Free surface lump wave dynamics of a saturated superfluid Helium film with nontrivial boundary condition at the substrate surface

Abhik Mukherjee

REVIEW 4 major objections 4 minor 57 references

Free surface lump wave dynamics of a saturated superfluid Helium film with nontrivial boundary condition at the substrate surface

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A weak superfluid leak into the substrate forces the film's free surface waves into the forced Kadomtsev–Petviashvili I equation, with lump waves that can stop, reverse, and oscillate.

desk verdict Novel fKP derivation for leaky superfluid films is undone by algebraic errors in the lump dynamics sections; the derivation may survive, the advertised physics does not. read the letter →

arxiv 1908.01512 v2 pith:VFQMZG44 submitted 2019-08-05 nlin.PS

Abhik Mukherjee This is my paper · ORCID

classification nlin.PS MSC 35Q5335Q5176B1576B0776Y05

keywords superfluidheliumfilmsforcedKadomtsev-Petviashviliequationlumpwavesfreesurfacethirdsoundporoussubstrateboundaryconditionreductiveperturbationmethodhydrodynamics

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the free surface of a saturated superfluid helium-4 film obeys a forced Kadomtsev–Petviashvili I (KP-I) equation whenever a weak, localized downward superflow leaks into the substrate. The forcing term is set by the leak velocity at the bottom boundary, so the same equation that describes unforced lump waves now carries the leak's effect. If true, the result gives an analytic handle on how a porous substrate changes surface-wave speed and amplitude in a system where surface tension dominates, which matters because measurements of third sound in such films are easier when wave speeds can be tuned. The paper works out exact lump solutions for time-only leaks and perturbative lump solutions for rapidly varying space-time leaks, and shows the leak can make a lump stop, reverse, or shake at the origin.

What carries the argument

The load-bearing object is the bottom boundary condition $\partial\varphi/\partial z|_{z=0} = C(x,y,t)$, expanded at order $\epsilon^{5/2}$, together with the recursive solution of Laplace's equation in the film. That recursion produces a velocity potential whose second series is driven by $C$; feeding it into the kinematic and dynamic free-surface conditions under the scalings $\bar x = \epsilon^{1/2}\xi$, $\bar y = \epsilon y$, $\bar t = \epsilon^{3/2}t$ yields the forced KP-I equation. The exact time-only solution works through the shift $\bar U = U + \int f\,dT$ with $d^2 a/dT^2 = 6f(T)$; the space-time solution works through a fast-time perturbation series in $\epsilon_1$ that separates an unforced KP-I part $V_0$ from leak-driven corrections $G_1, G_2$.

What would settle it

Measure or compute the actual downward superflow $C(x,y,t)$ for a saturated film on a porous substrate from pore size, pore pressure, and the film's chemical potential gradient, and check whether $C/\epsilon^{5/2}$ stays finite and bounded as $\epsilon\to 0$; if the leakage scales differently, the forced KP-I equation is not the governing equation. Alternatively, in a constant-leak experiment the model predicts a lump that reverses at $T_c = 1/C_k$ and a linearly decaying background, so observing no reversal or a non-linear background decay at the predicted time would falsify the central claim.

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Extended reading notes

Core claim

Starting from incompressible, irrotational superfluid hydrodynamics with surface tension and van der Waals forces, and replacing the usual impermeable bottom boundary condition by $\partial\varphi/\partial z = C(x,y,t)$ at $z=0$, the paper derives, through reductive perturbation theory, that the first-order surface displacement $a_1$ (rescaled to $U$) satisfies $\partial U/\partial T + 6U\,\partial U/\partial X + \partial^3 U/\partial X^3 - 3\,\partial^2/\partial Y^2 \int U\, dX = -f$, with $f$ proportional to the first-order downward superfluid velocity $C^{(1)}$. For a saturated film thicker than the critical thickness $d_c = \sqrt{\rho\alpha/\sigma}$, the coefficients make this a forced KP-I equation. The paper then shows that when $f$ depends only on time, a shift $U \mapsto U + \int f\,dT$ and a time-dependent translation turn the forced equation into the unforced KP-I equation, so the exact one-lump solution is inherited with a modified trajectory; in particular a constant leak gives a lump that decelerates, halts at $T_c = 1/C_k$, reverses, and rides on a secularly decaying background. For space-time dependent leaks that vary rapidly compared with the lump evolution, a two-time-scale perturbation expansion yields $U = V_0 + \epsilon_1 V_{0X}(1+\cos\Gamma\tau) + \tfrac{1}{2}\epsilon_1^2 V_{0XX} + O(\epsilon_1^3)$, with the forcing tied self-consistently to the initial data.

