A universal asymptotic regime in the hyperbolic nonlinear Schrödinger equation
AA universal asymptotic regime in the hyperbolicnonlinear Schr¨odinger equation
Mark J. Ablowitz, Yi-Ping Ma, Igor Rumanov ∗ Department of Applied Mathematics, University of Colorado, Boulder, CO 80309 USA
October 16, 2018
Abstract
The appearance of a fundamental long-time asymptotic regime in the two space onetime dimensional hyperbolic nonlinear Schr¨odinger (HNLS) equation is discussed. Basedon analytical and extensive numerical simulations an approximate self-similar solutionis found for a wide range of initial conditions – essentially for initial lumps of small tomoderate energy. Even relatively large initial amplitudes, which imply strong nonlineareffects, eventually lead to local structures resembling those of the self-similar solution,with appropriate small modifications. These modifications are important in order toproperly capture the behavior of the phase of the solution. This solution has aspectsthat suggest it is a universal attractor emanating from wide ranges of initial data. ∗ corresponding author; e-mail: [email protected] a r X i v : . [ n li n . PS ] J un Introduction
The (2+1)-dimensional hyperbolic nonlinear Schr¨odinger (HNLS) equation, i∂ Z Φ + ( ∂ xx − ∂ yy )Φ + | Φ | Φ = 0 , (1 . Z corresponds to propagation distance, x to thetransverse spatial direction, y to a retarded time and Φ to the complex amplitude of theelectromagnetic field [1]. In water wave applications Z typically is related to physical timeshifted by the group velocity while x and y correspond to the horizontal dimensions; seee.g. [4]. For other applications x , y and Z (e.g. plasma waves [51]) are related to the threephysical spatial axes. It is important in all these applications to study the various long-timepropagation regimes in HNLS equation.An exact similarity solution of eq. (1.1) was found in [47] and independently in [4]. It hasthe form Φ = Λ Z exp (cid:18) i (cid:20) s + θ − Λ Z (cid:21)(cid:19) , s = x − y Z , (1 . s defined above is the similarity variable.The main result of this paper is to show that the solution eq. (1.2), with a remarkablysmall modification, appears at long times (i.e. large Z ) in the central zone of the ( x, y )-planefor a wide range of initial conditions. This is basically the case for all initial conditions suchthat some lump of energy is initially present in the neighborhood of the origin in the ( x, y )-plane; hence we term this solution as universal. The form of the modified similarity solutionis found to be Φ = Λ Z exp (cid:18) i (cid:20) s + θ + η − Λ Z (cid:21)(cid:19) , (1 . A )where θ , Λ , η depend on the initial conditions. This is more fully discussed in the sectionsbelow. We only remark that we do not consider either rapidly varying or large initial data;this is consistent with the derivation of the HNLS from physical principles where all termsare of the same order.Apart from the linear and nonlinear stability analysis cf. [54, 44, 8, 39, 40] (and also numer-ous references therein) which are usually dedicated to the stability of exact one-dimensional so-lutions of the 1D-NLS equation (e.g. solitons) (and thus also exact solutions of the HNLS eq.),2here are relatively few analytical results in the literature regarding the general behavior of so-lutions of the HNLS equation, though some results can be found in [37, 33, 7, 24, 9, 11, 30, 45].Studies conducted in the 1970s-1990s are summarized in the reviews [44, 8] and thebook [51] where both elliptic NLS and HNLS in various dimensions are considered.Since the seminal work [54] it is known that one-dimensional solitons in multidimensionalNLS equations are unstable with respect to transversal perturbations, see also e.g. [5]. Re-cently there has been more research regarding the types of instability and the growth ratesof various instabilities in the HNLS eq.; see e.g. [40, 39] and their associated experimentaldemonstration [25, 26, 27].It is known that nonelliptic NLS equations do not admit localized traveling wave so-lutions [24]. This is consistent with our observations that lumps of energy in the HNLSeq. eventually disperse, and in turn, lead to the universal asymptotic solution described here.The underlying structure can consist of many localized hyperbolas. Sometimes we observe anumber of such hyperbolic structures with different centers, partly superimposed. These maybe preceded by more intricate structures at intermediate “times” Z .Apart from analyzing the development of instabilities of one-dimensional solutions [25,26, 27], there has been some numerical and experimental research on the HNLS equation [33,46, 42, 32, 13, 36, 30]. There has also been a number of studies of the (3+1)-D HNLS eq.,see e.g. [43, 8, 12, 13, 17] and references therein. The 3D case has attracted researchers dueto a wide variety of applications ranging from short (femtosecond) laser beams in condensedmedia [43, 12, 15, 17, 50], cyclotron waves in plasma [41, 47, 37, 10, 12, 13] and high energynonlinear electromagnetic phenomena [49].Given its numerous applications (surface waves in deep water, optical pulses in planarwaveguides etc.) and its fundamental role as an intrinsically simple (2+1)-dimensional nonlin-ear equation, the HNLS eq. is a laboratory for novel types of phenomena and its behavior canshed light on related problems such as the more complicated (3+1)-D HNLS eq. Indeed manyprocesses associated with the (3+1)-D HNLS eq. such as pulse splitting, multi-filamentation,fragmentation, so-called snake and neck instabilities and nonlinear X-waves, have qualitativecounterparts in the HNLS eq. [32, 12, 13, 25, 30, 26, 27].Interestingly, there is still controversy as to whether there exists a catastrophic collapse,or blowup with infinite singularity