Short-time large deviations of the spatially averaged height of a KPZ interface on a ring

Using the optimal fluctuation method, we evaluate the short-time probability distribution $P (\bar{H}, L, t=T)$ of the spatially averaged height $\bar{H} = (1/L) \int_0^L h(x, t=T) \, dx$ of a one-dimensional interface $h(x, t)$ governed by the Kardar-Parisi-Zhang equation $$ \partial_th=\nu \partial_x^2h+\frac{\lambda}{2} \left(\partial_xh\right)^2+\sqrt{D}\xi(x,t) $$ on a ring of length $L$. The process starts from a flat interface, $h(x,t=0)=0$. Both at $\lambda \bar{H}<0$, and at sufficiently small positive $\lambda \bar{H}$ the optimal (that is, the least-action) path $h(x,t)$ of the interface, conditioned on $\bar{H}$, is uniform in space, and the distribution $P (\bar{H}, L, T)$ is Gaussian. However, at sufficiently large $\lambda \bar{H}>0$ the spatially uniform solution becomes sub-optimal and gives way to non-uniform optimal paths. We study them, and the resulting non-Gaussian distribution $P (\bar{H}, L, T)$, analytically and numerically. The loss of optimality of the uniform solution occurs via a dynamical phase transition of either first, or second order, depending on the rescaled system size $\ell = L/\sqrt{\nu T}$, at a critical value $\bar{H}=\bar{H}_{\text{c}}(\ell)$. At large but finite $\ell$ the transition is of first order. Remarkably, it becomes an"accidental"second-order transition in the limit of $\ell \to \infty$, where a large-deviation behavior $-\ln P (\bar{H}, L, T) \simeq (L/T) f(\bar{H})$ (in the units $\lambda=\nu=D=1$) is observed. At small $\ell$ the transition is of second order, while at $\ell =O(1)$ transitions of both types occur.

Atypically large fluctuations in macroscopic systems out of equilibrium continue to attract great interest from statistical physicists.Although a universal description of such fluctuations is unavailable, there has been much progress in studies of particular systems.One of the main theoretical tools in this area is known under different names in different areas of physics: the optimal fluctuation method (OFM), the instanton method, the weak-noise theory, the macroscopic fluctuation theory, etc.This method relies on a saddle-point evaluation of the pertinent path integral of the stochastic process, conditioned on the large deviation.The method is based on a model-specific small parameter (often called "weak noise"), and it brings about a conditional variational problem.The solution of this problem -a deterministic, and in general time-dependent, field -describes the "optimal path" of the system: the most probable system's history which dominates the contribution of different paths to the statistics in question.
Among multiple applications of the OFM, we focus on one set of problems which has attracted attention in the last two decades : short-time large deviations of a stochastically growing interface as described by the onedimensional Kardar-Parisi-Zhang (KPZ) equation [23] where ξ(x, t) is a white noise with ⟨ξ(x, t)⟩ = 0 , ⟨ξ(x, t)ξ(x ′ , t ′ )⟩ = δ(x − x ′ )δ(t − t ′ ) . ( Here we employ the OFM to study a KPZ interface on a ring of length L, i.e. with periodic boundary conditions at x = 0 and x = L.The interface is initially flat, h(x, t = 0) = 0 , and we are interested in evaluating the probability density function (PDF) P H, L, T of the spatially averaged surface height at a final time t = T > 0, which is much shorter than the characteristic nonlinear time of Eq. ( 1), τ NL = ν 5 /D 2 λ 4 .The short-time limit allows one to employ the OFM in a controlled manner [1-5, 7, 9-12, 14-20, 22], as we will reiterate shortly.The problem, defined by Eqs. ( 1)-( 4), continues the line of studies of Refs.[10,22] of finite system-size effects (which turn out to be quite dramatic) in large deviations of height of the KPZ interface.Upon rescaling t → tT , x → (νT ) 1/2 x, h → νh/λ and ξ → νT 3 −1/4 ξ, Eq. (1) becomes with rescaled noise strength ε = Dλ 2 T 1/2 /ν 5/2 on a ring of rescaled length ℓ = L/ √ νT .The PDF of the rescaled average height H at final time t = 1 can then be written as a path integral with action functional where L(h, ∂ t h) is the Lagrangian.The OFM assumes a weak-noise limit ε → 0, when the path integral (6) can be evaluated by the saddle-point method, while the Jacobian J[h] does not contribute in the leading-order.In this limit, the PDF P ( H, ℓ, ε) is dominated by the optimal path of the system, that is by the most likely history h(x, t) conditional on a given average height at t = 1: uniform optimal path single-peak optimal path intermediate region 1.An overview of our main results for the large-deviation problem (8) on the ( H, ℓ) phase diagram.In the light blue region, which extends to all H < 0, the global minimum of the action functional ( 7) is attained at the spatially uniform solution (19).In the light orange region, the least-action solution h(x, t) is instead a spatially non-uniform path with a single maximum.For small domain sizes ℓ → 0 it is given by Eq. ( 67), and for large domain sizes it is described in Sec.V A. For small ℓ, there is a second order DPT from the uniform to the peaked solution, as discussed in Sec.III and Sec.VI.For large ℓ, a (weakly) first order transition takes place instead, at H approximately given by Eq. ( 64).In the intermediate region transitions of both first and second order occur.Here some numerical results are available, see Fig. 7, but a complete understanding is lacking.
Hence, the PDF can be determined, up to pre-exponential factors, from the solution of this constrained minimization problem.Here we will solve this minimization problem numerically, for different H and ℓ, and analytically in the asymptotic limits of large and small ℓ1 .
It will be convenient to present our results by setting ν = λ = D = 12 .Then the weak-noise scaling (8) reads Note that the limit ε → 0 at fixed ℓ corresponds to the short-time limit T → 0 and small-length limit L → 0 with L/ √ T = const.When instead T goes to zero at L = const, one has both ε → 0 and ℓ → ∞.The latter limit turns out to be most interesting, and it is analyzed here in detail.It is natural to expect that for any H, when ℓ → ∞, the action S( H, ℓ) should exhibit a large-deviation form leading to and this is what we indeed observe here.Less expectedly, we also find that the rate function f ( H) exhibits, at a critical value H = Hc (ℓ), a dynamical phase transition (DPT) which is accidentally second-order.By that we mean that the rate function at the critical point becomes continuously differentiable only in the limit of ℓ → ∞.At arbitrary large but finite ℓ the large-deviation form (10) breaks down.We show, however, that the action S( H, ℓ) still exhibits a DPT at a critical point H = Hc , but this DPT is of (weakly) first order and the optimal path at the critical point changes discontinuously via a subcritical bifurcation.
For small ℓ a truly second-order DPT is observed as predicted earlier [10,22].At intermediate values of ℓ = O(1) DPTs of both types occur.In the latter regime analytical results are unavailable as of yet, and we present some numerical results.All the DPTs that we found in this system occur because of a loss of optimality of a path that is uniform in space.The loss of optimality takes the form either of a subcritical bifurcation (for the first-order DPTs), or a supercritical bifurcation (for the true second-order DPTs).We summarize our main results on the ( H, ℓ) phase diagram of the system in Fig. 1.
The remainder of this paper is structured as follows.In Sec.II we formulate the OFM equations and boundary conditions, present a simple uniform solution of these equations, previously studied in Refs.[10,22], and argue that it describes the optimal path of the system at all λH < 0. Supercritical bifurcations of the uniform solution have been recently studied in Ref. [22].Still, for convenience of further discussion, we briefly rederive them in Sec.III.Section IV includes our results of numerical minimization of the action functional (7) in different regions of the ( H, ℓ) phase diagram.These numerical results provided valuable insights into the nature of optimal paths of the interface which led us to develop asymptotic analytical solutions of the OFM problem for large ℓ that we present in Sec.V.The asymptotic solution for small ℓ is briefly discussed in Sec.VI.We summarize and discuss our main results in Sec.VII.A description of numerical algorithms that we use here is relegated to the Appendix.

