A Normal Form for the Onset of Collapse: the Prototypical Example of the Nonlinear Schrodinger Equation
S.J. Chapman, M.E. Kavousanakis, I.G. Kevrekidis, P.G. Kevrekidis
aa r X i v : . [ n li n . PS ] A ug A Normal Form for the Onset of Collapse:the Prototypical Example of the Nonlinear Schr¨odinger Equation
S Jon Chapman
Mathematical Institute, University of Oxford, AWB, ROQ, Woodstock Road, Oxford OX2 6GG
M. Kavousanakis
School of Chemical Engineering, National Technical University of Athens, 15780, Athens, Greece
I.G. Kevrekidis
Department of Chemical and Biomolecular Engineering &Department of Applied Mathematics and Statistics,Johns Hopkins University, Baltimore, MD 21218, USA
P.G. Kevrekidis
Department of Mathematics and Statistics, University of Massachusetts, Amherst MA 01003-4515, USA andMathematical Institute, University of Oxford, AWB, ROQ, Woodstock Road, Oxford OX2 6GG (Dated: August 21, 2020)The study of nonlinear waves that collapse in finite time is a theme of universal interest, e.g.within optical, atomic, plasma physics, and nonlinear dynamics. Here we revisit the quintessentialexample of the nonlinear Schr¨odinger equation and systematically derive a normal form for theemergence of blowup solutions from stationary ones. While this is an extensively studied problem,such a normal form, based on the methodology of asymptotics beyond all algebraic orders, unifiesboth the dimension-dependent and power-law-dependent bifurcations previously studied; it yieldsexcellent agreement with numerics in both leading and higher-order effects; it is applicable to bothinfinite and finite domains; and it is valid in all (subcritical, critical and supercritical) regimes.
Introduction.
The nonlinear Schr¨odinger (NLS)model [1–4] has, arguably, been one of the most centralnonlinear partial differential equations (PDEs) withinMathematical Physics for the last few decades. Its wideappeal stems from the fact that it is a ubiquitous enve-lope wave equation arising in a variety of diverse physicalcontexts. Its applications span water waves [5–7], non-linear optical media [8, 9], plasma physics [10] and morerecently the atomic physics realm of Bose-Einstein con-densates and their variants [11, 12].The solitonic waveforms of the NLS model have beencentral to all of the above investigations. A similarlyprominent feature of the NLS model is its finite-time,self-similar blowup in higher (integer) dimensions or forhigher powers of the associated nonlinearity. Indeed, thelatter manifestation of lack of well-posedness has beencentral to both books [3, 13, 14] and reviews [15–17] andhas been the objective of continued study not only inthe physical literature, but also in the mathematical one;see, e.g., [18–20] and [21, 22] for only some recent ex-amples (and also references therein). Importantly forour purposes, these focusing aspects have become acces-sible to physical experiments. On the one hand, thereis the well-developed field of nonlinear optics, where notonly the well-known, two-dimensional collapsing wave-form of the Townes soliton has been observed [23], butalso more elaborate themes have been touched upon in-cluding the collapse of optical vortices [24], the loss ofphase information of collapsing filaments [25] or the ma- nipulation of the medium to avert optical collapse [26].On the other hand, there is the flourishing area of Bose-Einstein condensates where the Townes soliton has re-cently been announced [27]. Here, collapsing waveformsin higher dimensions had been experimentally identifiedearlier [28, 29] and the ability to manipulate the nonlin-earity [30] and the initial conditions [31] has continuedto improve in recent times.The emergence of collapsing solutions out of solitonicones is a topic that has been long studied since the earlyworks of [32, 33] and summarized in numerous