\int dr\,r^2 \left[ \frac12\dot{\phi}^2 + \frac12\phi'^2 + V_\phi(\phi) \right],$$ $$E_\chi = 4\pi\int dr\,r^2 \left[ \frac12\dot{\chi}^2 + \frac12\chi'^2 + V_\chi(\chi) \right],$$ $$E_{\rm int} = 4\pi\int dr\,r^2g\phi^2\chi^2,$$

and

$$E_{\rm tot}=E_\phi+E_\chi+E_{\rm int}.$$

The two-field paper finds that $\chi$ growth may stop when the $\phi$ oscillon loses enough energy to leave the $\chi$ instability band. They define a critical oscillon energy $E_0^{\rm osc}$ associated with the boundary of the instability region. Below that, the nonperturbative $\chi$ growth can shut off, and the remaining oscillon may return to single-field-like decay. (2507.13276Decay and lifetime of oscillons .pdf)

11. Recommended workflow for your project

1. Build a catalog of $\phi$-localized objects using $\rho_\phi$, not field amplitude alone.

2. For each object, compute $\mathbf{x}_\phi$, $R_\phi$, ellipticity, and local energy.

3. Independently build a catalog of $\chi$-localized objects using $\rho_\chi$.

4. Pair $\phi$ and $\chi$ candidates using center separation and overlap integral.

5. Classify every object as $\phi$-only, $\chi$-only, or co-localized $\phi+\chi$.

6. Fit the $\phi$ profile. Use a Gaussian profile if you want a simple phenomenological description; use the numerical oscillon profile family $\psi_\omega(r)$ if you want closer contact with 2507.13276.

7. For each fitted $\phi$ oscillon, run the fixed-$\phi$ linear $\chi$ calculation to determine whether an unstable localized $\chi$ mode exists and to extract $\bar{\chi}(r)$.

8. Compare the actual lattice $\chi$ profile near the $\phi$ center with $\bar{\chi}(r)$. If the projection is strong and residual is small, use the actual $\chi$ amplitude. If not, treat the local $\chi$ as noise/radiation and scan over plausible $\chi_0$.

9. Run full radial two-field simulations only for objects that pass the isolation, sphericality, and co-location tests.

10. Validate by convergence tests in $\Delta r$, $\Delta t$, box size, damping layer, and by comparing the fitted radial initial energy with the original 3D local energy.

12. Which parts are literature-supported?

The broad strategy is literature-supported.

The 1907.00611 paper supports the idea of extracting oscillon shapes from 3D lattice simulations and then using fitted spherical profiles as initial data for radial lifetime simulations. It identifies oscillons through energy overdensity, fits Gaussian profiles, and then performs spherically symmetric simulations to estimate lifetimes. (1907.00611v2(1).pdf)

The 2507.13276 paper supports the fixed-$\phi$-background procedure for $\chi$: evolve $\chi$ on a prescribed inhomogeneous oscillon background, let transients dissipate, extract the growing/relaxed $\chi$ mode, and then run the full coupled two-field simulation with backreaction. (2507.13276Decay and lifetime of oscillons .pdf) (2507.13276Decay and lifetime of oscillons .pdf)

The specific center-matching diagnostics I proposed—energy-weighted barycenters, overlap integral, monopole fraction, residual tests, and explicit classification into $\phi$-only, $\chi$-only, and co-localized objects—are my recommended extensions for your two-field 3D dataset. They are not spelled out in the two papers, but they are the natural consistency checks required before reducing a genuinely 3D two-field configuration to a one-center radial simulation.