II. OFM EQUATIONS AND UNIFORM SOLUTION
At a technical level, the main objective of this work is to determine the minimum action S( H, ℓ) as a function of the rescaled average height H and rescaled system size ℓ.In this section, we present the necessary conditions for minimizers of the action functional (7) -the OFM equations and the boundary conditions.We argue then that a simple spatially uniform solution of the ensuing OFM problem is always optimal for H < 0.
The first-order necessary conditions for a minimizer of the action functional ( 7) can be represented as a pair of Hamilton's equations for the optimal history of the interface h(x, t) and the conjugate momentum density p = ∂L/∂(∂ t h).These equations were derived in many papers [1-5, 7, 9-12, 14-20, 22], and they take the form The "momentum density" p(x, t) describes the (rescaled) optimal realization of the external noise ξ(x, t) that drives the interface conditional on a specified H.In the present case Eq. ( 12) and ( 13) should be complemented by the periodic boundary conditions at x = 0 and x = ℓ, by the initial condition and by the final-time condition which follows from the demand that a boundary term at t = 1, originating from an integration by parts, should vanish for any h(x, 1).The parameter Λ is a Lagrange multiplier which needs to be chosen so as to impose the rescaled final-time condition Once the optimal path is determined, the action S( H, ℓ) can be determined from the equation which follows from Eqs. (7) and (12).By differentiating the action S( H, ℓ) = S[h(x, t; H, ℓ)] of the optimal profile h = h(x, t; H, ℓ) with respect to H using the chain rule, one can show that Λ is related to the action via If the action S( H, ℓ) is a strictly convex function of H, there is a bijective relation between Λ and H, and it suffices, for the purpose of calculating the action, to only determine H(Λ) and use Eq. ( 18).This shortcut is very convenient and holds for many large-deviation calculations [24].
There is an obvious exact solution of the OFM equations and the boundary conditions: which describes a uniformly growing flat interface.We will often call this branch of solutions branch 1.By virtue of Eq. ( 8), whenever the uniform solution (19) is the optimal one, we have a Gaussian PDF for H up to pre-exponential factors.Of most interest, however, are the regions of parameters H and ℓ, for which the uniform solution is suboptimal.As we will see, the loss of optimality can occur via either a supercritical, or a subcritical bifurcation.
First of all, we can argue that, for negative H, the uniform solution ( 19) is always optimal.Using the evident conservation law of Eq. ( 13), we can rewrite the action (7) for any solution of the OFM equations as Also, integrating both sides of Eq. ( 12) with respect to t from 0 to 1 and with respect to x over the ring, and using the periodic boundary conditions and the conservation law (20), we obtain One can easily see from Eqs. ( 21) and ( 22) that, at negative Λ (or H) any inhomogeneity in the momentum density p both increases the action S, and decreases the average height | H| in comparison to their values for the uniform solution.Therefore, any nonuniform solution here is sub-optimal.In contrast to this, for Λ > 0 (or H > 0), an inhomogeneity increases both S, and H in comparison to the uniform solution.A competition between these two opposite effects may give rise to non-uniform solutions with lesser action than the uniform one, as we will indeed see in the following.

III. BIFURCATIONS OF THE UNIFORM SOLUTION
In this brief section we carry out a linear stability analysis of the uniform solution (19).We find that, for sufficiently large positive H, the uniform solution can continuously and supercritically bifurcate to a non-uniform solution.The first spatial Fourier mode to become unstable as H increases depends on the rescaled system size ℓ in a nontrivial way and is determined from Eq. ( 26).This equation has also been obtained in Ref. [22] by calculating the leading-order prefactor correction to the asymptotic scaling in Eq. ( 8) through Gaussian integration of fluctuations around the uniform solution (19).
At first order of a perturbation theory around the uniform solution (19) we have Here the wave number q spans the set 2πm/ℓ for m = 1, 2, . . . .Substituting the expressions (23) into Eqs.( 12) and ( 13) and neglecting higher-order terms, we obtain the following system of linear ordinary differential equations: It has solutions proportional to e iωt , where FIG. 2. The supercritical bifurcation points H = Hc vs. the wave number q as predicted by Eq. ( 27) for n = 1, 2 and 3.Only discrete values of q = 2πm/ℓ with m = 1, 2, . . .are allowed.The lowest curve corresponds to n = 1.The black dot indicates the global minimum of Hc vs. q, so that Hc ≥ 4.603 is a necessary condition for a supercritical bifurcation of the uniform solution for any ℓ.
Using the boundary conditions ( 14) and (15), we obtain the following relationship between q and H = Hc (q) at the bifurcation points: Note that the trivial solution H = q 2 of Eq. ( 26) does not correspond to a valid non-uniform solution due to the boundary conditions at t = 0 and 1.The resulting dependence H(q) can be expressed in a parametric form where, for given ℓ, only values of q = 2πm/ℓ with m = 1, 2, 3, . . .are allowed.The first three branches of Eq. ( 27) are shown in Fig. 2. As one can see, the first instability appears for n = 1, and a necessary condition for the instability, for any ℓ, is Hc ≥ 4.603.When ℓ → ∞, the first instability of the uniform solution will occur, at Hc ≃ 4.603, for a very high mode m ≃ 1.343 ℓ/2π.For finite ℓ, one can find the bifurcation point on the n = 1 branch of Eq. ( 27) numerically.Finally, for ℓ → 0, the first instability occurs for the m = 1 mode at H ≃ (2π/ℓ) 2 in agreement with Ref. [10].