reviewsand books [3, 13, 14]. Nevertheless, remarkably, a normalform—a prototypical model equation compactly describ-ing the relevant bifurcation, namely the onset of collaps-ing solutions out of non-collapsing ones—does not exist,to the best of our knowledge. Recent attempts to captureeven the well-known log-log law of the critical case and itscorrections [18] will confirm that. It is known that at thecritical point at which collapse emerges, σd = 2, where σ is the exponent of the nonlinearity and d the spatialdimension of the NLS model, a symmetry enabling self-similar rescaling of the solution towards becoming singu-lar at a finite time (the so-called pseudo-conformal invari-ance) arises. Beyond this critical point, solitary wavesbecome unstable and, in a form somewhat reminiscentof the traditional pitchfork bifurcation, two collapsingbranches of solutions emerge [34]. Yet, this is no ordi-nary pitchfork like, e.g., the one experimentally probedin BECs in double-well potentials [35]. Here, pseudo- σ -1-0.500.51 G FIG. 1. Variation of the blowup rate G , as a function of σ for d = 1, domain size K = 50. PDE results (black lines) ob-tained from Eq. (3) are in excellent agreement with the O ( G )asymptotic solution (red lines). The solitonic branch ( G = 0)is stable up to σ = 2 (solid line), and becomes unstable for σ > G >
G <
0) is illustrated withdash-dotted line and open squares. conformal symmetry breaks and, thus, collapse phenom-ena will not follow the standard cubic pitchfork normalform, but rather are associated with the exponentially-small, beyond-all-algebraic-orders phenomenology of therelevant symmetry breaking. Our aim is to go beyond theheuristic (steady state only) arguments of earlier stud-ies [32, 33] and present a systematic derivation of theassociated normal form. Key features of our analysis are: • We unify the case of general nonlinearity exponentand that of arbitrary dimension, offering a result broadlyapplicable in the above physical settings of interest. • Our analysis captures both the case of the critical log-log collapse and the supercritical t − / collapse . • Crucially, we capture not only the leading collapse or-der but also systematically the higher-order corrections . • We find excellent agreement with computations of the stationary solutions and of the dynamical evolution.
Problem Formulation & Asymptotic Analysis.
Uponexposing the general formulation of the problem, we willsolve it separately in the near and far fields. The far fieldhas a turning point , resulting in an exponentially-smallreflection back towards the near field [36]. Matching withthe near field solution yields our onset of collapse normalform, bearing this exponentially small contribution.We start with the NLS in dimension d and nonlinearitypower determined by the exponent σ as:i ∂ψ∂t + ∂ ψ∂r + ( d − r ∂ψ∂r + | ψ | σ ψ = 0 . (1)We will perturb around the critical (radially symmetric)case dσ = 2 [3, 13]. Introducing the well-known stretchedvariables [3, 13, 34] ξ = rL , τ = Z t d t ′ L ( t ′ ) , ψ ( r, t ) = L − /σ e i τ v ( ξ, τ )leads toi ∂v∂τ + ∂ v∂ξ + ( d − ξ ∂v∂ξ + | v | σ v − v − i G (cid:18) ξ ∂v∂ξ + 1 σ v (cid:19) = 0 , (2)where the blowup rate G = − LL t = − L τ /L . In thisdynamic change of variables, and in order to close thedynamics in this “co-exploding” frame (upon determin-ing G ( τ )), we impose a pinning condition of the form[34] Z ∞−∞ Re( v ( ξ, τ )) T ( ξ ) d ξ = C, for some constant C and some (essentially arbitrary)“template function” T , to enable us to uniquely identifythe solution v and the blowup rate G . In our numericalexamples we choose T = δ ( ξ −
2) [37]. Finally, we write v ( ξ, τ ) = V ( ξ, τ )e − i G ( τ ) ξ / to give (using G ′ ≡ d G/ d τ )i ∂V∂τ + G ′ ξ V + ∂ V∂ξ + ( d − ξ ∂V∂ξ + | V | σ V − V − i( dσ − G σ V + G ξ V = 0 . (3) Near Field.
