STRANG SPLITTING METHODS FOR A QUASILINEAR SCHRÖDINGER EQUATION - CONVERGENCE, INSTABILITY AND DYNAMICS 1. INTRODUCTION.

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1 STRANG SPLITTING METHODS FOR A QUASILINEAR SCHRÖDINGER EQUATION - CONVERGENCE, INSTABILITY AND DYNAMICS JIANFENG LU AND JEREMY L. MARZUOLA ABSTRACT. We study the Strang splitting scheme for quasilinear Schrödinger equations. We establish the convergence of the scheme for solutions with small initial data. We analyze the linear instability of the numerical scheme, which explains the numerical blow-up of large data solutions and connects to analytical breakdown of regularity of solutions to quasilinear Schrödinger equations. Numerical tests are performed for a modified version of the superfluid thin film equation. 1. INTRODUCTION Consider a general quasi-linear Schrödinger equation (1) i u t = u + u f ( u ) + ug ( u ) g ( u ), for f, g : R R. Such equations can be written as i u t + a j k (u) j k u = F (u, u), () u(0, x) = u 0 (x) u : R R d C m with small initial data in a space with relatively low Sobolev regularity but with some extra decay assumptions. Here a : C m (C m ) d R d d, are smooth functions which we will assume satisfy F : C m (C m ) d C m a(y, z) = I d +O( y + z ), F (y, z) = O( y 3 + z 3 ) near (y, z) = (0,0). Quasilinear equations of this form have arisen in several models. See [40 for a thorough list, but we mention here works related to the superfluid thin-film equation [31, modeling ultrashort pulse lasers [14, 15, and time dependent density functional theory [11. The model we will consider here numerically equates to setting g (s) = f (s) = s, and hence (3) i u t = u + u u + u ( u ). This is a pseudo-attractive version of the superfluid thin-film equation, which is given by (4) i u t = u + u u u ( u ) and can be seen as a leading order contribution to the ultra-short pulse laser models from [14, 15. Existence of solutions to quasilinear equations have been studied analytically in several case, Date: March 31,

2 JIANFENG LU AND J.L. MARZUOLA see [14, 15, 7, 8, 9, 35, 36, 40 and many others. The reason we choose to study (3) is that while similar to (4) in that it is guaranteed to have small data local well-posedness from [36 and hence can be used to verify our numerical convergence results for general quasilinear models, the dynamics of (3) can lead to a breakdown of regularity due to a non-positive definite conserved energy. The model (4) on the other hand has a positive energy quantity and as a result much more stable dynamics. The nonlinear flow will allow interesting singularities to form in the evolution for large enough initial data. In particular, we observe blow-up at a particular amplitude threshold, but these singularities are representative of a breakdown of regularity in the higher derivatives and hence not the standard self-similar style blow-up from the semilinear Schrödinger equation. Such a threshold was observed as an obstruction to local well-posedness using Nash-Moser type arguments in [3. We show analytically that this mechanism for instability is inherited by the Strang-Splitting scheme through a rigorous convergence result and analysis of a finite frequency approximation. Moreover, we observe numerically that this threshold for ill-posedness arises in several different types of initial configuration and is rather robust. However, we note that this threshold is not the numerically observed sharp threshold for long-time well-posedness, as indeed the dynamics are able to drive nearby solutions to this critical amplitude. These features of (3) will be explored in Section. Let us consider the nonlinear part of the equation (5) i v t = v f ( v ) + v g ( v ) g ( v ). Taking the complex conjugate, we have i sv t = sv f ( v ) + sv g ( v ) g ( v ). We calculate i t v = i sv t v + i v t sv (6) = v f ( v ) + v g ( v ) g ( v ) v f ( v ) v g ( v ) g ( v ) = 0, and hence under the evolution (5) the amplitude is conserved. This will be a key property used to develop the numerical scheme. To construct a stable numerical scheme, we consider a Strang splitting method for the quasilinear Schrödinger equation, which is a composition of the exact flows of the differential equations (7) i t u = u and (8) i t u = u f ( u ) + ug ( u ) g ( u ).