IV. NUMERICAL RESULTS
Now we proceed with a numerical solution of the minimization problem in Eq. ( 8) for different H and ℓ.The numerical methods that we used are described in the Appendix.In addition to confirming the supercritical bifurcations of the uniform solution that we discussed in Sec.III, we will uncover important subcritical bifurcations and get insight into non-perturbative optimal paths which will be studied analytically in Secs.V and VI.
We start with the simpler case of small ℓ.Choosing a moderately small value ℓ = π/8 and numerically minimizing the action (A1) for different Λ, we obtain the rate function S( H, ℓ) and Lagrange multiplier Λ( H) shown in Fig. 3.The spatially uniform solution (19), corresponding to branch 1 of the action, is seen to become unstable close to H ≃ (2π/ℓ) 2 as stated in Sec.III, and there is a continuous (second-order) DPT to a spatially nonuniform solution.Indeed, the (m = 1)-spatial Fourier mode of the profile becomes unstable at this point.One such spatially nonuniform solution close to the transition point is shown in Fig. 4. As H increases, the optimal solution turns, for most of the time 0 < t < 1, into a stationary "cnoidal" solution for p which drives an h-profile which is non-uniform in x, but is uniformly translating in the vertical direction.The same solution appears in the problem of the one-point height distribution for the KPZ equation on a ring [10], and we use it in Sec.VI to calculate the theoretical curves in Figs. 3  and 4, which match the numerical results quite well.
Next, we turn to the more complicated and interesting case of large ℓ.For ℓ = 16π the minimization of the augmented action (A2) leads to the results for the rate function S( H) and Lagrange multiplier Λ( H) shown in Fig. 5.In addition to branch 1 we observe two other branches of solutions.Branch 2 is observed to the right of a narrow  26).The dotted green line corresponds to the asymptotic S( H) = ( 8√ 2/3) H3/2 , see Eq. ( 73), the gray line corresponds to the spatially uniform solution (19), and the orange line corresponds to Eqs. ( 70) and (72).12) and ( 13) for H = 270.319and a moderately small rescaled system size of ℓ = π/8 with numerical resolution nx = 64 and nt = 8000.The numerically found action S = 14322.081deviates by 0.009% from the predicted small-ℓ result S = 14320.806for this H as given by Eq. (70).The action for the spatially uniform solution (19) for the same H and ℓ would be S = 14347.685.The maximum height at final time is H = 270.357.Left: Optimal height profile h(x, t) at different times t, with the maximum at each t subtracted in order to emphasize deviations from spatial homogeneity.The prediction (67) for intermediate times is indicated by the black dashed line and agrees well with the numerical solution for t = 0.25, t = 0.5 and t = 0.75.The inset shows that the growth of the maximum maxx h(x, t) in time is still linear as predicted.Center: conjugate momentum density p(x, t) with Λ = 261.057subtracted, compared to the analytical result (68) indicated by the black dashed line.Right: Spatial maximum maxx p(x, t) over time to visualize the long lifetime of the stationary cnoidal solution with some small boundary layers in time at t = 0 and t = 1.
transition region close to H ≃ 4. On this branch the action S( H) is approximately a linear function, while Λ is almost constant.Further, for much larger H, there is a smoothed-out second-order transition from branch 2 to a third branch 3 with a different scaling behavior.The optimal paths for branches 2 and 3 are shown in Fig. 6.They consist of strongly localized large-amplitude stationary solitons of p that drive an outgoing almost triangular structure of h (or two antishocks of V (x, t) = ∂ x h(x, t), see Sec.V.The solution, corresponding to branch 2, clearly emerges via a subcritical, rather than supercritical bifurcation.Strikingly, the soliton has a well-defined life time which is very close to 1/2.The difference between branches 2 and 3 is that, for branch 3, the two edges of the triangular structure of h(x, t) collide before the final time t = 1 is reached, while for branch 2 they do not.
These crucial findings will guide our stationary-soliton-based asymptotic theory for large ℓ that we develop in Sec.V.There we give an analytical description of the optimal paths for branches 2 and 3, which are the only relevant  31), with Λ( H) then obtained from Eq. ( 18).The solid vertical lines indicate the theoretical critical points in the large-ℓ limit at H ≃ 4 and H ≃ ℓ 2 /6.The dashed vertical line shows where the spatially uniform solution becomes unstable in the (m = 11) spatial Fourier mode, as given by the minimization of Eq. ( 27) over the allowed wave numbers.The dotted green line corresponds to the asymptotic behavior S( H) = ( 8√ 2/3) H3/2 , see Eq. (58).A better resolved transition region close to H = 4 is shown in Fig. 7.
ones for large ℓ.There we establish a first-order transition at H ≃ 4 for large but finite ℓ and show that it becomes "accidentally" second order in the limit of ℓ → ∞.We also find that the smoothed-out second-order transition from branch 2 to branch 3 occurs at H = ℓ 2 /6.The resulting analytical predictions, indicated by the lines in Figs. 5 and  6, are in good agreement with numerics at large, but finite ℓ.
At moderate ℓ the transition region where the spatially uniform solution (19) of branch 1 becomes sub-optimal is quite complex, as one can appreciate from Fig. 7.We see that, in general, there are both first and second order transitions in this region: The uniform solution becomes linearly unstable for some m > 1, leading to second-order transitions, but there is also a competition with the (subcritical) one-soliton solution.The subcritical scenario clearly wins for sufficiently large ℓ.Indeed, for ℓ = 32π we observe only a first-order transition from the spatially uniform to the soliton solution, while the linear instability becomes irrelevant.
Note that, for branch 2, in addition to stationary single-soliton solutions of the OFM equation, discussed so far, there are also stationary multi-soliton solutions consisting of two or more (almost) non-interacting strongly localized stationary solitons of p and corresponding expanding triangles of h.One such solution, which we observed numerically, is shown in the top row of Fig. 8.We found, however, that such solutions always have a larger action than the onesoliton solution for the same ℓ and H. Therefore, the one-soliton solution indeed seems to provide the optimal solution.In the limit ℓ → ∞, these multi-soliton solutions -a soliton gas -would contribute to the pre-exponential factor for P( H, ℓ), but pre-exponential factors are beyond the scope of this paper.Additionally, in the bottom row in Fig. 8, we show an optimal path for ℓ = 16π and close to H = 4, which emerges through linear instability of the (m = 11)-mode.Later on, however, it is overtaken by the one-soliton solution.