Motivated by pseudo-conformal invari-ance, we aim to solve (3) in the limit G → dσ → a posteriori ) that the solutionevolves exponentially slowly (in G ), and that σ and d areexponentially close to σ c , d c satisfying d c σ c = 2 [3, 13].Thus, the second from the left and from the right terms in Eq. (3) can be neglected for now. We look for a solution: V = e iΦ( τ ) ( V G ( ξ, τ ; G ( τ )) + V exp ( ξ, τ )) (4)where V G is the (real) regular algebraic expansion in G , V exp is exponentially small in G , and the exponentially-slowly-varying phase Φ is determined by the pinning con-dition. - - - - l og ( G ( σ − )) /G e π / G G ( σ − ) / σ /G FIG. 2. Collapsing solution branch for d = 1, domain size K = 50. Top panel: the leading-order asymptotic solution(black) is shown against a stationary numerical solution of(2) (red). The two lines essentially coincide. The weak undu-lations are due to the sinusoidal term in (18). Bottom panel: exponentially scaled illustration of the same result to showthe accuracy of our higher-order analysis. The asymptoticsolutions shown are leading order (blue), accurate to O ( G )(yellow), accurate to O ( G ) (green); the full numerical resultis in black. To obtain the near-field solution, we expand the solu-tion in powers of G as V G = ∞ X n =0 G n V n , Φ = ∞ X n =0 G n +1 Φ n ; (5)this gives the leading-order equation ∂ V ∂ξ + ( d c − ξ ∂V ∂ξ + V σ c +10 − V = 0 , (6)the solution of which is the critical ground-state soliton.The next order V then satisfies: ∂ V ∂ξ + ( d c − ξ ∂V ∂ξ + (cid:18) d c + 1 (cid:19) V σ c V − V = − ξ V , with V ′ (0) = 0, and V → ξ → ∞ . Far Field.
The above expansion (5) breaks down atlarge distances. In the far field we rescale ξ = ρ/G togive G ∂ V G ∂ρ + G ( d c − ρ ∂V G ∂ρ + | V G | σ c V G − V G + ρ V G = 0 . The exponential decay of V G renders it exponentiallysmall in the far field, allowing us to neglect the nonlinearterm V σ c +1 G . We now look for a WKB-solution as: V G ∼ G k e φ ( ρ ) /G ∞ X n =0 A n ( ρ ) G n . (7)At leading order this gives the eikonal equation:( φ ′ ) = 1 − ρ ⇒ φ = − Z ρ (cid:18) − ¯ ρ (cid:19) / d¯ ρ (8)(so that V G is decreasing in ρ ). Note the turning pointat ρ = 2 from Eq. (8). The amplitude equation for A then leads to: A = a ρ ( d c − / ( − φ ′ ) / = 2 / a ρ ( d c − / (4 − ρ ) / , for some constant a , which we will determine by match-ing with the near field. As ρ →
0, the far field yields: G k e φ ( ρ ) /G A ∼ a G k e − ρ/G ρ ( d c − / . (9)As ξ → ∞ , the near field expression is dominated by: V ( ξ ) ∼ A d c e − ξ ξ ( d c − / = A d c G ( d c − / e − ρ/G ρ ( d c − / , (10)for some dimension-dependent constant A d c . We note,in particular, the values A = 12 / (from the quinticNLS exact soliton solution [3]), while A ≈ .
518 [18].Matching (9) with (10) gives k = ( d c − / a = A d c .For ρ > φ ′ = i (cid:18) ρ − (cid:19) / has a finite Hamiltonian. Thus for ρ > V G = αG k e i φ ( ρ ) /G ∞ X n =0 B n ( ρ )(i G ) n , (11)for some constant α , where φ = Z ρ (cid:18) ¯ ρ − (cid:19) / d¯ ρ, B ( ρ ) = 2 / a ρ ( d c − / ( ρ − / . The fact that only one of the oscillatory exponentialsis present in ρ > exponentially small reflec-tion back towards the near field, which we will obtain byanalysing the turning point region. This is a key featureof our exponential asymptotics analysis.
Turning Point.