3 STRANG SPLITTING FOR QLS 3 More concretely, we approximate u(t n ) with t n = nτ for a step size τ > 0 by u n via u n+1/ = e i τ u n ; (9) u + n+1/ = u n+1/ exp( iτ ( f ( u n+1/ ) + g ( u n+1/ ) g ( u n+1/ ) )) ; u n+1 = e i τ u + n+1/. We note that the scheme is explicit and symmetric, thanks to the amplitude preserving property (6) of (5). One can use a Fourier pseudo-spectral method for the spatial discretization, and hence the flow exp( i τ ) can be efficiently calculated using fast Fourier transform (FFT), and the flow (8) amounts to changing the phase of the solution on each mesh point. Due to the advantage of being structure-preserving, the Strang splitting [43 and higher order splitting schemes (e.g. [44, 50) have been widely applied to a variety of nonlinear Schrödinger equations, mainly semilinear Schrödinger equations, modeling monochromatic light in nonlinear optics, Bose-Einstein condensates, as well as envelope solutions for surface wave trains in fluids. See for example [, 3, 5, 6, 7, 8, 9, 1, 17, 19,, 3, 37, 38, 39, 4, 49. While we focus on the Strang splitting scheme for quasilinear Schrödinger equations, let us also mention that many other time discretization approaches to solve non-linear evolution equations have been developed, including Crank-Nicholson type schemes (see e.g. [41 and also [1 and [0 for applications in studying numerical blow-ups and nonlinear scattering), Magnus expansion approaches ([34 and also the recent review article [10), exponential time-differencing schemes (see e.g. [13, 4), implicit-explicit methods (see e.g. [4), the comparison study in [45, and many others. The convergence of splitting schemes for semilinear Schrödinger equation was analyzed in [16, 18, 1, 33, 4, 46. In the present work, we extend the previous works to the quasi-linear Schrödinger equation. We will prove the convergence of the time-splitting method to the original evolution for the superfluid thin-film equation. The analysis follows the ideas in the seminal contribution by Lubich in [33, where the main tools are the calculus of Lie derivatives. We will emphasize on the regularity of the time flow, for which the behavior of the quasilinear Schrödinger equation is different from the semilinear ones. This is a Lie theoretic approach to the continuous time approximation and Sobolev-based well-posedness results of the second author with J. Metcalfe and D. Tataru in order to model small initial data solutions of finite time intervals [35, 36. The scheme is symplectic and is stable within a range of parameters, motivated by the analysis in [36, where the analysis is done purely in Sobolev spaces H s for s sufficiently large. In addition the Strang splitting method converges in the order τ for time-step τ. Moreover, we are able to extend a linear instability observed in a quasilinear Schrödinger equation in [3 to the numerical scheme used to approximate it, justifying the accuracy of a numerically observed blow-up. We study the dynamics of this blow-up solution using both Gaussian and plane-wave configurations of initial data to observe that the threshold for instability is not the sharp global well-posedness threshold for the equation and can indeed be reached through frequency dynamics on lower-amplitude solutions. This is not the standard blow-up