V. LARGE-ℓ ASYMPTOTICS: RISE AND FALL OF THE SOLITON A. General description of the solution
Guided by our numerical solutions and by the previous works on the one-point KPZ height statistics on the line [4] and on a ring [10], here we find approximate asymptotic solutions of Eqs. ( 12)-( 15) which give rise to two nontrivial branches (we call them branches 2 and 3) of the large-deviation function S( H) for large ℓ.As we found, for both branches the maximum one-point height of the interface H = max h(x, t = 1) turns out to be very large: H ≫ 1.Therefore, in addition to the strong inequality ℓ ≫ 1, we can also use the strong inequality H ≫ 1.This allows us to construct "inviscid" asymptotic solutions in different regions of space, separated by discontinuities of proper types.Like their numerical counterparts, the analytical solutions exhibit two distinct stages in time, with an abrupt  12) and ( 13) for two mean heights H = 144.198(top row) and H = 507.56(bottom row) at ℓ = 16π with numerical resolution nx = 2048 and nt = 4000.Both solutions are one-soliton solutions for branch 2 and 3, respectively.The action of the solution in the top row is S = 28723.766,which deviates by 0.4% from the predicted large-ℓ result S = 28590.660given by Eq. ( 50).The action for the bottom row solution is S = 102754.528,so it deviates by 0.2% from the predicted value S = 102522.498for this mean height given by Eq. ( 56).The qualitative difference of these solutions is whether the two edges of the growing triangle for h collide.The left column shows the optimal height profile h(x, t) at different times t, together with the prediction (44), ( 45) and (55) for the final time t = 1.The center column shows the corresponding conjugate momentum density p(x, t).The right column shows the spatial maximum maxx p(x, t) over time to visualize the lifetime of the stationary soliton solution and compare it to the analytical expressions τ = 1/2 for branch 2 and Eq.(59) for branch 3 (dashed vertical lines).
transition between them at some branch-dependent intermediate time 0 < t = τ < 1 which we will determine.
For 0 < t < τ the solution has the form of a strongly localized stationary soliton of p(x, t) and "antishock" of V (x, t) = −∂ x h(x, t) which were previously identified in the problem of one-point height statistics on the line [4,5] and on a ring [10].The characteristic width, O(1/ √ H), of the soliton-antishock structure is much less than unity.Outside of the soliton-antishock one has p(x, t) ≃ 0. As a result, Eq. ( 13) is obeyed trivially and, at distances ≳ 1 from the soliton, h(x, t) follows the deterministic KPZ dynamics which is equivalent to the Burgers equation for the field V (x, t) = −∂ x h(x, t).In addition, the diffusion term in Eq. ( 29) can be also neglected at large distances [4], and one arrives at the inviscid Hopf equation The stationary soliton-antishock structure drives an almost triangular configuration of h(x, t) which is expanding outwards [4].The height of the triangle grows linearly with time, while its two edges propagate with a constant speed as "ordinary" shocks of V (x, t) obeying Eq. ( 29) or, when treated as discontinuities, obeying Eq. ( 30) [4].The positions of these shocks at t = 1 determine the boundaries of the "impact region" of the soliton-antishock structure.When the size of the impact region, which scales as O( √ H) [4], is shorter than the rescaled system size ℓ (this happens , where the transition between branches 1 and 2 at large ℓ is predicted, see Eq. (31).Using the numerical minimization techniques described in the Appendix, we search for (possibly multiple distinct local) minimizers of the action functional (7) for given ℓ and H.When there are more than one minimizers for the same ℓ and H, the one with the least action gives the true optimal solution.For a better visualization shown is the difference between the numerically computed action of the found solutions and the action ℓ H2 /2 of the spatially uniform solution.The vertical dashed lines indicate the smallest H where a spatial Fourier mode q = 2πm/ℓ of the uniform solution first becomes unstable according to Eq. ( 27) (m = 3 for ℓ = 4π, m = 5 for ℓ = 8π, m = 11 for ℓ = 16π, and m = 21 for ℓ = 32π).For ℓ = 4π, 8π and 16π, the rate function displays both a second-order transition at the predicted point, and a first-order transition at slightly larger H where the one-soliton solution (see the top row of Fig. 6), described theoretically in Sec.V B, becomes optimal.At the largest ℓ = 32π, only a first-order transition from the uniform to the one-soliton solution is observed, while oscillating solutions are irrelevant.See Sec.V D for a more detailed analysis of the transition region at large but finite ℓ.Note that for ℓ = 16π one can also see, around H = 4.7, another oscillating solution with a second superimposed wave number.Numerical resolutions used: nx = 128 and nt = 2000 for ℓ = 4π, nx = 512 and nt = 4000 for ℓ = 8π, nx = 1024 and nt = 4000 for ℓ = 16π, and nx = 2048 and nt = 4000 for ℓ = 32π.when H is not too large, see below), there is also an external region where the uniform solution p(x, t) = Λ = const and V (x, t) = 0 holds, see Eq. (19).The external uniform solution holds for all times 0 < t < 1, and it contributes to the large-deviation function of H.In the inviscid limit the regions of zero and nonzero p are divided by a stationary discontinuity.This regime corresponds to branch 2.
Branch 3 appears when, due to the periodicity of the system, the ordinary shocks of V (x, t) collide with each other before the final time t = 1 is reached.In this case the impact region of the soliton-antishock structure extends to the whole system, and a region of the uniform solution does not appear.
For the solution to obey the boundary condition (15), the p-soliton must turn into a constant p = Λ at t = 1.Remarkably, as we have seen in our numerical results for large ℓ, the soliton rapidly decays in the vicinity of a welldefined time t = τ < 1.For both branches 2 and 3, the subsequent dynamics, at τ < t < 1, gives only a subleading contribution (which we neglect, alongside with other subleading contributions) to the maximum one-point height H and to the action.This stage is important, however, for determining H.We can qualitatively understand this nontrivial temporal structure of the solutions from the viewpoint of action minimization: First, for 0 ≤ t ≤ τ , the interface is efficiently driven upward by a stationary p-soliton, in the same manner as for the one-point height PDF of the KPZ equation on the line [4] and on a ring [10].Then, quickly suppressing the soliton at an intermediate time  12) and ( 13) for ℓ = 16π.The top row (obtained with resolution nx = 2048 and nt = 4000) shows a two-soliton solution for the same value of H = 144.198as in the top row of Fig. 6.The action of the two-soliton solution, S = 28746.682,is larger than that for the one-soliton solution for the same ℓ and H.The bottom row shows a spatially oscillating solution, originating from linear instability of the m = 11 mode of the spatially uniform solution for H ≳ 4.607 as predicted by Eq. ( 27).Here the resolution is nx = 1024 and nt = 4000.As can be seen from Fig. 7 (bottom left), this family of solutions can be optimal for small H and moderate ℓ, but it becomes irrelevant as large ℓ.0 < τ < 1 and evolving the interface according to the almost free KPZ dynamics for τ < t ≤ 1 increases considerably the average height H for a negligible additional cost in terms of action.The optimal value of τ is the one that minimizes the action for a given H.
As an overview, we present here the action S( H, ℓ) at leading order for large ℓ, as will be derived in subsections V B and V C: where the function Φ(. . . ) is defined in Eq. ( 57) and obeys Φ(z → ∞) → 8 √ 2/3.The first line in Eq. ( 31) comes from the uniform solution (19).The first two lines manifestly reveal the large-deviation scaling (11), while the third line does not.Now we proceed to a more detailed description of the solutions, and we will start with branch 2.