Load-bearing premise

The derivation requires that the downward superfluid velocity into the substrate be weak in the precise sense $C = \epsilon^{5/2}C^{(1)}$ — small enough that the film depth stays essentially constant, yet large enough to appear as a forcing term at the KP order — and the paper gives no independent physical estimate from pore geometry or pressure to justify that ordering.

Editorial extensions

If this is right

A constant downward superflow of strength $C_k$ predicts a lump that turns around at $T_c = 1/C_k$, so the leak's magnitude can be read off from the reversal time.
The secular background term $-T C_k$ means a persistent leak slowly lowers the film height, so at long times the lump rides on a falling baseline.
A sinusoidal leak $f = \sin(\Omega T)$ makes the lump's velocity along $X$ oscillate and its height at the origin oscillate with $\Omega$, imprinting the forcing function on the wave trajectory.
For rapidly varying space-time leaks, the first-order correction is $V_{0X}(1+\cos\Gamma\tau)$, so the leak's effect appears as a periodic distortion of the lump's spatial derivative rather than a change of its core profile.
Because the unforced part $V_0$ satisfies KP-I, the stability of lump solutions carries over to the forced problem whenever the leak is weak and satisfies the assumed ordering.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The ordering $C = \epsilon^{5/2} C^{(1)}$ is a postulate with no independent physical estimate; if a realistic porous-substrate calculation gave a different power of $\epsilon$, the same boundary condition would lead to a different forced equation, so the specific form of the forced KP-I equation is a prediction to be checked against pore-scale parameters.
The exact time-only solution suggests a testable diagnostic: measuring the turning time $T_c$ for a film on a porous substrate would directly measure the scaled leak strength, and observing whether the background decays linearly in $T$ would discriminate this model from a simple damping term.
The same reductive scheme could be applied to thin films below the critical thickness, where surface tension is negligible; there one would expect a forced KdV equation with a leak-induced phase shift, connecting this analysis to the (1+1)-dimensional shallow-water leak results the paper cites.
The paper's admitted neglect of pore size, pore pressure, and time-dependent van der Waals coefficients means the forcing function $f$ is effective rather than derived; calibrating $f$ from substrate properties would turn the reversal and shaking predictions into quantitative ones.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Bottom line: this is a genuinely new physical set-up -- saturated 4He film with a downward leak into the substrate -- and the reductive perturbation derivation of the forced KP-I equation looks plausible and is worth taking seriously. But the lump dynamics sections, which are the advertised pay-off, contain algebraic errors that invalidate the main claims. The derivation follows a standard route from [15] and the trick of removing f(T) by a shift and accelerating frame works, so that part may survive correction. Credit where it's due: the choice of scaling C ~ eps^{5/2} is at least consistent with the standard KP ordering, though no physical estimate is offered to justify it.