formation in the (3+1)-D HNLS eq. [34, 20, 12, 55, 56, 13].In this respect, the situation with the HNLS eq. is clearer. There is a simple convincing,though non-rigorous, argument regarding the absence of catastrophic collapse in this case,see e.g. [20]. The HNLS eq. without the second-derivative defocusing term ( ∂ yy Φ in eq. (1.1))is the integrable 1D NLS eq. which is known to have no collapse (and possess multisolitonsolutions). Adding the second-derivative term which causes defocusing in the transversaldirection should only improve the situation, further dispersing the energy. There are somerigorous quantitative arguments [9], based on virial inequalities (i.e. inequalities for secondmoments or variances and their Z -derivatives), which show that total collapse, i.e. finite3nergy concentration on sets of measure zero (points or lines) is impossible for the (2+1)-DHNLS eq. This does not rigorously rule out a blowup singularity formation but, together withthe qualitative argument above, makes it much less plausible.Virial-based arguments like those in [9] suggest that the HNLS eq. favors structuresstretched along the defocusing direction leading to hyperbolas in the ( x, y )-plane. This issimilar to X-wave phenomena cf. [15, 16, 30] which exhibit characteristic X-shapes in the ( x, y )-plane formed by the lines x = ± y and characteristic hyperbolas asymptotic to these lines.While such existing exact solutions have infinite energy (i.e. infinite L -norm (cid:82) (cid:82) | Φ | dxdy ),their finite-energy counterparts have been observed in experiments on electromagnetic beampropagation; they appear in the central cores of the beams and split at sufficiently largepropagation distance Z , see e.g. [15, 16]. When the phenomenon is described by the HNLSequation [12], the end result of their splitting must be the hyperbolic structure which weobserve numerically and describe analytically below.Recently some exact X-wave solutions of the HNLS eq. and the (3+1)-D HNLS equationwith an additional supporting potential have been found [19] but they still have infinite energy.Also, some infinite energy standing wave solutions were proven to exist in [35]. The existence ofcertain types of bounded and continuous hyperbolically radial standing and self-similar waveswas established in [30] where the asymptotics of such solutions at large hyperbolic radius,i.e. large | x − y | , are computed. The region we consider here is different: x ± y (cid:46) Z i.e. the center zone of the pulse. The scenario, also confirmed by our numerical studies is thatinitial localized lumps asymptotically tend to a similarity solution valid in the above centralzone and falling off sharply (exponentially) beyond it. Although the similarity solution hasinfinite energy, other small amplitude solutions, which can be obtained by WKB methods andmatched to these similarity solutions, exist in regimes away from the core; cf. e.g. [5].A number of exact solutions based on symmetry reductions using Lie group invariancemethods, have been constructed for both the NLS and HNLS eq. [52, 23, 22, 21, 14, 38]; thiswas also recently revisited in application to HNLS eq. in [28]. We summarize some of thiswork in Appendix A. Some of these similarity reductions and exact solutions may be relevantat intermediate stages before reaching the long-time regime. They may allow for a betterquantitative description of various phenomena like self-focusing, splitting, pattern formation.The descriptions available so far in literature [7, 10, 8, 12] are only approximate and animproved understanding might be reached by obtaining more sophisticated exact solutions.This question deserves further, more systematic study and we plan to consider it elsewhere.We emphasize, however, that our current results are of importance for these questions sincethey allow one to select among the many complicated transient solutions those which areasymptotically close to the universal regime described here.Of the approximate methods applied to all types of NLS equations, variational methods,though not rigorous, have proven to be very popular. They were used to construct approxi-mate solutions for both the HNLS [7, 45] and the (3+1)-D HNLS equations [10, 12, 13]. They4ere used in [7, 10] to quantitatively understand the self-focusing and pulse-splitting phenom-ena and in [45] to investigate possible mechanisms of generating rogue waves in HNLS. Whilethe range of validity of a variational ansatz remains to be rigorously established, the resultsof using the Gaussian ansatz of [7, 45] for HNLS can be obtained from a usual approximatesolution where the validity and the precision of the approximation are completely clarified,see Appendix B. Thus, the variational approximation can be useful e.g. at long times in Z ,and can be related to the universal solution eq. (1.2A), e.g. the amplitude there also maydecay as the inverse of time or propagation distance Z . However, some limitations of thisapproach are exposed when we compare the phases in Appendix B.It is well-known that similarity solutions play a crucial role in the long time asymptoticsolution of certain integrable nonlinear dispersive wave equations [5, 2]. Equations which arenot known to be integrable, such as the HNLS equation have been less intensively studiedfrom this point of view. For example, the one dimensional integrable NLS equation iu z + u xx + σ | u | u = 0 (1 . u ( x, z ) = Az / exp( iθ ) where θ = x z + σA log z + θ (1 . σ = − z → ∞ the solution tended to the abovesimilarity solution in the central core region. Ablowitz and Segur showed how to includesuitable perturbations and solitons (when σ = +1) cf. [5].It was also shown that similarity solutions played key roles in the long time limit of otherwell-known integrable PDEs, e.g. the Korteweg-deVries (KdV) and modified KdV (mKdV)equations [6, 48]. In the case of the mKdV equation u t − u u x + u xxx = 0 (1 . u ( x, t ) = w ( η ) / (3 t / ) where w satisfies the 2nd Painlev´e equation w (cid:48)(cid:48) − ηw − w = 0 (1 . u tends (up to the factor 3 t / ) to a solution of the Painlev´e equation (1.6)(in the case of KdV, a related ODE) in the long time limit. Indeed, Ablowitz, Kruskal andSegur [3] showed that the decaying solution of mKdV equation had the following property.Corresponding to the boundary condition w ( η ) ∼ r Ai ( η ) , as η → + ∞ (1 . η ) is the well-known Airy function, there were three types of behavior as η → −∞ .5) For | r | < w ( η ) ∼ d ( − η ) / sin (cid:18)