4 4 JIANFENG LU AND J.L. MARZUOLA through re-scaling of a nonlinear state, but is a frequency instability of sorts that causes high frequencies to grow exponentially in a method akin to a backwards heat map. The result is laid out as follows. We begin with a numerical study of the equation with of the modified superfluid film equation in 1d (3) using the Strang splitting scheme (9). To understand the convergence of the scheme, we prove the convergence of the numerical scheme for small data in Section 3, by using Lie theoretic results and necessary multilinear estimates. To understand the blow-up behavior observed for large enough data, we analyze the stability and instability of the scheme in Section 4. In Section 5, we discuss the regularity of the time flow of the quasilinear Schrödinger equation and the time-splitting scheme. In [3, an L threshold for local wellposedness was observed through use of Frechet derivatives in a Nash-Moser scheme. We will show this similarly arises in analysis about exact plane-wave solutions on the torus using analysis similar to that of [49. In order to establish the differentiability of the numerical solution with respect to time to sufficiently high accuracy, we rely on bounds in a much stronger topology in space.. NUMERICAL RESULTS To test the Strang splitting scheme (9) for quasilinear Schrödinger equations and numerically study the regularity breakdown, we consider the modified superfluid thin film equation (see (3) and also [31, 40) given by (10) i u t + u xx = u u + ( u ) xx u on the domain ( π, π with periodic boundary condition. We have done similar computations for the ultra-short pulse laser equation as described in [14, 15, but no further interesting features of the numerical analysis arose, so we do not present them here for clarity of exposition..1. Symmetric Gaussian initial condition. Let us consider initial conditions given by (11) u(0, x) = ae x /(σ ), on the domain ( π,π with periodic boundary condition. Here σ is the width and a is the amplitude of the Gaussian profile. We calculate the solution up to time T = π/4 with parameters a = 1/5 and σ = 1/5 for the initial condition. A Fourier pseudo-spectral method with N = 56 spatial grid points is used. To verify the second order accuracy of the time splitting scheme, we choose different number of time steps and estimate the error by comparing the numerical solutions to a solution with N t = The results in Table 1 and Figure 1 confirm the second order convergence. Provided the solution remains in H 1, the PDE (10) conserves mass and energy given by (1) M(u) = u dx, E(u) = 1 u x dx + 1 u 4 dx 1 (13) ( u ) x dx. 4 4

5 STRANG SPLITTING FOR QLS 5 N t L -norm H 1 -seminorm e e e e e e e e e e 06 TABLE 1. Numerical error of the time splitting scheme for initial data (11) with a = 1/5 and σ = 1/5 at T = π/ error L error H 1 error N t FIGURE 1. Log-log plot of the numerical error in Table 1 measured in L norm and H 1 seminorm. The numerical scheme conserves the mass conservation law. While there is no energy conservation law [, Table 1, numerically the energy is observed to remain conserved with very high accuracy, which is confirmed in Figure. Due to the nonlinearity of the equation (10), the problem becomes more stiff for initial condition with larger amplitude or derivatives. For the family of Gaussian initial data (11), this means to increase a or reduce σ. We next consider the example with a = 0.65, σ = 1/10 and T = π/4. The problem is considerably more difficult than the previous choice of parameters. We refine the spatial discretization to N = 51 to resolve the oscillatory profile of the solution. The numerical error can be found in Table. We still observe second order accuracy, though in this case, the time step size cannot be too large, otherwise the numerical scheme becomes unstable. We remark that to make the scheme more stable, it is possible to apply Fourier spectrum truncation to eliminate spurious Fourier components of the numerical solution, as introduced in [30. At each time step, we set to zero all Fourier coefficients with amplitude below a certain threshold δ times the maximum amplitude of Fourier coefficients. In practice, for this example, we find the threshold δ = 1e 3 makes the scheme stable with N t = 000 (recall that the solution is not stable for N t = without Fourier truncation). On the other hand, the filtering might introduce inconsistency to the numerical results.

6 6 JIANFENG LU AND J.L. MARZUOLA conserved quantity Mass Energy t FIGURE. Mass and energy conservation for the numerical solution. The initial condition is given by (11) with a = 1/5 and σ = 1.5. The numerical solution is calculated up to time T = 4π and with N t = 3000 time steps. Energy and mass are recorded every 100 time steps. N t L -norm H 1 -seminorm unstable e e e e e e e e e e 04 TABLE. Numerical error of the time splitting scheme for initial data (11) with a = 0.65 and σ = 1/10 at T = π/4. The error is estimated by comparing with the numerical solution with N t = If we further increase the amplitude of the initial condition, the numerical results indicate a blow-up behavior for the PDE. We increase the amplitude to a = 0.65 while keeping σ = 1/10. The numerical solution is calculated up to T = Figure 3 shows max u(, t) as a function of t for different choices of time step sizes. The sudden jump and exponential increase of the magnitude of maximum around t = indicates a numerical blow-up of the solution. Note that the onset point of the behavior does not depend on the choice of time step size, indicating that this is not due to numerical instability of the time integration. Here we have chosen N = 4096 spatial grid points. The blow-up behavior persists for further refinement of the spatial discretization. To investigate more closely the above observed blow-up, we study the solution around x = 0 and the time when the blow-up occurs. Note that since the initial condition is symmetric, the