The problems begin in Sec. III.A. For constant forcing f = C_k, the lump center is X_c = 3T(1 - C_k T). Its derivative vanishes at T = 1/(2C_k), not at the claimed T_c = 1/C_k. The velocity V(T) = 3(C_k T - 1) printed in the paper is simply not the derivative of the argument that appears in Eq. (28). For f = sin(Omega T), the center is X_c = (6/Omega^2) sin(Omega T) + 3T; its velocity is 3 + (6/Omega) cos(Omega T), which is nonnegative for Omega = 2 and strictly positive for Omega = 4. So the claimed continuous direction reversal is not realized for the very parameters plotted. In Sec. III.B, Eq. (50) defines F = -Gamma V0XX sin(Gamma tau). The defining relation G1 = - integral integral F dX dtau gives G1 = -V0X cos(Gamma tau), not +V0X cos(Gamma tau) as written in Eq. (51). Because of that sign, the perturbative solution (52) does not satisfy Eq. (23) with forcing (50). These are load-bearing errors: the paper's central physical conclusions, the reversal time, the direction-change dynamics, and the perturbative profile, are all affected. They are also easy to spot, which makes the paper risky in its current form.

The literature review is acceptable, and the author is candid about the crude hydrodynamics and the need for microscopic detail. The appendix's exact lump solution follows [34] and is likely correct as mathematics, but it is not connected to the leaky-film forcing in a physically transparent way.

For whom? Someone working on forced KP equations or third-sound in porous substrates might want to see the fKP derivation, but I would not cite this version for any lump dynamics. If I were editor, I would not accept as is. I'd send it to a referee with a request to verify the algebra, expecting a major revision. The derivation is novel enough to deserve one serious look despite the errors.

Referee Report

4 major / 4 minor

Summary. The paper considers a saturated superfluid 4He film of thickness d on a substrate with a weak downward superfluid leakage velocity C(x,y,t) at z=0. Using a reductive perturbation expansion with C=O(ε^{5/2}), it derives a forced Kadomtsev–Petviashvili-I equation for the free-surface disturbance, Eq. (23), with forcing f derived from C. For time-dependent forcing f=f(T), it obtains an exact lump solution through a shift-and-translation transformation of the unforced KP-I lump. For constant and sinusoidal f, it analyzes the lump's motion and claims direction reversal. For space-time dependent forcing, it presents a two-time-scale perturbative solution with the forcing chosen self-consistently from the unforced lump, and it plots the resulting profiles. The paper concludes that the leakage velocity changes the lump wave speed and can reverse its direction of motion.

Significance. If the results were correct, the paper would provide a new exactly solvable and perturbative description of lump dynamics in a superfluid film with a porous substrate, extending earlier KP models for 4He films. The reduction in Sec. II follows a standard reductive-perturbation pattern and is plausibly correct, and the exact-solution transform for f=f(T) is a legitimate technique. The paper also cites relevant prior work on perturbed KP equations. However, the central solution analysis contains algebraic errors that invalidate the reported reversal dynamics and the displayed perturbative solution. In addition, the paper does not justify the physical ordering that makes the leakage appear at KP order. Despite the interesting setup and a reasonable formal derivation, the main claims about lump dynamics are not currently supported.

major comments (4)

[III.A, Eq. (28) and items 1–3] For f=Ck, Eqs. (24) and (28) give the lump center X_c=3T(1−CkT). Its time derivative is 3(1−2CkT), not the stated V(T)=3(CkT−1). The reversal therefore occurs at T=1/(2Ck), not at the reported Tc=1/Ck. The claims about the time of stopping, the direction reversal, and the interpretation of Fig. 4 are quantitatively incorrect.
[III.A.2, f=sin(ΩT)] For f=sin(ΩT), a(T)=−(6/Ω²)sin(ΩT), so the lump center is X_c=(6/Ω²)sin(ΩT)+3T and its velocity is 3+(6/Ω)cos(ΩT). For Ω=2 this velocity is nonnegative and for Ω=4 it is strictly positive; hence the claimed continuous reversal of direction (text near Fig. 5 and Fig. 6) is not realized by the solution. The sinusoidal forcing produces an oscillatory velocity modulation, not a directional reversal.
[III.B, Eqs. (50)–(52)] With F=−Γ V0XX sin(Γτ), the definition G1=−∫∫F dX dτ gives G1=−V0X cos(Γτ), not +V0X cos(Γτ) as in Eq. (51). Consequently U1=V1+G1=V0X(1−cos(Γτ)), and the displayed perturbative solution (52) with the factor (1+cos(Γτ)) does not satisfy Eq. (23) with forcing (50) at O(ϵ1). The perturbative lump profile and the plots in Fig. 7 are therefore not solutions of the stated forced equation.
[II, item 5 and Eq. (11)] The ordering C=O(ε^{5/2}) and the assumption that the film height d remains constant are postulates. No estimate from material parameters of porous substrates is given to show that the leakage velocity lies in this window. Since the forcing term in the final fKP equation arises only at this order, the derivation establishes a mathematical model but not its physical applicability to the claimed experimental situations.