23 ( − η ) / − d log( − η ) + θ (cid:19) (1 . d = − π log(1 − r ); the formula for θ = θ ( r ) is more complicated; see [48].ii) For | r | = 1 (critical), w ( η ) ∼ Sgn ( r ) (cid:18) ( − η/ / − ( − η ) − / / + O (( − η ) − / ) (cid:19) (1 . | r | > w ( η ) ∼ Sgn ( r ) (cid:18) η − η − η η − η ) + O (( η − η ) ) (cid:19) (1 . η = η ( r ). Subsequently, Hastings and McLeod [29] studied case (ii) in detail.What is clear from the above is the important role similarity solutions play in long timeevolution of nonlinear dispersive wave equations.The plan of the paper is the following. In section 2 we compute the perturbative solution ofthe HNLS equation for the Gaussian lump initial condition keeping first order in nonlinearityand determine the form of corrections to its exact solution eq. (1.2) which is relevant for long-time (large Z ) asymptotics. Section 2 also shows how focusing and defocusing can be describedfor moderate amplitudes in the the HNLS equation. In section 3 we present the general large Z asymptotics for both the linear and nonlinear equation, assuming in the last case that thesolution falls off as 1 /Z in the central region. Section 4 presents extensive numerical resultsdemonstrating the appearance of the solution eq. (1.2A) with the corrections discussed insections 2 and 3. Section 5 is dedicated to the discussion of the results. In appendix Awe present some exact similarity reductions of the HNLS equation which may be useful forunderstanding the complicated intermediate dynamics of the HNLS eq. prior to the long-timeregime. Appendix B shows what kind of approximation underlies the variational approachsubject to a Gaussian ansatz; cf. [7, 45], and how it relates to the solutions (1.2-1.2A). It is instructive to compute the first order corrections due to the nonlinearity to the exactsolution of the linearized HNLS equation (see also [1]) i.e. to considerΦ ≈ Φ L + Φ n , (2 . L satisfies 6 ∂ Z Φ L + ( ∂ xx − ∂ yy )Φ L = 0 , (2 . n is found as the first order perturbation to Φ L from i∂ Z Φ n + ( ∂ xx − ∂ yy )Φ n = −| Φ L | Φ L . (2 . x, y, Z = 0) = Φ L ( x, y, Z = 0) = A e − x − y , A constant, then the exact solution of eq. (2.2) isΦ L ( x, y, Z ) = A √ Z + 1 e − x y Z · e iZ ( x − y Z = A L ( x, y, Z ) e iθ L ( x,y,Z ) . (2 . n from eq. (2.3) andeq. (2.4) with Φ n ( x, y, Z = 0) = 0Φ n ( x, y, Z ) = iA (cid:90) Z dZ (cid:48) (cid:112) (16 Z (cid:48) + 1)(16 Z (cid:48) + 9)( R + J ) · e − R ( x y R J · e iJ ( x − y R J , (2 . R = R ( Z (cid:48) ) = 3(16 Z (cid:48) + 1)4(16 Z (cid:48) + 9) , J = J ( Z (cid:48) , Z ) = Z − Z (cid:48) Z (cid:48) + 9 . (2 . Z becomes large. It is then convenient to rewrite eq. (2.5) asΦ n ( x, y, Z ) = I − I , where I = iA (cid:90) ∞ dZ (cid:48) (cid:112) (16 Z (cid:48) + 1)(16 Z (cid:48) + 9)( R + J ) · e − R ( x y R J · e iJ ( x − y R J (2 . I = iA Z (cid:90) ∞ du (cid:112) ( u + 1 /Z )( u + 9 /Z )(( R/Z ) + ( J/Z ) ) · e − R ( x y R J · e iJ ( x − y R J , (2 . u = 4 Z (cid:48) /Z . In the first integral I , we change integration variable to ζ = 16 Z (cid:48) + 116 Z (cid:48) + 9 , Z (cid:48) = (cid:115) ζ − − ζ . (2 . I can be rewritten as I = iA e i ( x − y Z Z (cid:90) / dζ (cid:112) ζ (1 − ζ )(9 ζ − · e − x y ζ Z g ( ζ,Z ) + i ( x − y u − u /Z )4 Z g ( ζ,Z ) (cid:112) g ( ζ, Z ) , (2 . g ( ζ, Z ) = 1 − u Z + u Z , u = u ( ζ ) = (cid:112) (9 ζ − − ζ )4 , u = u + 9 ζ . The last formula is convenient to expand in inverse powers of Z . Restricting the considerationto the central zone x + y (cid:46) Z , we find I = iA e i ( x − y Z Z (cid:90) / dζ (cid:112) ζ (1 − ζ )(9 ζ − ·· (cid:18) u Z − x + y ) ζ Z + i ( x − y ) u Z + 3 u − u Z + O (cid:18) Z (cid:19)(cid:19) . (2 . I , since J = Z (cid:18) − Z u (1 + 9 / ( Z u )) (cid:19) , R = 34 (cid:0) Z u (cid:1)(cid:0) Z u (cid:1) , it is easy to see that, for large Z , I is expanded as I = iA e i ( x − y Z Z (cid:90) ∞ duu (cid:18) O (cid:18) Z (cid:19)(cid:19) = iA e i ( x − y Z Z (cid:18) O (cid:18) Z (cid:19)(cid:19) . (2 . I and I , we obtainΦ n = iA e i ( x − y Z Z (cid:18) C − Z − C ( x + y )16 Z + i ( x − y )12 Z + C Z + O (cid:18) Z (cid:19)(cid:19) , (2 . C , C and C are numerical constants given by C = (cid:90) / dζ (cid:112) ζ (1 − ζ )(9 ζ − ≈ . , C = (cid:90) / √ ζdζ (cid:112) (1 − ζ )(9 ζ − ≈ . ,C = 116 (cid:90) / dζ (cid:112) ζ (1 − ζ )(9 ζ − (cid:18) (9 ζ − − ζ ) − ζ (cid:19) ≈ − . . (2 . /Z for Z (cid:29)