7 STRANG SPLITTING FOR QLS 7.5 max u t x max u t x 10 3 FIGURE 3. max u(, t) as a function of time for Gaussian initial condition with a = 0.65 and σ = 1/10. Five time step sizes are taken, correspond to N t = 10000,0000,40000,80000, and (blue, green, red, cyan, and purple curves) for total simulation time T = π. The bottom panel zooms in the region near the numerical blow-up. The dashed horizontal line indicates the level /. solution has the symmetry u(x, t) = u( x, t). Hence, we plot the absolute value of the solution in Figure 4. The numerical simulation indicates that the solution develops a focusing peak at x = 0 with amplitude close to /. The numerical results suggest that the solution to the PDE becomes unstable for this family of initial conditions when the amplitude reaches around /. To further confirm this, we compare the results for the initial condition with a = 0.65 and σ = 1/10, the solution stays below the amplitude of / as in Figure 5. Numerically, no blow-up is observed for a = The instability for large time step size is caused by pollution in the Fourier spectrum, but not the intrinsic instability of solutions to the PDE.

8 8 JIANFENG LU AND J.L. MARZUOLA 0.75 Abs u(x, t) x FIGURE 4. Snapshots of the absolute value of the numerical solutions around the numerical blow-up. The solution squeezes as time increases and leads to a blow-up. The reference horizontal line is plotted at the value / max u t x 10 3 FIGURE 5. The maximum magnitude of u as a function in time for initial condition with a = 0.65 and σ = 1/10. Compare with the quite different behavior in Figure 3... Plane wave initial conditions. The only exact solution we are aware of for the superfluid equation (10) is the family of wave trains: (14) u(x, t) = a expi (kx ωt). This is a solution to (10) provided that (15) ω = k + a. Since u(x, t) = a for the solution (14) at any x and t, the splitting error of the Strang splitting scheme vanishes, as the potential commutes with operator.

9 STRANG SPLITTING FOR QLS 9 We study the instability by adjusting the amplitude a of the initial data u(x,0) = a exp(i kx) of the solution (14). Figure 6 shows the simulation results for two solutions with initial conditions given by (14) with a = / 10 8 and a = /+10 8 respectively. Even though the amplitudes of the two solutions only differ by 10 8, the behavior of the numerical solutions are completely different. While the numerical solution for the former is stable and accurate, the local truncation error kicks off instability in the latter case. This indicates again that / is the threshold of instability. FIGURE 6. Numerical solution for (14) with a = / 10 8 and a = /+10 8 respectively. The solution to the PDE is unstable when the amplitude is larger than /. We also study multiple Fourier mode solutions to observe if non-local interactions can vary the blow-up profile. Hence, given a pseudospectral discretization scheme keeping the first N Fourier modes, we take initial data of the form (16) u(x,0) = a k j x, j =1expi

10 10 JIANFENG LU AND J.L. MARZUOLA for 0 k 1 k j N. The blow-up behavior of these solutions become more complicated, in particular, oscillations begin to factor around the blow-up after an initial exponential growth of the maximum amplitude, see Figure 7. However, it seems that generically / is still a threshold for blow-up. max u max u t t 1 max u t max u t x 10 3 FIGURE 7. max u(, t) as a function of time for multiple Fourier Mode initial condition with a = 0.65 and frequencies,8 in the -mode setting (top), and,8,14,0 in the 4-mode setting (bottom). Four time step sizes are taken, correspond to N t = 10000,0000,40000, and (blue, green, red, and cyan curves) for total simulation time T = The right panel zooms in the region near the numerical blow-up. The dashed horizontal line indicates the level /. 3. CONVERGENCE OF TIME-SPLITTING SCHEME In this section, we prove the convergence of the Strang splitting scheme for small data. We suppose that the solution u(t) to the modified superfluid thin film equation (10) in 1d is in H 7 for 0 t T, and set Theorem 1. The numerical solution u n given by the Strang splitting scheme (9) with time step size τ > 0 has a second-order error bound in H 1 (17) u n u(t n ) H 1 C (m 7,T )τ, where t n = nτ T and m 7 = max 0 t T u(t) H 7.