Writing ρ = 2 + G / s , the equationnear the turning point becomes, to leading order,d V G d s + sV G = 0 , with solution V G = λ Ai( − s ) + µ Bi( − s ), where Ai and Biare Airy functions of the first and second kind, respec-tively. The asymptotic expansions of Ai and Bi give: V G ∼ λ e − − s ) / / √ π ( − s ) / + µ e − s ) / / √ π ( − s ) / as s → −∞ , (12) V G ∼ e s / / √ πs / (cid:16) λ e − i π/ + µ e i π/ (cid:17) + e − s / / √ πs / (cid:16) λ e i π/ + µ e − i π/ (cid:17) as s → ∞ . (13)Matching with (7) and (11) gives α = e i π/ and λ = i µ = a i √ πG / e φ (2) /G . Including both WKB solutions in ρ < V G ∼ (cid:16) e φ ( ρ ) /G + γ e − φ ( ρ ) /G (cid:17) G k ∞ X n =0 A n ( ρ ) G n , (14) where matching with (12) gives γ = i2 e φ (2) /G = i2 e − π/G . Exponentially small correction to the nearfield. As ρ → γ e − φ ( ρ ) /G G k ∞ X n =0 A n ( ρ ) G n ∼ a γG ( d c − / e ρ/G ρ ( d c − / . (15)This term will match with the exponentially small cor-rection to the near field. In the original near-field scaling,using Eq. (4) neglecting time derivatives and quadraticterms in V exp , but keeping all the exponentially-smallterms, gives ∂ V exp ∂ξ + ( d c − ξ ∂V exp ∂ξ + V σ c G (cid:0) σ c V ∗ exp + ( σ c + 1) V exp (cid:1) − V exp + G ξ V exp = − ( d − d c ) ξ ∂V exp ∂ξ − i ∂V G ∂τ + Φ ′ V G − G ′ ξ V G − σ − σ c ) V σ c +1 G log V G + i( dσ − G σ V G , where Φ ′ = dΦ / d τ . We now use V exp = U exp + i W exp and separate into real and imaginary parts. Since V G satisfiesthe homogeneous version of the equation for W exp , this enables a solvability condition: multiplying that equation by ξ d c − V G , integrating from 0 to R , and using (6), we obtain: ξ d c − V G ( R ) ∂W exp ∂ξ ( R ) − ξ d c − W exp ( R ) ∂V G ∂ξ ( R ) = − Z R ξ d c − V G ∂V G ∂τ − ξ d c − ( dσ − G σ V G d ξ (16)As R → ∞ we evaluate the boundary terms by matchingusing (15), givinglim R →∞ ξ d c − V G ( R ) ∂W exp ∂ξ ( R ) − ξ d c − W exp ( R ) ∂V G ∂ξ ( R ) ∼ lim R →∞ a e − R (cid:0) a Im( γ )e R (cid:1) − (cid:0) a Im( γ )e R (cid:1) ( − a e − R )= 2 a Im( γ ) . Now Z ∞ ξ d c − V G d ξ ∼ Z ∞ ξ d c − ( V + G V + · · · ) d ξ ∼ b + 2 G c + · · · , say, where b = Z ∞ ξ d c − V d ξ, c = Z ∞ ξ d c − V V d ξ. Thus the solvability condition (16) ultimately results in:2 c d G d τ = ( dσ − b σ − A d c e − π/G G . (17) which is the normal form for the onset of collapse . Inprinciple a , b , and c are leading terms in power se-ries expansions in G above, and we can calculate the fullpower series. In some of our numerical examples we in-clude the O ( G ) and O ( G ) corrections to these terms.One can discern similarities of Eq. (17) with the pitchforkbifurcation normal form: the natural bifurcation param-eter is r = ( dσ − G , it canbe seen that for all r < G = 0 is the only equilibriumbranch of solutions. When r >
0, the dynamics tendstowards the non-trivial (stable, collapsing) steady statesolution of Eq. (17). Changing the sign of G and τ andof the imaginary part W (in Eq. (3)), we obtain the fi-nal branch of this unusual pitchfork bifurcation diagram,a solution that is a mirror image but is stably collaps-ing in negative (rather than positive) time, i.e., “comingback from infinity”. These are some of the intriguing by-products of unfolding the original Hamiltonian dynamicalsystem of Eq. (1) into the dissipative renormalized frameof Eq. (3). Moreover, a key feature of this collapse nor- - - - U , W x U , W x FIG. 3. Comparison of the numerical solution (black, dashed)with the asymptotic solution accurate to O ( G ) for K = 50,for the real ( U , red) and the imaginary ( W , blue) parts of thesolution. The main plot shows the near field and the insetshows the far field. mal form is its exponentially small (large) nonlinear term(rather than the usual cubic in the standard pitchfork),yielding a nearly vertical bifurcation for G = G ( σ ), asshown in Fig. 1. Notice that our analysis is still valid for r = 0. Finite Domain.