minor comments (4)

[II, Sec. III] The phrases 'in-compressible' and 'pertubative' should be corrected to 'incompressible' and 'perturbative', and the spelling of 'Helium' should be made consistent.
[Fig. 3(a) caption] The caption contains 'at at T=1'; the duplicate word should be removed.
[III.B, Eq. (48)] The derivation of Eq. (48) is not shown; the way the initial condition fixes ∫R dX should be explained explicitly, since this relation is used to determine the forcing function.
[III.A, III.B] The term 'Damping Function' is used for a velocity that, in the sinusoidal case, can accelerate the wave; a more descriptive name would avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the fKP derivation is a self-contained multiple-scale expansion, and no prediction reduces by construction to a fitted input or self-citation.

full rationale

The derivation is a self-contained multiple-scale expansion from the incompressible irrotational superfluid equations (1) with the modified bottom boundary condition (2), following the standard recursion (7) and reductive perturbation (9)-(13); no parameter is fitted to data and no external result is renamed as an input. The fKP equation (23) follows algebraically from (14)-(19) with the ordering C = ε^{5/2} C^(1) (Eq. 11), which is an explicit smallness postulate rather than a circular input. The exact f(T) solution (27) is obtained by a legitimate variable shift satisfying a''(T) = 6f(T) in (24), mapping the forced equation to the unforced KP-I equation; this is a standard equivalence transformation, not a circular redefinition. The Appendix's exact solution follows the cited non-self method of Yong-Ma-Huang [34] with the nonholonomic constraint (53), and the perturbative construction in Sec. III.B determines the forcing self-consistently from the initial data via Eq. (48), a recognized self-consistent-source technique rather than a fitted prediction. The author's self-citations [19,20,25,26] are contextual and not load-bearing for the central derivation. Suspected algebraic errors in the lump-center reversal time (Sec. III.A) and in the sign of G1 in Eqs. (50)-(52) are correctness concerns, not circularity; the paper's own admissions of crudeness and missing microscopic details (Sec. IV) are acknowledged limitations, not circular reductions. Therefore no circular step is exhibited.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new particles, forces, or conserved quantities. Its only new ingredient is the prescribed bottom velocity C(x,y,t), which is an input function rather than an invented dynamical entity. The free parameters are limited to arbitrary constants in the lump solution, and the main axiomatic burden is the ad hoc ordering and boundary condition for the leakage.

free parameters (1)

lump parameters k1 and m1 = k1=0, m1=1 in plots
Arbitrary parameters in the one-lump solution; they are chosen by hand for illustrations and do not affect the existence of the solution, but they do set the specific wave trajectories shown.

assumptions (6)