11 STRANG SPLITTING FOR QLS 11 Remark. The small data local existence in H s for the cubic quasilinear nonlinear terms is established in Marzuola-Metcalfe-Tataru [35, 36 (See also the works of Poppenberg [40, Kenig-Ponce- Vega and Kenig-Ponce-Rolvung-Vega [5, 6, 7, 8, 9). In particular, in the case of equation (3) in dimension d, there exists a local in time solution in H σ as long as σ > d+5 provided u 0 is sufficiently small. Therefore, the regularity assumption m 7 < holds for sufficiently small data in H σ for σ sufficiently large (and hence in L ). This is far from sharp however and much work must be done to explore the threshold between well-posedness and blow-up. Proof. We will actually prove the result for arbitrary spatial dimension d. The convergence proof follows a Lie theoretic idea of Lubich [33 for semilinear nonlinear Schrödinger equations. Before we begin let us take (18) m k = max u(t) H 0 t T k, k max( 7, d ɛ ), for and ɛ > 0 such that u the solution to (1) with small initial data can be defined in H k using [35, 36. We can approximate the solution through the continuous time generators of the split step equations: (19) (0) i ψ = ψ i ψ = V [ψψ where (1) V [ψ = ψ + ( ψ ). The generators of the split step method can thus be described as exponential maps of the vector fields given by () (3) ˆT (ψ) = i ψ, ˆV (ψ) = iv [ψψ = i [ ψ + ( ψ )ψ. The key estimate we will require is of the type (4) (uv)w H s C u H s+ 4+d +ɛ v s+ 4+d H +ɛ w H s for s > 0 using the L L L L Hölder s inequality and the Sobolev embedding for L. Note, we could likely sharpen our results with multilinear estimates such as (5) (uv)w H s C u H s+ 6+d 3 v H s+ 6+d 3 w H s+ d 3 for s 0 using L 6 L 6 L 6 L Hölder s inequality and the Sobolev embedding for L 6 as well as moving to L p based spaces and applying the techniques of Strichartz estimates, etc. However, for simplicity of exposition we will simply use the Sobolev embedding estimate (4). Before computing Lie derivatives, we want to understand the stability of the evolution generated by ˆV. To do this, we study (6) i ν = V [ψν, ν(0) = ψ.

12 1 JIANFENG LU AND J.L. MARZUOLA For ψ sufficiently regular, it is possible to show that the evolution varies continuously with the choice of initial data in a weak topology. In particular, we can show that given i ν = V [ψν, ν(0) = ψ, i µ = V [φµ, µ(0) = φ, then by looking at the difference of these two evolutions, expanding V [φµ V [ψν = (V [φ V [ψ)µ V [ψ(ν µ) and applying (4) we have (7) µ(t) ν(t) H s ψ φ H s +C 1 t ψ φ H s+ d+4 +ɛ +C t 0 µ(s) ν(s) H s, for any ɛ > 0 and s > d+4 chosen sufficiently large to control the evolution, where C 1, C both depend upon M s = max 0 t T { φ H s, ψ H s }. As a result, a Gronwall type argument shows (8) µ(t) ν(t) H s e C 0τ ψ φ H s+ d+5 where C 0 depends upon m k. Now, to compare the full evolution to the split-step method, we must compute the Lie commutators between generating vector fields: (9) Hence, we observe [ ˆT, ˆV ψ = ( ψ ψ ( ψ )ψ ) [ ψ( ψψ ψ ψ) [ ( ψ ψ)ψ (ψ ψ)ψ + ( ψ ) ψ. (30) [ ˆT, ˆV (ψ) H 1 C ψ 3 max(5, 4+d H +). In addition, we then can easily compute (31) [ ˆT,[ ˆT, ˆV (ψ) H 1 C ψ 3 max(7, 6+d H +). Setting the vector field Ĥ = ˆT + ˆV, the underlying idea is that the evolution of the full quasilinear Schrödinger equation given by the exact evolution (3) ψ(τ) = exp(τd H )Id(ψ 0 ) when well defined can be compared through a double Duhamel expansion to the split-step generator (33) ψ SS (τ) = exp( 1 τd T )exp(τd V )exp( 1 τd T )Id(ψ 0 ), the error terms of which can be written using the Lie commutators. Indeed, the error estimates come from successive application of the quadrature first order error formula (34) τf ( 1 τ 1 τ) f (s)ds = τ κ 1 (θ)f (θτ)dθ and the second-order error formula (35) τf ( 1 τ 1 τ) f (s)ds = τ 3 κ (θ)f (θτ)dθ