Usually, when numerically simulat-ing (1) or (3) the domain is truncated to some large butfinite domain [0 , K ]. In that case both oscillatory WKBsolutions are present in ρ >
2, and the ratio of theiramplitudes is determined by the position of the bound-ary and the nature of the boundary condition. A similaranalysis can be performed, and the result is a more com-plicated expression for the coefficient γ , the prefactor ofthe reflection term at the turning point. For example,imposing a Neumann condition on v at ξ = K results inIm( γ ) = (1 − ν )e − π/G − ν sin(2 φ ( KG ) /G )) + ν ) , (18)where ν = p ( KG ) − − KG p ( KG ) − KG .
We see that as K → ∞ , ν → γ ) → e − π/G / Numerical Verification.
Equation (17) predicts the ex-istence of a stable branch of solutions bifurcating from dσ = 2. We compare this prediction with direct numeri-cal simulations of (2) by fixing d = 1 and varying σ closeto σ c = 2. The relevant bifurcation diagram can be seenin Fig. 1. Here, we compare the PDE results obtaineddirectly from Eq. (2) with the normal form of Eq. (17)finding excellent agreement between the two. The defini-tive comparison of the full NLS results with those of ournormal form is illustrated in Fig. 2. The top panel clearlyshowcases the exponential nature of the relevant bifurca-tion over of the associated ODEand PDE data in excellent agreement between the two.Notice that the finite nature of the computation leads tosome nearly imperceptible oscillations in the top panel
100 200 300 400 τ ξ -0.015-0.01-0.00500.0050.01 τ G FIG. 4. K = 50, σ = 2 . ξ − τ space) of | v | − | V | .The inset shows the evolution of G ( τ )for the numerical solution (red), and O ( G ) asymptotic so-lution (black). The renormalized NLS reaches a steady-statesolution after τ ≈ of the figure, also observed but not commented in earlierworks [32, 34]. The full power of our methodology is re-vealed when factoring out the exponentially small leadingorder by rescaling through e π/G as shown in the bottompanel of Fig. 2. In addition to the leading-order behav-ior we present the first- and second-order corrections, il-lustrating how they progressively match in a remarkably quantitative fashion the PDE results. To complement thequality of the match, we show in Fig. 3 how we can cap-ture not only the rate of collapse, but also near perfectlyboth the real and the imaginary parts of the profile ofthe associated solution U + i W .Lastly, we note that our methodology not only offersa tool for capturing the statics (i.e., the equilibrium col-lapse branch and its spatial profile), but also enables anexcellent capturing of the associated dynamics as shownin Fig. 4. Here, in addition to the spatio-temporal evo-lution of the field in the ( ξ, τ ) variables, the evolutionof the collapse rate G ( τ ) towards its stable asymptoticvalue is observed in the inset, and compared against thenumerical solution showing excellent agreement. Conclusions.
In the present work we have revisitedthe fundamental problem of the collapse of a nonlinearSchr¨odinger equation. We have offered a unified per-spective of the emergence of the self-similar solutionsvia a mathematically compact, yet quantitatively accu-rate normal form that combines the famous log-log be-havior at the critical point, the emergence of a stableself-similarly collapsing branch past that point, the expo-nentially small (large) breaking of the pseudo-conformalinvariance of the critical point, the Hamiltonian natureof the original model and the dissipative features of therenormalized dynamics. In our view this constitutes ageneric and broadly applicable (in optics, BECs and be-yond) normal form associated with the onset of collapse.The identification of this normal form prompts numer-ous exciting questions for the future, such as, e.g., theexamination of the stability of the collapsing solutions orthe examination of a potential normal form for general-ized Korteweg-de Vries equations [38] and their travelingwaves that are of broad relevance to water waves andplasmas. This analysis may also pave the way for thestudy of self-similar periodic orbits that have recentlyemerged in interfacial hydrodynamics [39].This material is based upon work supported by the USNational Science Foundation under Grants No. PHY-1602994 and DMS-1809074 (PGK) and by the US AROMURI (IGK). PGK also acknowledges support from theLeverhulme Trust via a Visiting Fellowship and thanksthe Mathematical Institute of the University of Oxfordfor its hospitality during part of this work. [1] M.J. Ablowitz and P.A. Clarkson,
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