1, we findΦ L = A e i ( x − y Z Z (cid:18) − x + y Z − Z − i ( x − y )64 Z + O (cid:18) Z (cid:19)(cid:19) . (2 . Z asymptotics of Φ for the Gaussian initialcondition, Φ ≈ Φ L + Φ n = A e i ( x − y Z Z (cid:18) iC A − iA Z −− x + y Z (cid:18) C iA (cid:19) − A ( x − y )96 Z − − C iA Z + O (cid:18) Z (cid:19)(cid:19) . (2 . µ + iν = ρe iσ , (2 . µ = − x + y Z − A ( x − y )96 Z − Z + O (cid:18) Z (cid:19) , (2 . ν = C A − A Z − C A ( x + y )128 Z + C iA Z + O (cid:18) Z (cid:19) . (2 . ρ = (1 + µ ) + ν , σ = arctan ν µ . (2 . ∼ /Z in ρ and σ , we find ρ = 1 + C A − C A Z −− x + y Z (cid:18) C C A (cid:19) − A ( x − y )48 Z − Z (cid:18) − (72 C C + 1) A (cid:19) + O (cid:18) Z (cid:19) , (2 . σ = arctan C A (cid:16) − A Z + ( C − C ) A ( x + y )128 Z + C A ( x − y )768 Z + (cid:0) C + C (cid:1) A Z (cid:17) C A / − C A · (48) Z (1 + ( C A / ) + O (cid:18) Z (cid:19) . (2 . A , we should keeponly the terms up to order ∼ A in the last formulas. Then they simplify to ρ = 1 − x + y Z − A ( x − y )48 Z − Z + O (cid:0) A (cid:1) + O (cid:18) Z (cid:19) , (2 . σ = C A − A Z + ( C − C ) A ( x + y )128 Z + ( C + 32 C ) A Z + O (cid:0) A (cid:1) + O (cid:18) Z (cid:19) . (2 . Ae iθ with amplitude and phase given by A ≈ A Z (cid:18) − x + y Z − A ( x − y )96 Z − Z (cid:19) , (2 . θ ≈ C A x − y Z − A Z + ( C − C ) A ( x + y )128 Z + ( C + 32 C ) A Z . (2 . − A ( x − y )96 Z in theamplitude eq. (2.25). Its sign shows compression in the focusing x -direction and decompressionin the defocusing y -direction as expected. (We note that in Fig. 3 of [1] the axes Y and T should be relabelled since there Y is the focusing and T is the defocusing direction.)The previous formulae imply that, for the Gaussian lump initial condition of moderateamplitude A , we have the following theoretical parameters in the asymptotics:Λ = A , θ = C A ≈ . A , η = Λ − A
48 = A , (2 . Z asymptotics – linear and nonlinear The linear problem has a similarity solution which describes the central x − y region for longtime. Indeed, the linear problem 10 ∂ Z Φ + ∂ xx Φ − ∂ yy Φ = 0 (3 . k, l, Z ) the Fouriercomponent of Φ in xy -space, one gets ˆΦ( k, l, Z ) = ˆΦ ( k, l ) e − i ( k − l ) Z where ˆΦ ( k, l ) = ˆΦ( k, l, k → k/ √ Z and l → l/ √ Z , one canwrite the inverse Fourier transform restoring Φ( x, y, Z ) asΦ( x, y, Z ) = 14 π Z (cid:90) (cid:90) ˆΦ ( k/ √ Z, l/ √ Z ) e − i ( k − l ) e i ( kx + ly ) / √ Z dkdl, (3 . Z . In eq. (3.2), we expand ˆΦ ( k/ √ Z, l/ √ Z )in a Taylor series around the origin,ˆΦ ( k/ √ Z, l/ √ Z ) = ˆΦ (0 , ∂ k ˆΦ (0 , k √ Z + ∂ l ˆΦ (0 , l √ Z + ∂ kk ˆΦ (0 , k + 2 ∂ kl ˆΦ (0 , kl + ∂ ll ˆΦ (0 , l Z + . . . (3 . e i ( kx + ly ) / √ Z and getΦ( x, y, Z ) = 14 π Z (cid:90) (cid:90) e − i ( k − l ) dkdl (cid:18) ˆΦ (0 ,
0) + ∂ k ˆΦ (0 , k √ Z + ∂ l ˆΦ (0 , l √ Z ++ ∂ kk ˆΦ (0 , k + 2 ∂ kl ˆΦ (0 ,
0) + ∂ ll ˆΦ (0 , l Z + . . . (cid:33) (cid:18) i ( kx + ly ) √ Z − ( kx + ly ) Z + . . . (cid:19) = 14 π Z (cid:90) (cid:90) e − i ( k − l ) dkdl (cid:18) ˆΦ (0 , (cid:18) − k x + l y Z (cid:19) ++ ∂ k ˆΦ (0 , ik xZ + ∂ l ˆΦ (0 , il yZ + ∂ kk ˆΦ (0 , k + ∂ ll ˆΦ (0 , l Z + O (cid:18) Z (cid:19)(cid:33) , the last equality being true due to the survival of only even powers of k and l under theintegration. This implies asymptotics of the formΦ( x, y, Z ) = C Z (cid:18) C Z + ( C x + C y ) Z + i ( x − y )4 Z + O (cid:18) Z (cid:19)(cid:19) , (3 . C , C , C and C are constants depending on the IC Φ ( x, y ). Exponentiating theexpression in the parentheses one finally obtainsΦ( x, y, Z ) ≈ C Z e i ( x − y Z + C Z + ( C x + C y ) Z , C = ˆΦ (0 , π ,C = − i ( ∂ kk ˆΦ (0 , − ∂ ll ˆΦ (0 , (0 , , C = ∂ k ˆΦ (0 , (0 , , C = − ∂ l ˆΦ (0 , (0 , , (3 . (0 , (cid:54) = 0. For symmetric ICs C = C = 0. It should be noted that the above derivation requires the initial data in Fourier spaceto be sufficiently smooth. This is not always the case and later we make a further commentabout this, see the remark about noise in Fourier space in the discussion of the numerics.We see that this solution to the linear problem, which is valid for all lump type initialconditions with ˆΦ (0 , (cid:54) = 0, is approximately the same as the nonlinear similarity solutioneq. (1.2). However we will see that the additional contribution in the phase in eq. (1.2) canmake a significant difference. Without this term the error in the phase of the solution can bequite substantial. If we assume that the solution falls like 1 /Z at large Z as we observe in all cases numerically,then it is convenient to express Φ = φ/Z in the HNLS (1.1). Then HNLS takes form i∂ Z φ − iφZ + ∂ xx φ − ∂ yy φ + | φ | Z φ = 0 . (3 . Z , the solution of eq. (3.6) has the series expansion φ ( x, y, Z ) = ∞ (cid:88) n =0 φ n ( x, y ) Z n (3 . Z . Substituting eq. (3.7) into eq. (3.6) we find the linear wave equation ∂ xx φ − ∂ yy φ = 0 (3 . /Z which implies that in general φ ( x, y ) = φ + ( x + y ) + φ − ( x − y ), wherefunctions φ + and φ − are arbitrary. The next order gives ∂ xx φ − ∂ yy φ = iφ , (3 . x ± = x ± y , 12 = i (cid:18) x − (cid:90) x + φ + ( u ) du + x + (cid:90) x − φ − ( u ) du (cid:19) + g + ( x + ) + g − ( x − ) , (3 . g + and g − . The first term coming from original nonlin-earity appears only at second order in 1 /Z which reads4 ∂ x + x − φ = 2 iφ − | φ | φ . (3 . linear PDEs of the form ∂ x + x − φ n = F ( { φ j , j < n } ) allowing one to findin principle each φ n in terms of the previous coefficients of the series (3.7). This shows theconsistency of expansion (3.7) at large Z and its generality since we have a sufficient numberof arbitrary functions in the solution. However, the universal regime implies that a wide rangeof initial conditions leads to φ = Λ e iθ = const. (3 . φ · i ( x − y ) / Z which indeed universally appears in the asymptotics. As we alsoobserve numerically, one should take g + + g − = i ( η − Λ ) φ for a large variety of ICs, where η is a real constant. Then, exponentiating the correction φ , we obtainΦ ≈ Λ Z exp (cid:20) i (cid:18) θ + x − y Z + η − Λ Z (cid:19)(cid:21) . (3 . s ∼ O (1) (cid:28) Z . This solution is observed for a large class of ICs –virtually all that contain a lump of energy around the center x = y = 0. The numerical simulations in this paper employed the ETD2 scheme (exponential time dif-ferencing, spectral in space and second-order in time) proposed in [18]. The computationdomain is taken to be a square of size L and the number of gridpoints N was chosen suchthat L/N = 400 / / / . Z ) step was 0 .