B. Branch 2
Due to a translational symmetry of the problem ( 12)-( 15), we can place the soliton-antishock structure at x = 0 (see Fig. 9) so that, to the leading order, H ≃ h(0, τ ).As explained above, at H ≫ 1, the p-soliton can be considered as a point-like object.We will only need the value of its "mass", dx p(x, t) which, by virtue of Eq. ( 13), is conserved.Using the explicit expression for the soliton, p(x, t) = p s (x) = 2c cosh −2 ( c/2 x) [4], where c = H/τ , we obtain x t 1 1/2 0 uniform solution uniform solution p=h=0 p=h=0 p-soliton FIG. 9. A space-time map of the optimal path of the system which determines the large-deviation function S( H) of branch 2 in the large-ℓ limit.The thick blue line shows the position of the stationary p-soliton and V -antishock.The thick red lines show the positions of propagating ordinary shocks, see Eq. ( 33).The dashed red lines show the positions of propagating weak discontinuities caused by the rapid decay of the p-soliton at t = τ = 1/2, see Eq. ( 42).The weak discontinuities catch up with the (twice as slow) ordinary shocks at t = 1.This condition selects τ = 1/2.The boundaries of the impact region of the soliton-antishock are shown by the dashed black lines.Outside of the impact region, the uniform solution (45) holds.The regions between the dashed black lines and the thick dark lines correspond to the trivial solution (35).In the regions between the dashed and thick red lines, there remain linear h-profiles of the original "triangular solution", see Eq. (36).Inside the gray triangle the solution is described by Eqs. ( 40) and ( 41) leading to Eq. ( 44).
The base of the triangular structure of the h-profile is equal to while the triangle's height is Let us denote the total size of the impact region of the soliton-antishock structure by 2a 1 , where a 1 ≡ a(t = 1).In the region a(t) < |x| < a 1 we have The triangular profile of h on the interval 0 < |x| < a(t) is described by the expressions [4] p where As one can see from Eqs. ( 33) and ( 37), the ordinary shocks propagate with the speed Ṽ /2, as to be expected from Eq. ( 29) or (30) [25].
After the rapid decay of the soliton at t = τ , the "post-soliton" solution (in the region to be determined) can be described by the ideal hydrodynamic equations corresponding to the inviscid limit of Eqs. ( 12) and ( 13): The V -antishock now plays the role of a discontinuity which undergoes a decay starting from t = τ .In the leading order we can neglect the −∂ x p term, so that Eq. ( 38) becomes the Hopf equation (30).Its solution is Plugging Eq. ( 40) into Eq.( 39) and using the "final" condition (15) on p(x, t = 1), we obtain The solution ( 40) and ( 41) holds at t > τ and |x| ≤ a d (t).The boundaries of this region, represent weak discontinuities, moving with the speed Ṽ -that is twice as fast as the ordinary shocks at x = ±a(t), see Eq. (33).Our simulations show that the weak discontinuities catch up with the shocks at t = 1.The corresponding condition can be written as a d (1) = a 1 , and it yields τ = 1/23 Therefore, during the second stage of the dynamics, 1/2 < t < 1, V (x, t) is described by the following expressions: Using the relation V (x, t) = −∂ x h(x, t), we can obtain the h-profile at any time 1/2 < t < 1 by integrating Eq. ( 43) over x.The result describes a parabolic profile of h at |x| < a d (t), flanked by the linear profiles at a d (t) < |x| < a 1 corresponding to the triangular structure of h(x, t) of the first stage the dynamics.At t = 1 the parabolic profile takes over the whole interval |x| < a 1 , and we obtain At |x| > a 1 the uniform solution holds: Now we evaluate the contributions of the uniform solution to the action, ∆S u , and to the average height, ∆ Hu , at t = 1.As ℓ goes to infinity, we can neglect the difference between the total system length ℓ and the length of the domain of uniform solution ℓ − 2a 1 , and obtain The leading-order contribution of the soliton-antishock solution to the action is [4] ∆S s = 8 √ 2 3 This contribution comes from the first stage of the process, 0 < t < 1/2, while the second stage gives only a subleading contribution which we neglect.The second stage, 1/2 < t < 1 does contribute to H, however.Using Eq. ( 44), we obtain What remains to be done is to determine Λ, to collect the contributions to S and H, and to eliminate H in favor of H and ℓ.In order to determine Λ, we use the local conservation of p(x, t) evident in Eq. ( 13).Because of this local conservation law, the total soliton "mass", see Eq. (32), must be equal to the integral of the solution (41) for p(x, t) over x from -a 1 to a 1 .This condition yields a remarkably simple result: Λ = 4, a constant value (up to small subleading corrections).Combining Eqs. ( 46)-( 48), we obtain x t 1  0 p=h=0 p=h=0 p-soliton FIG. 10.A space-time map of the optimal path of the system which determines the large-deviation function S( H) of branch 3 in the large-ℓ limit.The notations are similar to those of Fig. 9.
Eliminating H, we arrive at the leading-order result for the large-deviation function of H for branch 2 in the limit of large ℓ, which was announced in the second line of Eq. ( 31): This expression obeys the large-deviation scaling (10).As was to be expected, the actions of branch 1 and 2 coincide at H = Hc = 4. Noticeably, their first derivatives with respect to H also coincide at this point.In addition, using Eq. ( 18), we see that Eq. ( 50) is consistent with Λ = 4, independently of H, for branch 2. We will look into these peculiarities more carefully in Sec.V D. One applicability condition of Eq. ( 50) is the strong inequality H ≫ 1.Using the first relation in Eq. ( 49), we can rewrite this strong inequality in terms of H and ℓ ≫ 1: This condition limits H from below.A condition on H from above distinguishes branch 2 from branch 3. It demands that the ordinary shocks of V (x, t) do not collide with each other until t = 1 4 .This condition can be written as 2 √ H < ℓ or, using Eq. ( 49), Now we proceed to a description of branch 3.