standard math Superfluid is incompressible and irrotational, so the velocity potential satisfies Laplace's equation (Eq. 1).
This is the standard starting point for superfluid hydrodynamics and potential flow, taken from prior work [6,15].
domain assumption Free-surface boundary conditions (Eqs. 3 and 4) use linearized surface tension and a truncated van der Waals expansion g1 a - (g2/(2d)) a^2.
These conditions are standard for small-amplitude long waves on saturated films, following Nakajima et al. and Sreekumar and Nandakumaran, but they restrict the validity to small disturbances.
ad hoc to paper The bottom boundary condition is replaced by a prescribed vertical velocity C(x,y,t) at z=0 (Eq. 2), representing a weak downward superflow into the substrate.
This nontrivial boundary condition is the new physical input; no microscopic model of the porous substrate is given, and the magnitude of C is not tied to pore parameters.
ad hoc to paper The reductive perturbation ordering assumes C = O(epsilon^{5/2}) (Eq. 11), with the film height d treated as constant on the wave-dynamics timescale.
This ordering is chosen so that the leakage appears exactly as a forcing term at the KP order. A different leakage strength would change the asymptotic equation.
domain assumption The film is saturated with thickness d larger than the critical thickness dc = sqrt(rho alpha / sigma), so surface tension dominates the dispersion.
This is the regime considered in [6,15]; it fixes the sign of the KP-I dispersion coefficient.
standard math The known one-lump solution of the unforced KP-I equation (Eq. 26) and the transformation (Eq. 24) for time-dependent forcing are used without re-derivation.
The KP-I lump solution is a standard result from Satsuma and Ablowitz [21]; the transformation is a straightforward change of variables.

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Pith. "Pith review of Free surface lump wave dynamics of a saturated superfluid Helium film with nontrivial boundary condition at the substrate surface." pith.science (2026). https://pith.science/paper/VFQMZG44

@misc{pith2026190801512,
  author       = {Pith},
  title        = {Pith review of: Free surface lump wave dynamics of a saturated superfluid Helium film with nontrivial boundary condition at the substrate surface},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VFQMZG44}},
  note         = {Machine review of arXiv:1908.01512}
}

read the original abstract

In this article, the free surface wave dynamics of a saturated superfluid Helium film is considered under the condition that there exists a very weak downward localized superfluid flow into the substrate. For saturated film, the effect of surface tension plays a decisive role in the surface wave evolution dynamics of the system. The free surface evolution is shown to be governed by forced Kadomtsev Petviashvili-I equation, with the forcing function depending on downward superfluid velocity at the substrate surface. Exact as well as perturbative free surface lump wave solutions of the (2+1) dimensional nonlinear evolution equation are obtained and the effect of the leakage velocity function on the lump wave solutions are shown.

Figures

Figures reproduced from arXiv: 1908.01512 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: FIG.4 : We plot the lump wave solution (28) at the origin ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5: 3D plot of the solution (27) in [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 6.** Figure 6: FIG. 6: We plot the lump wave solution (27) at the origin ( [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]

**Figure 7.** Figure 7: FIG.7: We plot the full perturbative solution of the lump wave (52) at the origin ( [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

57 extracted references · 57 canonical work pages

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4 for diﬀerent constant values of Ck

Now if we concentrate on the wave ﬂuctuations at the origin i.e at X = Y = 0, we can get the plots of FIG. 4 for diﬀerent constant values of Ck. We can see that, whenT increases the amplitude decreases at origin (X =Y = 0) because the wave moves away from origin towards positiveX axis. After T =Tc as we have discussed, the wave reverses it’s motion and st...

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The acceleration of the superﬂuid ﬁlm due to ﬁnite temperature gradient gives a small cor- rection factor [8], hence it is neglected

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Since, we have considered saturated ﬁlm (∼ 10−7 cm), the eﬀect of surface tension is not neglected

Van der Waals force is the nonlinear force acting on the superﬂuid ﬁlm [15]. Since, we have considered saturated ﬁlm (∼ 10−7 cm), the eﬀect of surface tension is not neglected

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We consider that the vertical superﬂuid velocity at the bottom boundary is very weak so that the interesting surface wave dynamics appear much earlier time, hence the height of the ﬁlm d does not change signiﬁcantly

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We consider that the leakage of the superﬂuid into the substrate is localized in space i.e, it vanishes at space inﬁnities (x,y−→±∞)