01 in these simulations.Corresponding to each of the initial conditions in the table below there are five figures.In the top row, in the leftmost figure the real part of the numerical solution of eq. (1.1) isplotted in the x, y plane at the final value of time Z = Z max in the simulation. In the top row,center figure, the real part of the exact solution eq. (1.2) is plotted for the same Z with fitted13mplitude and phase constants Λ and θ based on the numerical solution, and the absolutevalue of their difference is shown next in the top right figure; there is very good agreement inthe central spot, for all these initial conditions. The chosen initial conditions include variousGaussian lumps which cover a wide range of parameters – amplitude and the widths along x and y axes, as well as some other functions. In some cases to the initial conditions a smallamount of randomness was added (10%). Apart from an expected spread in the slope of thenumerical line in the right figure of the bottom row the results are largely the same. This isdiscussed more fully below.In the left figure of the bottom row of two figures, the maximum amplitudes of the abovetwo solutions are plotted together versus log Z ; in each case they approach each other ratherfast as Z grows. The agreement is already excellent when Z ∼
10, for the properly chosenamplitude Λ parameter of eq. (1.2) specified to agree with numerics asymptotically at large Z . Thus, the amplitude is well described by the exact similarity solution. The parameter θ in eq. (1.2) was also specified to fit the numerics. In the right figure of the bottom two figures,the differences ∆Θ = θ − θ − s taken at the center x = y = 0, and therefore also s = 0,were plotted versus 1 /Z together for the above two solutions. There one observes gradualapproach to a straight line in almost all cases; however, in most cases the slopes are seen tobe different for the numerical and the exact solution. The slope of the line corresponding tothe numeric solution on the center phase plot is equal to η − Λ , from which, with alreadyknown (determined by amplitude fitting) Λ , the parameter η is found.Thus, the phase exhibits a significant error. This discrepancy is the motivation to con-sider the corrected approximate analytic solution presented in section 3, with the additionalparameter η . It is in turn determined from the numerics.The parameters computed from the numerical data are presented in the following table(there are more cases presented here than in figures - due to restrictions on space):14nitial condition Λ θ Z max η − Λ η e − x − y e − x − y e − x − y . e − x − y e − x − y . · e − x − y . · e − x − y · . e − . x − . y · . e − . x − . y · . e − . x − . y · . e − . x − . y e − x − y e − x − y . e − ( x +1) − y + e − ( x − − y ) 0.25 0.105 16 0.2464 0.30890 . e − x − ( y +1) + e − x − ( y − ) 0.25 0.105 16 -0.232 -0.1695sech( x + y ) 0.395 0.42 24 -0.0498 0.10622 e − x − y cos(2( x + y )) 0.0746 0.607 16 -0.0345 -0.02892 . | x − y | ) e − x − y x + iy ) e − x − y As one might expect, corresponding to initial conditions with larger energy, whether due tolarger amplitude or width, larger Z are required to achieve the same degree of agreementbetween the numerical and the asymptotic solution described by eq. (1.2) or eq. (1.2A). Theactual solution is closer to the exact similarity solution eq. (1.2) (i.e. the correction parameter η is smaller by absolute value) when initially one has a moderate lump of energy with themaximum density at the center. When the initial amplitude is much bigger or the lump ismuch more narrow than the ones presented in the table/figures, more accurate numerics arerequired. Previous numerical investigations, e.g. [10, 12, 32], also found that for larger initialamplitudes of HNLS high resolutions are required to obtain reliable results. In this work wedo not investigate large or rapidly varying functions. This is in the spirit of the asymptoticderivation of the HNLS equation.From the numerical values of the parameters presented in the table, one can see that formost initial conditions featuring a localized lump of energy at the center, the parameter η turns out to be of the same order as Λ . The absolute value of their difference is smaller fornarrower initial beams while it becomes of the same order as the parameters themselves for15nitial widths ∼ | η | is less than Λ .We see in Figs. 15–20 that an initial lump with additional noise (which was taken to beof moderate amplitude ten times smaller than that of the deterministic part) leads to similarpictures as the corresponding lump without noise. But the amplitude and phase undergorandom shifts so that after many (100) realizations we obtain thick curves for the numericalamplitude and especially numerical phase. The amplitude shifts due to the randomness aresmaller (cf. the numerical curves in figs. 15–20). The average amplitude and phase areconsistent with the corresponding values without noise. When the noise is added to everyspatial grid point, it creates relatively large effective gradients which lead to the wide spreadof the phase curves around the average. The amplitude curves, however, remain close tothe mean curve even in this case. As different realizations of the random noise presentthe various possible initial conditions in a neighborhood of their average, these results areespecially indicative of the main point we emphasize: essentially all initial conditions withoutlarge gradients lead to the same universal asymptotic regime that we exposed here. Remark.
The picture turns out to be very different if one adds a similar type of noisein the spectral (Fourier) space instead. Then the asymptotics become drastically modifiedand we observe oscillations of significant amplitude. This occurs for both the HNLS eq. andits linearized version. Therefore the discrepancy with the asymptotics eq. (1.2A) can beunderstood looking at the derivation of the asymptotic formula for the linear case in section3.1. There it was necessary for the Fourier transform of the IC to be smooth enough inorder for its Taylor expansion at the origin in spectral space to be valid. The spectral noise,however, makes this function rough. In contrast, for the noise in physical space consideredabove, the IC in Fourier space turns out to be smooth which explains the discrepancy in thelarge Z behavior.If there is a hole rather than a lump at the center of the xy -plane in the initial condition,then one sees the hole spreading at later times and the values of the solution become tinyaround the origin. Still, even in such cases where the most energy is well away from thecenter, the hyperbolic structure of the solution (1.2) can be observed (see fig. 14, the intitialcondition ( x + iy ) e − x − y ). This phenomenon is a feature of both the full HNLS equationand its linearized version.For relatively small initial amplitudes, the reshaping of the wave packet can be well de-scribed by considering nonlinearity as a perturbation to the linearized equation. This way onecan quantitatively understand the dumbbell shapes often observed forming from the initialround beam both in two and three spatial dimensions [33, 34, 32, 12, 1]. We analyzed thissituation in section 2 for an initial Gaussian beam and showed, as expected, that the beam iscompressed in the focusing x -direction and decompressed in the defocusing y -direction. Forthe Gaussian beam of small or moderate amplitude, we have theoretical values, in the firstperturbative approximation in the initial amplitude A , for the parameters Λ , θ and η , see16q. (2.27). Their numerical values are presented in the table below: A Λ θ η A ∼ A it rapidly worsens. Still the qualitative agreement with the universal regimeeq. (1.2A) often exists even for larger amplitudes.An interesting observation is that in the HNLS with initial lump of energy that is large/wide,rings of low amplitude are observed to develop and move away from the center. Later theycan disconnect, reconnect with parts of other structures in various ways forming intricatepatterns. These complex processes or sometimes a simpler initial deformation of the wavebeam (energy lump) are followed later by outbursts of energy from the central region toboth directions of the defocusing coordinate axis ( y -axis in our case) corresponding to beamsplitting.Thus, if the initial condition has relatively large energy or is substantially different froma single packet of small energy, much more complicated pictures than we show in this paperappear at intermediate times Z ∼ . − Z . Still on the edgesof these sometimes exotic patterns one clearly sees the development of the same familiarhyperbolic structure described above. Based on our numerical findings the universal regimewith a central hyperbolic structure eventually develops even for initial lumps of relativelylarge amplitude. We believe that the phenomena discussed in literature like spiky hyperbolicstructures numerically observed in [32, 12, 13, 55, 56] as well as observed X-waves [15, 16,17, 36, 30] correspond to intermediate regimes just at the onset of the hyperbolic long-timeasymptotic structure, at least in the (2+1)-D case considered here. We expect these waves toeventually develop into the universal regime, perhaps with many centers as we also observed inour simulations. We also note that X-waves are known to appear in both linear and nonlinearsituations; this can be also be said about our universal hyperbolic structure. The main conclusion is that the similarity solution eq. (1.2) with the corrections described byeq. (1.2A) appears universally in the central zone of the HNLS at long times/large propagationdistances. As long as there are no large or rapidly varying initial data the universal regimeoutlined here is expected to be observed in the long-time limit. This universal behavioralso may help select among many existing large energy solutions of the HNLS equation at17ntermediate times; this also might be relevant to the transient behavior observed in differenttypes of beam propagation.Our results are supported by analytical estimates and numerical computations. Analyt-ically we consider the nonlinear term as a perturbation of gaussian initial conditions andconsider linear and nonlinear problems via their long time limits. Numerically we considera wide range of initial conditions including random initial data. We also investigate the av-eraged variational method in the context of a Gaussian ansatz in Appendix B. We find thatwhile the method reproduces the similarity solution (1.2) it does not reproduce the importantmodification of the phase in (1.2A).