C. Branch 3
When the inequality ( 52) is violated, the two outgoing ordinary shocks of V (x, t) collide with each other and merge at x = ±ℓ/2 (which is the same point of the ring) at some t < 1. Upon the merger, a single stationary shock appears, see Fig. 10.Now the impact region of the soliton-antishock is the whole system: 2a 1 = ℓ, and the external region of the uniform solution, characteristic of branch 2, does not appear here.Most of the general formulas, derived in the context of branch 2, remain valid for branch 3.In particular, here too τ is determined by the condition that the weak discontinuities catch up with the ordinary shocks at t = 1.The only difference is that a 1 = ℓ/2 now.Solving the equation we obtain so that τ depends on H and ℓ.Unsurprisingly, Eq. ( 54) yields τ = 1/2 in the boundary case H = ℓ 2 /4, when the size 2a 1 of the impact region of the soliton-antishock in an infinite system is equal to the system size ℓ.When H goes to infinity, τ approaches 1.
We will not repeat here all expressions for h(x, t), V (x, t) and p(x, t) in different regions, and present only the expression for h(x, 1): with τ from Eq. (54).Using this expression, we can evaluate H.The action S remains the same as in the first equality in Eq. ( 47), and we obtain Eliminating H from these relations and using Eq.(54), we arrive at a leading-order result for the large-deviation function S( H, ℓ) in the limit of large ℓ and very large H, which was announced in the third line of Eq. ( 31): In terms of H, the condition H > ℓ 2 /4 becomes, in the leading order, H > ℓ 2 /6.As a result, the function Φ(z) is defined for z ≥ 1/6, and Φ(1/6) = 4 √ 6.A graph of Φ(z) is depicted in Fig. 11.In the limit of H ≫ ℓ 2 ≫ 1 Eq.(57) yields The leading-order term of this expression coincides with the action for a single-point height H [4].This is to be expected, because for very large H, τ approaches 1, and the difference between H and H becomes relatively small.
For completeness, we also present the optimal transition time τ in Eq. ( 54) in terms of H and ℓ:

D. Dynamical phase transition
In this subsection we resolve the nature of the DPT between branches 1 and 2, which corresponds to the subcritical bifurcation from the uniform solution (19) to the leading-order soliton solution discussed in Sec.V B. To this end we will have to focus on subleading corrections that we have previously ignored.We will also present the large-deviation scaling of P( H, L, T ) in the limit of T → 0 at fixed L, in the physical units.
As we have already noticed, the actions S 1 ( H, ℓ) and S 2 ( H, ℓ), described by the first and second lines of Eq. ( 31), coincide at H = Hc = 4 together with their first derivatives ∂S 1 ( H, ℓ)/∂ H and ∂S 2 ( H, ℓ)/∂ H at Hc = 4.It would be incorrect, however, to conclude from here that the DPT between branches 1 and 2 at H = Hc is of second order.Indeed, the supercritical first bifurcation of the uniform solution (19) to a solution with a single maximum of h(x, 1) -the one with q = 2π/ℓ in Eq. ( 26) -actually occurs, as ℓ → ∞, at much larger H ≃ ℓ 2 /16 ≫ 4. Furthermore, as follows from numerical minimization of Eq. ( 26), instability of any Fourier mode around the uniform solution can only occur at H ≃ 4.60334 (for q ≃ 1.34336).It is not surprising, therefore, that at large but finite ℓ, and at a slightly shifted transition point Hc > 4 where the actions of branches 1 and 2 are equal, the optimal paths h(x, t) for branches 1 and 2, that we found numerically, are dramatically different, and their respective Lagrange multipliers Λ are not equal.The latter fact means, by virtue of Eq. ( 18), that at large ℓ we actually observe a first-order DPT, not a second-order one.
To make sense of these facts, we recall that Eq. ( 50) for the action of branch 2 is merely a leading order asymptotic at ℓ → ∞.Subleading terms, so far unaccounted for, should remove the degeneracy of the leading-order results by breaking the accidental continuity of the first derivative ∂S( H, ℓ)/∂ H at H = Hc , and rendering the corresponding bifurcation subcritical and the corresponding DPT first-order.The subleading terms should also account for a slight shift of the critical point Hc to the right from its leading-order value Hc = 4, as observed in our numerics.
Motivated by the large-H asymptotic of the upper tail of the exact short-time probability distribution of the onepoint height h(x = 0, t = 1) = H on the line, determined in Ref. [12], we can conjecture the following subleading terms of S 2 ( H, ℓ) at large ℓ: where B > 0 and C are numerical constants O(1), which are independent of ℓ.The condition B > 0 is necessary for the equation to have a solution for Hc close to 4 at large ℓ.
To verify Eq. ( 60), we plotted in Fig. 12 our large-ℓ numerical results for S 2 ( H, ℓ) − 4 H − 8 ℓ / √ H versus H.A fair plateau at large H is observed, with B ≃ 5.3 > 0 found by fitting.Now, keeping the first subleading term in Eq. ( 60) and the leading-order dependence of H on H in Eq. ( 49), we can rewrite Eq. (60) in terms of H and ℓ: Now Eq. (61) for the critical point becomes Its approximate solution, Hc = 4 + 6 1/5 B 3/5 ℓ −2/5 + . . ., 12. Numerical data from high-resolution optimal path computations at rescaled system sizes ℓ = 8π and 16π.Assuming an asymptotic expansion (60), the ordinate should be given by the subleading terms B + C/H with constants B and C that are independent of ℓ and H. Accordingly, we performed least-squares fit with this functional form to the data for 20 ≤ H ≤ 125.We observe a fair agreement for the two rescaled system sizes and determine B ≃ 5.3.For the smaller domain size ℓ = 8π there are small oscillations which may come from sub-subleading terms not included in Eq. (60).describes a small ℓ-dependent positive shift of the critical point from the leading-order value 4.This Hc corresponds to of the branch-2 solution at the critical point.We observe that, for this solution, H → ∞ as ℓ → ∞, guaranteeing applicability of our theory at large ℓ.Going back to the large-deviation scaling (10), we notice that there is now a small but finite jump ∼ ℓ −2/5 of the derivative ℓ −1 ∂S/∂ H of the effective rate function at the shifted critical point.The transition between branches 1 and 2, therefore, is of first order at large but finite ℓ.Such transitions -with a finite but small jump of the first derivative of the free energy (or the action) are usually called weakly first order transitions [26].By virtue of Eq. ( 18), the subleading correction in Eq. (62) also removes the degeneracy of the leading-order result Λ = 4 by adding to it a small ℓ-dependent correction that goes to zero as ℓ → ∞.
Using Eq. (62), we plotted in Fig. 13 the actions of branches 1 and 2, normalized by ℓ H2 , in the vicinity of the H = Hc .It is clearly seen that the subleading correction removes the degeneracy and makes the DPT first-order.Furthermore, the predicted Hc from Eq. (64) for ℓ = 32π, which is Hc ≃ 4.6, is close to our numerical result Hc ≃ 4.57.for this ℓ, see Fig. 7.
Note that our arguments in favor of the expansion (60) are far from rigorous.In particular, we cannot exclude a very slow (for example, logarithmic) dependence of the coefficient B on H in Eq. ( 60) based only on the numerical evidence.However, our main conclusion about the first-order DPT between branches 2 and 3 seems robust.
To conclude this section, we present our large-deviation results, described by the first two lines of Eq. ( 31), in the physical units.Recall that, by taking the limit T → 0 at fixed L, we have both ε ∝ T 1/2 → 0 and ℓ → ∞.In this limit only the first two lines of Eq. ( 31) are relevant, and we obtain5 − lim As we elaborated in this subsection, the DPT in Eq. ( 66) at w = 4 can be called an "accidental" second order DPT in the sense that the optimal paths, that are responsible for the two branches in Eq. ( 66), transition into each other discontinuously, and that the differentiability of the rate function at the critical point emerges only in the limit T → 0 at fixed L.