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For the fast surface wave dynamics, the other physical parameters like the Van der Waals coeﬃcients are also taken to be constant. Since we consider irrotational superﬂow and in-compressible superﬂuid hence from continuity equation we get, ▽2φ(x,y,z,t ) = 0, (1) where φ(x,y,z,...
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(21) The sign of ˜B, depends on the size of the thickness d, and changes sign at the critical thickness dc = √ρα σ , which in this case is of the order of 10−7 cm. Hence in case of a very thin ﬁlm consisting of a few atomic layers [5], the surface tension eﬀect is completely n...
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˜T}2 +m2 1( ˜Y + 6k1 ˜T )2 + 1 m2 1 {{ ˜X +k1 ˜Y + 3(k2 1−m2
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For unforced KP-I equation (when f = 0), the 3D plot of the lump solution (27) in the X−Y plane at a given time is shown in FIG

f = Constant As the simplest case we consider f = constant, which means there is a constant downward superﬂuid velocity at the bottom surface. For unforced KP-I equation (when f = 0), the 3D plot of the lump solution (27) in the X−Y plane at a given time is shown in FIG. 2. We...
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We can see an interesting feature of the lump wave dynamics from Eq. (28). We see that the velocity of the wave is time dependent i.e, V (T ) = 3(CkT− 1). When T increases from zero, the lump wave propagates along positive direction of X axis
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For Tc = 1 Ck (for the given choice of parameters), the velocity becomes zero and the wave stops instantaneously
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For further increase of time, the wave moves towards opposite direction i.e, towards negative X axis and continues its motion
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A decaying background term (−TCk) appears in (28) which comes from the second term of Eq. (27). It is not interesting because it does not propagate with time. It can be explained as, the wave heightU continuously decreases due to a constant leakage (f =Ck) of superﬂuid into th...
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Depending on the real physical condition, the functional form of f may be diﬀerent

f = sin (ΩT ) where Ω is constant parameter Now as an extension to the previous study, we consider a time dependent sDF. Depending on the real physical condition, the functional form of f may be diﬀerent. As an example we consider a sinusoidal term as f = sin (ΩT ) where Ω is ...
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Following [32], we introduce a fast time scale, τ = T/ϵ1 in the calculation where ϵ1 is a small parameter

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O(1/ϵ1) : U0Xτ = 0

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [15]

4 for diﬀerent constant values of Ck

Now if we concentrate on the wave ﬂuctuations at the origin i.e at X = Y = 0, we can get the plots of FIG. 4 for diﬀerent constant values of Ck. We can see that, whenT increases the amplitude decreases at origin (X =Y = 0) because the wave moves away from origin towards positiveX axis. After T =Tc as we have discussed, the wave reverses it’s motion and st...

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[2] [34]

A.Gopakumar and J.Sreekumar, Phys.Rev.B, 49 (1994) 4323

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[3] [1]

The acceleration of the superﬂuid ﬁlm due to ﬁnite temperature gradient gives a small cor- rection factor [8], hence it is neglected

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[4] [2]

Since, we have considered saturated ﬁlm (∼ 10−7 cm), the eﬀect of surface tension is not neglected

Van der Waals force is the nonlinear force acting on the superﬂuid ﬁlm [15]. Since, we have considered saturated ﬁlm (∼ 10−7 cm), the eﬀect of surface tension is not neglected

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[5] [3]

The superﬂuid is assumed to be in-compressible and it’s motion to be irrotational. 3

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[6] [4]

The free surface disturbance a and the equilibrium height d of the ﬁlm is such that a/d≪ 1

Reductive perturbation technique is used following [15] with the nontrivial bottom boundary condition (2) to derive the nonlinear dynamical equation. The free surface disturbance a and the equilibrium height d of the ﬁlm is such that a/d≪ 1

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[7] [5]

We consider that the vertical superﬂuid velocity at the bottom boundary is very weak so that the interesting surface wave dynamics appear much earlier time, hence the height of the ﬁlm d does not change signiﬁcantly

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[8] [6]

We consider that the leakage of the superﬂuid into the substrate is localized in space i.e, it vanishes at space inﬁnities (x,y−→±∞)

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