Acknowledgments
This research was partially supported by the the NSF under grant CHE 1125935 and the U.S.Air Force Office of Scientific Research, under grant FA9550-16-1-0041.
Appendix A: Some exact reductions of (2+1)-D HNLS
A larger class of reductions of the HNLS eq. (1.1) is obtained if we consider the followingansatz. Letting Φ = Ae iθ with A = Λ( ξ, η ) R ( Z ) , θ = σ ( ξ, η ) + ( α Z + α ) ξ β Z + β ) η γ Z + γ ) ξη µ Z + µ + µ ∗ R ) ξ + ( ν Z + ν + ν ∗ R ) η + h ( z ) , ( A ξ = C x + C yR ( Z ) + ξ ( Z ) , η = C x + C yR ( Z ) + η ( Z ) , ( A R ( Z ) = HZ + H Z + R , α = − C H ∆ , β = − C H ∆ , γ = C H ∆ , ( A C = C − C , C = C − C , C = C C − C C , ∆ = C − C C = ( C C − C C ) (cid:54) = 0 . ( A Z de-pendence. When eq. (A1) is substituted into eq. (1.1), its imaginary part eq. (B1) multipliedby Λ( ξ, η ) becomes a conservation law, 18 ξ [Λ ( C ∂ ξ σ + C ∂ η σ + C ξ + C η + C )] + ∂ η [Λ ( C ∂ ξ σ + C ∂ η σ + C ξ + C η + C )] = 0 , ( A C = C α + C γ − H / , C = C α + C γ ,C = C γ + C β , C = C β + C γ − H / . ( A C and C are at this point arbitrary and functions ξ ( Z ), η ( Z ) are determinedby equations whose solutions are written out later. The real part of eq. (1.1), after thesubstitution of ansatz (A1) then yields C [ ∂ ξξ Λ − Λ( ∂ ξ σ ) ] + C [ ∂ ηη Λ − Λ( ∂ η σ ) ] + 2 C [ ∂ ξη Λ − Λ ∂ ξ σ∂ η σ ]++Λ[Λ − C ξ + C η + C ) ∂ ξ σ − C ξ + C η + C ) ∂ η σ − K ξ − K η − K ξη − K ξ − K η − K ] = 0 , ( A C and C must satisfy HC = H ( C µ − C ν )+ H ( C ν − C µ ) / , HC = H ( C ν − C µ )+ H ( C µ − C ν ) / , ( A K i , i = 1 , . . . , K = C α + C γ + 2 C α γ − α H + α R , ( A K = C γ + C β + 2 C β γ − β H + β R , ( A K = C α γ + C β γ + C ( α β + γ ) − γ H + γ R , ( A K = 2( α C + γ C ) − µ H + 2 µ R , K = 2( γ C + β C ) − ν H + 2 ν R , ( A K is arbitrary. Finally, after using eq. (A8), the functions ξ and η take form ξ ( Z ) = 2 C ( ν H − ν H ) HR ( Z ) (cid:90) Z dZR ( Z ) − C µ ∗ + C ν ∗ ) ZR ( Z ) − C µ + C ν ) H , ( Z ) = 2 C ( µ H − µ H ) HR ( Z ) (cid:90) Z dZR ( Z ) − C µ ∗ + C ν ∗ ) ZR ( Z ) − C µ + C ν ) H , ( A h ( Z ) can be found from R h (cid:48) ( Z ) = C ( µ Z + µ + µ ∗ R ) + C ( ν Z + ν + ν ∗ R ) +2 C ( µ Z + µ + µ ∗ R )( ν Z + ν + ν ∗ R ) −− C ( µ Z + µ + µ ∗ R ) − C ( ν Z + ν + ν ∗ R ) + K . ( A ξ ) and σ = σ ( ξ ) (without loss of generality, taking functions of η only gives identical results up torelabeling of constant parameters). Then eq. (A5) reduces to[Λ ( C σ (cid:48) + C ξ + C )] (cid:48) + C Λ = 0 , ( A d/dξ , together with the further restriction C = 0, i.e. C γ + C β =0. Eq. (A7) then implies that K = K = K = 0 and reduces to C [Λ (cid:48)(cid:48) − Λ( σ (cid:48) ) ] + Λ[Λ − C ξ + C ) σ (cid:48) − K ξ − K ξ − K ] = 0 . ( A U ( ξ ) such that U (cid:48) = Λ allows one to integrate eq. (A15) once and get U (cid:48) ( C σ (cid:48) + C ξ + C ) + C U = C I = const. , ( A C I is an arbitrary constant. Plugging σ (cid:48) from eq. (A17) into eq. (A16) multiplied by2Λ finally yields a third-order ODE for U , C (cid:18) U (cid:48)(cid:48)(cid:48) − ( U (cid:48)(cid:48) ) U (cid:48) (cid:19) + 2( U (cid:48) ) ++2 (cid:0) ( C − K ) ξ + (2 C C − K ) ξ + C − K (cid:1) U (cid:48) − C I − C U ) U (cid:48) = 0 . ( A ppendix B: the approximate solution of HNLS corre-sponding to the Gaussian variational ansatz Expressing Φ = Ae iθ in the HNLS eq. (1.1), we can rewrite the HNLS equation as two realequations for the amplitude A and the phase θ , ∂ Z A + A ( ∂ xx θ − ∂ yy θ ) + 2( ∂ x A∂ x θ − ∂ y A∂ y θ ) = 0 , ( B A∂ Z θ = ∂ xx A − ∂ yy A − A (( ∂ x θ ) − ( ∂ y θ ) ) + A . ( B A = Λ (cid:112) L ( Z ) R ( Z ) e − x L Z ) − y R Z ) , Λ = const. , L ( Z ) > , R ( Z ) > , ( B θ = U ( Z ) x + V ( Z ) y + σ ( Z ) , ( B U ( Z ) = 14 L dLdZ , V ( Z ) = − R dRdZ . ( B xy -plane. First, in the region | x | (cid:46) L ( Z ) , | y | (cid:46) R ( Z ),eq. (B2) is satisfied if the scaling functions L ( Z ) and R ( Z ) satisfy the system found in [7, 45]from variational principle, d LdZ = 16 L − L R , d RdZ = 16 R + 8Λ LR , ( B σ ( Z ) in eq. (B4) is chosen so that dσdZ = Λ LR − L + 2 R , ( B e − x L Z ) − y R Z ) ≈ − x L ( Z ) − y R ( Z ) , ( B | x | (cid:29) L ( Z ) , | y | (cid:29) R ( Z ), eq. (B2) canbe approximately satisfied if one uses there eqs. (B6) and (B7) with Λ = 0 in them andapproximates the exponent in eq. (B8) by zero i.e. neglects the last term A in eq. (B2).Thus, the error of the approximation here is bounded above by21 (cid:12)(cid:12)(cid:12) e − x L Z ) − y R Z ) − x L ( Z ) + y R ( Z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:18) x L ( Z ) + y R ( Z ) (cid:19) , ( B Z asymptotics of the approximate solution given by eqs. (B3)–(B8) in the central region, one can see that they are compatible with the the scales L ( Z )and R ( Z ) changing as L ( Z ) ∼ Z + O (1) and R ( Z ) ∼ Z + O (1), i.e. both approaching Z .This corresponds to the amplitude decreasing as 1 /Z which is consistent with all our foundasymptotics. Besides, as follows from eqs. (B4), (B5) and (B7), under such symmetric along x and y asymptotic scaling the phase of the solution behaves as θ = θ + x − y Z − Λ Z + O (cid:18) Z (cid:19) . The last expression shows that the important parameter η , see eq. (1.2A), which we foundboth analytically and numerically, is missing here. It could be recovered if we consider asym-metric scaling at large Z i.e. L ( Z ) ∼ C Z and R ( Z ) ∼ C Z with constants C (cid:54) = C . However,in section 2 we found nonzero η for symmetric scales which follow from symmetric GaussianIC. 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Journ. Exp. Theor. Phys. , 96:643–652, 2003.26igure 1: Top: Initial condition e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 2: Top: Initial condition 3 e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric
Journ. Exp. Theor. Phys. , 96:643–652, 2003.26igure 1: Top: Initial condition e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 2: Top: Initial condition 3 e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 4: Top: Initial condition 2 . e − . x − . y : Numerical solution, Exact similarity solutionand absolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric
Journ. Exp. Theor. Phys. , 96:643–652, 2003.26igure 1: Top: Initial condition e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 2: Top: Initial condition 3 e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 4: Top: Initial condition 2 . e − . x − . y : Numerical solution, Exact similarity solutionand absolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−6−5−4−3−2−10 exactnumeric ∆Θ exactnumeric Figure 6: Top: Initial condition 2 e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−6−5−4−3−2−10 exactnumeric ∆Θ exactnumericexactnumeric
Journ. Exp. Theor. Phys. , 96:643–652, 2003.26igure 1: Top: Initial condition e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 2: Top: Initial condition 3 e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 4: Top: Initial condition 2 . e − . x − . y : Numerical solution, Exact similarity solutionand absolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−6−5−4−3−2−10 exactnumeric ∆Θ exactnumeric Figure 6: Top: Initial condition 2 e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−6−5−4−3−2−10 exactnumeric ∆Θ exactnumericexactnumeric e − . x − . y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 24. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ exactnumeric ∆Θ exactnumeric Figure 8: Top: Initial condition sech( x + y ): Numerical solution, Exact similarity solutionand absolute value of their difference at Z = 24. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric
Journ. Exp. Theor. Phys. , 96:643–652, 2003.26igure 1: Top: Initial condition e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 2: Top: Initial condition 3 e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 4: Top: Initial condition 2 . e − . x − . y : Numerical solution, Exact similarity solutionand absolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−6−5−4−3−2−10 exactnumeric ∆Θ exactnumeric Figure 6: Top: Initial condition 2 e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−6−5−4−3−2−10 exactnumeric ∆Θ exactnumericexactnumeric e − . x − . y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 24. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ exactnumeric ∆Θ exactnumeric Figure 8: Top: Initial condition sech( x + y ): Numerical solution, Exact similarity solutionand absolute value of their difference at Z = 24. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 10: Top: Initial condition e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric
Journ. Exp. Theor. Phys. , 96:643–652, 2003.26igure 1: Top: Initial condition e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 2: Top: Initial condition 3 e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 4: Top: Initial condition 2 . e − . x − . y : Numerical solution, Exact similarity solutionand absolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−6−5−4−3−2−10 exactnumeric ∆Θ exactnumeric Figure 6: Top: Initial condition 2 e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−6−5−4−3−2−10 exactnumeric ∆Θ exactnumericexactnumeric e − . x − . y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 24. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ exactnumeric ∆Θ exactnumeric Figure 8: Top: Initial condition sech( x + y ): Numerical solution, Exact similarity solutionand absolute value of their difference at Z = 24. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 10: Top: Initial condition e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric . e − ( x +1) − y + e − ( x − − y ) initial condition: Numericalsolution, Exact similarity solution and absolute value of their difference at Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 12: Top: Two Gaussian peaks 0 . e − x − ( y +1) + e − x − ( y − ) initial condition: Numericalsolution, Exact similarity solution and absolute value of their difference at Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric
Journ. Exp. Theor. Phys. , 96:643–652, 2003.26igure 1: Top: Initial condition e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 2: Top: Initial condition 3 e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 4: Top: Initial condition 2 . e − . x − . y : Numerical solution, Exact similarity solutionand absolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−6−5−4−3−2−10 exactnumeric ∆Θ exactnumeric Figure 6: Top: Initial condition 2 e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−6−5−4−3−2−10 exactnumeric ∆Θ exactnumericexactnumeric e − . x − . y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 24. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ exactnumeric ∆Θ exactnumeric Figure 8: Top: Initial condition sech( x + y ): Numerical solution, Exact similarity solutionand absolute value of their difference at Z = 24. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 10: Top: Initial condition e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric . e − ( x +1) − y + e − ( x − − y ) initial condition: Numericalsolution, Exact similarity solution and absolute value of their difference at Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 12: Top: Two Gaussian peaks 0 . e − x − ( y +1) + e − x − ( y − ) initial condition: Numericalsolution, Exact similarity solution and absolute value of their difference at Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric . | x − y | ) e − x − y : Numerical solution, Exact simi-larity solution and absolute value of their difference at Z = 16. Bottom: Log-amplitude vs.log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 14: Top: Initial condition ( x + iy ) e − x − y : Numerical solution, Exact similaritysolution and absolute value of their difference at Z = 24. Bottom: Log-amplitude vs. log Z ,∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−10−8−6−4−20 exactnumeric ∆Θ exactnumericexactnumeric