VI. SMALL-ℓ ASYMPTOTICS
We found that our numerical results on the second-order DPT at small ℓ, shown in Figs. 3 and 4 and described in Sec.IV, can be understood in terms of a small-ℓ asymptotic solution of the OFM equations ( 12) and ( 13) which was previously found in the context of the one-point height distribution on a ring [10].In this solution the interface is driven by a stationary dn 2 profile (see below) of p.The solution represents a finite-amplitude generalization of a weak sinusolidal modulation with m = 1 which results from the second-order DPT from the uniform solution.This solution is given by the following expressions where K(k) is the complete elliptic integral of the first kind and dn(. . . ) is one of the Jacobi elliptic functions [27].
The elliptic modulus k ∈ (0, 1) is determined by H via the relation The action of this solution as a function of k is [10] At given ℓ ≪ 1, Eqs. ( 69) and (70) determine S as a function of H in a parametric form.The critical point H = (2π/ℓ) 2 corresponds to k = 0, when Eqs. (69) and (70) reduce to the uniform solution.k > 0 correspond to supercritical solutions.
In order to recast this dependence in terms of S( H, ℓ), we need to express H through H and ℓ.Although Eq. ( 67) is formally inapplicable at t = 1, asymptotically as ℓ → 0 we still have where we have used a product formula for dn [28].Using Eqs. ( 69) and (71), we obtain Equations ( 72) and (70) determine S = S( H, ℓ) and were used in Fig. 3 to draw the theoretical curves for the action and Lagrange multiplier (via Eq. ( 18)) at ℓ = π/8, which agree very well with the numerical action minimization results.Also shown is the asymptotic action as H → ∞, which agrees with Eq. ( 58) and can be obtained from Eqs. ( 70) and (72) by considering the limit k → 1 with E(k) → 1 and K(k) ≃ 1 2 ln 1 1−k .As one can see from Fig. 4, the asymptotic relation (71) is not yet satisfied for the moderately small ℓ = π/8: noticeably, the solution h(x, 1) at the final time deviates from Eq. (67).However, the numerically found action is already accurately described by Eqs. ( 70) and (72), because the difference between H and H is always subleading -at most O( √ H) -at small ℓ.

VII. SUMMARY AND DISCUSSION
We applied the OFM to evaluate analytically and numerically the short-time PDF P ( H, L, t = T ), and the optimal paths which dominate this PDF, of the KPZ interface on a ring.The short-time PDF has the scaling form (9), where ε ∼ T 1/2 plays the role of the weak-noise parameter.The phase diagram of the system represents the ( H, ℓ = L/ √ νT ) plane.We were especially interested in the DPTs that occur in this system at sufficiently large positive λ H > 0. We found that, depending on ℓ, these DPTs occur via either a supercritical, or a subcritical bifurcation of the "trivial" (uniform in space) optimal path of the KPZ interface.The supercritical bifurcations dominate at very small ℓ, the subcritical bifurcations dominate at very large ℓ.In these two limits we obtained asymptotic analytical solutions for the optimal paths of the system, evaluated the resulting action, and verified the analytical results numerically.We also found that, as T goes to zero at constant L, the PDF acquire a simple large-deviation form (11) and (66).Interestingly, the rate function f ( H) exhibits, at a critical value of H = Hc (ℓ), a DPT which is accidentally second-order.
In the (much more complicated) region of intermediate ℓ = O(1) we observed numerically both supercritical, and subcritical bifurcations of the uniform solution.This region of the phase diagram is presently out of reach of analytical theory.It would be very interesting, but challenging, to determine the complete phase diagram of the system in this region.In particular, it would be interesting to locate, somewhere between ℓ = 16π and ℓ = 32π, at least one critical point ( H * , ℓ * ) where the second order DPT curve H(2) c (ℓ) ends when it meets the first order DPT curve H(1) c (ℓ).These tasks will become more feasible if this problem, as described by Eqs. ( 12)-( 16), joins the list of similar large-deviation OFM problems for the KPZ equation which have been solved exactly by the inverse scattering method (ISM) [18,20].Indeed, as was previously found in Ref. [7], a canonical Hopf-Cole transformation brings Eqs. ( 12) and ( 13) into the nonlinear Schrödinger equation in imaginary space and time.Therefore, Eqs.(12) and (13) belong to a family of completely integrable models.The only problem (but potentially a big one) is to adapt the ISM to a finite system with periodic boundaries and to accommodate the problem-specific boundary conditions (14) and (16).The exact solution would provide a full analytic control of the subleading corrections to the action of branch 2, which are presently half-empiric.
It would be very interesting to explore the possibility of extending to the spatially averaged KPZ interface height some of the recent "stochastic integrability" approaches, which led, for selected initial conditions, to exact representations for the complete statistics of the one-point interface height [29][30][31][32][33][34][35][36][37].
Finally, we can try to put our results into a broader perspective of DPTs in large deviations of macroscopic systems far from equilibrium.The second-order DPT in the present periodic KPZ system occurs when the unstructured optimal height profile h(x, t) = Ht, uniformly translating in the vertical direction, gives way to spatially non-uniform height profiles via a supecritical bifurcation.This transition can be compared with second-order DPTs in large deviations of the current [38][39][40][41] and of the system activity [42] in some diffusive lattice gas models with periodic boundaries.In each of these systems a spatially uniform state gives way to a non-uniform state via a supercritical bifurcation.The structured states, however, have the form of simple travelling waves, whereas in the present system they are more complicated.
A second-order DPT is also observed in large deviations of the one-point height of a KPZ interface in infinite system with Brownian initial conditions [7,8,11,20].The underlying bifurcation is also supercritical, but the transition mechanism is quite different.Indeed, the optimal height profiles h(x, t) are non-uniform and essentially non-stationary both below and above the transition, while the transition is accompanied by a mirror symmetry breaking.
Similarly, the first-order DPT, that we reported here, differs from its counterpart in large deviations of the onepoint height of a KPZ interface at a shifted point in the KPZ equation with Brownian initial conditions [11].These differences between mechanisms of different DPTs of the same order call for a better understanding of DPTs in general.
equations and uniform solution III.Bifurcations of the uniform solution IV.Numerical results V. Large-ℓ asymptotics: Rise and fall of the soliton A. General description of the solution B

FIG. 4 .
FIG.4.Numerically found spatially slightly non-uniform solution of Eqs.(12) and(13) for H = 270.319and a moderately small rescaled system size of ℓ = π/8 with numerical resolution nx = 64 and nt = 8000.The numerically found action S = 14322.081deviates by 0.009% from the predicted small-ℓ result S = 14320.806for this H as given by Eq. (70).The action for the spatially uniform solution(19) for the same H and ℓ would be S = 14347.685.The maximum height at final time is H = 270.357.Left: Optimal height profile h(x, t) at different times t, with the maximum at each t subtracted in order to emphasize deviations from spatial homogeneity.The prediction (67) for intermediate times is indicated by the black dashed line and agrees well with the numerical solution for t = 0.25, t = 0.5 and t = 0.75.The inset shows that the growth of the maximum maxx h(x, t) in time is still linear as predicted.Center: conjugate momentum density p(x, t) with Λ = 261.057subtracted, compared to the analytical result (68) indicated by the black dashed line.Right: Spatial maximum maxx p(x, t) over time to visualize the long lifetime of the stationary cnoidal solution with some small boundary layers in time at t = 0 and t = 1.

FIG. 5 .
FIG.5.Analytical large-ℓ results of Sec.V (lines) vs. numerics (dots) for the rate function S( H) (left) and Lagrange multiplier Λ( H) (right) for a rescaled system size of ℓ = 16π.The numerical computations were performed at resolutions nx = 1024 and nt = 4000.The analytically found branches for the action S( H) and Lagrange multiplier Λ( H) at large ℓ are drawn as the colored lines according to the results summarized in Eq. (31), with Λ( H) then obtained from Eq. (18).The solid vertical lines indicate the theoretical critical points in the large-ℓ limit at H ≃ 4 and H ≃ ℓ 2 /6.The dashed vertical line shows where the spatially uniform solution becomes unstable in the (m = 11) spatial Fourier mode, as given by the minimization of Eq. (27) over the allowed wave numbers.The dotted green line corresponds to the asymptotic behavior S( H) = (8√ 2/3) H3/2 , see Eq. (58).A better resolved transition region close to H = 4 is shown in Fig.7.

FIG. 6 .
FIG.6.Numerically found least-action solutions of Eqs.(12) and(13) for two mean heights H = 144.198(top row) and H = 507.56(bottom row) at ℓ = 16π with numerical resolution nx = 2048 and nt = 4000.Both solutions are one-soliton solutions for branch 2 and 3, respectively.The action of the solution in the top row is S = 28723.766,which deviates by 0.4% from the predicted large-ℓ result S = 28590.660given by Eq. (50).The action for the bottom row solution is S = 102754.528,so it deviates by 0.2% from the predicted value S = 102522.498for this mean height given by Eq. (56).The qualitative difference of these solutions is whether the two edges of the growing triangle for h collide.The left column shows the optimal height profile h(x, t) at different times t, together with the prediction (44), (45) and (55) for the final time t = 1.The center column shows the corresponding conjugate momentum density p(x, t).The right column shows the spatial maximum maxx p(x, t) over time to visualize the lifetime of the stationary soliton solution and compare it to the analytical expressions τ = 1/2 for branch 2 and Eq.(59) for branch 3 (dashed vertical lines).

FIG. 7 .
FIG.7.Numerically computed action S( H) and Lagrange multiplier Λ( H) for different ℓ in the transition region not far from H = 4, where the transition between branches 1 and 2 at large ℓ is predicted, see Eq.(31).Using the numerical minimization techniques described in the Appendix, we search for (possibly multiple distinct local) minimizers of the action functional(7) for given ℓ and H.When there are more than one minimizers for the same ℓ and H, the one with the least action gives the true optimal solution.For a better visualization shown is the difference between the numerically computed action of the found solutions and the action ℓ H2 /2 of the spatially uniform solution.The vertical dashed lines indicate the smallest H where a spatial Fourier mode q = 2πm/ℓ of the uniform solution first becomes unstable according to Eq. (27) (m = 3 for ℓ = 4π, m = 5 for ℓ = 8π, m = 11 for ℓ = 16π, and m = 21 for ℓ = 32π).For ℓ = 4π, 8π and 16π, the rate function displays both a second-order transition at the predicted point, and a first-order transition at slightly larger H where the one-soliton solution (see the top row of Fig.6), described theoretically in Sec.V B, becomes optimal.At the largest ℓ = 32π, only a first-order transition from the uniform to the one-soliton solution is observed, while oscillating solutions are irrelevant.See Sec.V D for a more detailed analysis of the transition region at large but finite ℓ.Note that for ℓ = 16π one can also see, around H = 4.7, another oscillating solution with a second superimposed wave number.Numerical resolutions used: nx = 128 and nt = 2000 for ℓ = 4π, nx = 512 and nt = 4000 for ℓ = 8π, nx = 1024 and nt = 4000 for ℓ = 16π, and nx = 2048 and nt = 4000 for ℓ = 32π.

FIG. 8 .
FIG.8.Two types of sub-optimal solutions to Eqs.(12) and(13) for ℓ = 16π.The top row (obtained with resolution nx = 2048 and nt = 4000) shows a two-soliton solution for the same value of H = 144.198as in the top row of Fig.6.The action of the two-soliton solution, S = 28746.682,is larger than that for the one-soliton solution for the same ℓ and H.The bottom row shows a spatially oscillating solution, originating from linear instability of the m = 11 mode of the spatially uniform solution for H ≳ 4.607 as predicted by Eq. (27).Here the resolution is nx = 1024 and nt = 4000.As can be seen from Fig.7(bottom left), this family of solutions can be optimal for small H and moderate ℓ, but it becomes irrelevant as large ℓ.
FIG.12.Numerical data from high-resolution optimal path computations at rescaled system sizes ℓ = 8π and 16π.Assuming an asymptotic expansion (60), the ordinate should be given by the subleading terms B + C/H with constants B and C that are independent of ℓ and H. Accordingly, we performed least-squares fit with this functional form to the data for 20 ≤ H ≤ 125.We observe a fair agreement for the two rescaled system sizes and determine B ≃ 5.3.For the smaller domain size ℓ = 8π there are small oscillations which may come from sub-subleading terms not included in Eq. (60).

FIG. 13 .
FIG. 13.The rescaled actions Sν /(ℓ H2 ) for ν = 1 and ν = 2 versus H for a very long domain ℓ = 128π.The result for ν = 1 (branch 1) is shown by the blue lines, the prediction of Eq. (62) for ν = 2 (branch 2) is shown by the black lines.The solid lines correspond to the global minimum of the action, whereas the dashed lines correspond to local but not global minima of the action.The thin dotted line shows the asymptotic theoretical prediction for S2/(ℓ H2 ) = (4 H − 8) H−2 for ℓ → ∞.