Journ. Exp. Theor. Phys. , 96:643–652, 2003.26igure 1: Top: Initial condition e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 2: Top: Initial condition 3 e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 4: Top: Initial condition 2 . e − . x − . y : Numerical solution, Exact similarity solutionand absolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−6−5−4−3−2−10 exactnumeric ∆Θ exactnumeric Figure 6: Top: Initial condition 2 e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−6−5−4−3−2−10 exactnumeric ∆Θ exactnumericexactnumeric e − . x − . y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 24. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ exactnumeric ∆Θ exactnumeric Figure 8: Top: Initial condition sech( x + y ): Numerical solution, Exact similarity solutionand absolute value of their difference at Z = 24. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 10: Top: Initial condition e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric . e − ( x +1) − y + e − ( x − − y ) initial condition: Numericalsolution, Exact similarity solution and absolute value of their difference at Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 12: Top: Two Gaussian peaks 0 . e − x − ( y +1) + e − x − ( y − ) initial condition: Numericalsolution, Exact similarity solution and absolute value of their difference at Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric . | x − y | ) e − x − y : Numerical solution, Exact simi-larity solution and absolute value of their difference at Z = 16. Bottom: Log-amplitude vs.log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 14: Top: Initial condition ( x + iy ) e − x − y : Numerical solution, Exact similaritysolution and absolute value of their difference at Z = 24. Bottom: Log-amplitude vs. log Z ,∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−10−8−6−4−20 exactnumeric ∆Θ exactnumericexactnumeric e − x − y (1+0 . Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 16: Top: Initial condition e − x − y (1 + 0 . Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric
Journ. Exp. Theor. Phys. , 96:643–652, 2003.26igure 1: Top: Initial condition e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 2: Top: Initial condition 3 e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 4: Top: Initial condition 2 . e − . x − . y : Numerical solution, Exact similarity solutionand absolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−6−5−4−3−2−10 exactnumeric ∆Θ exactnumeric Figure 6: Top: Initial condition 2 e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−6−5−4−3−2−10 exactnumeric ∆Θ exactnumericexactnumeric e − . x − . y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 24. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ exactnumeric ∆Θ exactnumeric Figure 8: Top: Initial condition sech( x + y ): Numerical solution, Exact similarity solutionand absolute value of their difference at Z = 24. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 10: Top: Initial condition e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric . e − ( x +1) − y + e − ( x − − y ) initial condition: Numericalsolution, Exact similarity solution and absolute value of their difference at Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 12: Top: Two Gaussian peaks 0 . e − x − ( y +1) + e − x − ( y − ) initial condition: Numericalsolution, Exact similarity solution and absolute value of their difference at Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric . | x − y | ) e − x − y : Numerical solution, Exact simi-larity solution and absolute value of their difference at Z = 16. Bottom: Log-amplitude vs.log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 14: Top: Initial condition ( x + iy ) e − x − y : Numerical solution, Exact similaritysolution and absolute value of their difference at Z = 24. Bottom: Log-amplitude vs. log Z ,∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−10−8−6−4−20 exactnumeric ∆Θ exactnumericexactnumeric e − x − y (1+0 . Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 16: Top: Initial condition e − x − y (1 + 0 . Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y (1 + 0 . Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 18: Top: Initial condition 3 e − x − y (1+0 . Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric
Journ. Exp. Theor. Phys. , 96:643–652, 2003.26igure 1: Top: Initial condition e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 2: Top: Initial condition 3 e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 4: Top: Initial condition 2 . e − . x − . y : Numerical solution, Exact similarity solutionand absolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−6−5−4−3−2−10 exactnumeric ∆Θ exactnumeric Figure 6: Top: Initial condition 2 e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−6−5−4−3−2−10 exactnumeric ∆Θ exactnumericexactnumeric e − . x − . y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 24. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ exactnumeric ∆Θ exactnumeric Figure 8: Top: Initial condition sech( x + y ): Numerical solution, Exact similarity solutionand absolute value of their difference at Z = 24. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 10: Top: Initial condition e − x − y : Numerical solution, Exact similarity solution andabsolute value of their difference at Z = 16. Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric . e − ( x +1) − y + e − ( x − − y ) initial condition: Numericalsolution, Exact similarity solution and absolute value of their difference at Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 12: Top: Two Gaussian peaks 0 . e − x − ( y +1) + e − x − ( y − ) initial condition: Numericalsolution, Exact similarity solution and absolute value of their difference at Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric . | x − y | ) e − x − y : Numerical solution, Exact simi-larity solution and absolute value of their difference at Z = 16. Bottom: Log-amplitude vs.log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 14: Top: Initial condition ( x + iy ) e − x − y : Numerical solution, Exact similaritysolution and absolute value of their difference at Z = 24. Bottom: Log-amplitude vs. log Z ,∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−10−8−6−4−20 exactnumeric ∆Θ exactnumericexactnumeric e − x − y (1+0 . Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 16: Top: Initial condition e − x − y (1 + 0 . Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric e − x − y (1 + 0 . Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 18: Top: Initial condition 3 e − x − y (1+0 . Z = 16. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric x + y )(1 + 0 . Z = 24.Bottom: Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumeric Figure 20: Top: Initial condition sech( x + y )(1+0 . Z = 24. Bottom:Log-amplitude vs. log Z , ∆ θ = θ − θ − s vs. 1 /Z . logA logZ −1 0 1 2 3−5−4−3−2−101 exactnumeric ∆Θ exactnumericexactnumeric