13 STRANG SPLITTING FOR QLS 13 where κ 1 (θ) and κ (θ) are the Peano kernels for the midpoint rule and f (s) = exp((τ s)d T )D V exp(sd T )Id(ψ 0 ) and hence f (s) = e i s [ ˆT, ˆV e i (τ s) ψ 0, f (s) = e i s [ ˆT,[ ˆT, ˆV e i (τ s) ψ 0. Note, the Peano kernels are defined as the integral kernels of the linear transformation L : C k+1 [0,T R such that k f (j ) (0) L(f ) = f T j = 1 j! k! j =0 T 0 κ k (s)f (k+1) (s)d s, hence we observe that using the mid-point rule the f (t/) term vanishes explaining why there is not a quadratic term in (35), though the expressions (34) and (35) can still vary due to the nature of the error term expansions in both cases. Hence, it is essential that for the below we can prove that for our approximation we have f (s) C 3, which very much relates to the analyticity of the linear Schrödinger evolution kernel in the Strang splitting scheme as in particular a generic quasilinear Schrödinger flow cannot be shown to be more than C 0 by the purely dispersive techniques in [35, 36. We will come back to this in more detail in Section 5. Applying (3), (33), (30), (31) and (7) in succession as in [33 gives (36) u n u(t n ) H 1 C (m K0,T )τ for t n = nτ T and K 0 = max(7, d+6 + ɛ). In order to obtain the τ convergence here, it is important to compute the double commutator bound leading to (33) in order to expand out to 3rd order in the Lie derivatives. However, we note the same quadratic convergence would hold in L with only K 0 = max(5, d+5 + ɛ) as then the double Duhamel commutator would not be required and we would be mostly restricted by the well-posedness threshold for (1). Remark. So far we have considered the convergence of the time-splitting flow to the flow of the original PDE. We further discretize the spatial degree of freedom using a Fourier pseudo-spectral method. The convergence of the fully discretized scheme follows if we can show that the fully discretized scheme converges to the time-splitting flow. Though this is beyond the scope of our aim in this work. See for instance [18, 4, 47, 48 for analysis of fully discretized scheme for semilinear Schrödinger equations. Remark. Note, we have chosen a model example that demonstrates the techniques quite nicely in our choice of (19) and (0), but the flow of the metric term and the nonlinear term in () are both well defined and have well defined Lie derivatives provided one can prove the existence of

14 14 JIANFENG LU AND J.L. MARZUOLA solutions. Hence, the splitting process can be done in complete generality using the method (37) (38) i ψ = i g i j [ψ j ψ i ψ = V [ψ, ψψ. Each evolution would require time-dependent kinetic energy terms and multilinear estimates akin to those in [35, 36. However, this gives a nonlinear dependence in the Hamiltonian related to the evolution of (37). The nonlinearity causes troubles in designing structure preserving schemes and also difficulty associated with the regularity of the solution map with respect to time in order to approximate using Lie derivatives. 4. STABILITY AND INSTABILITY OF THE NUMERICAL SCHEME As discussed in Section, for large data, we observe blow-up behavior in the numerical study. In this section, we will investigate the numerical instability of the scheme, which shed some light on the blow-up behavior Linear stability analysis for wave train. For the uniform wave trains solution (14), we study the stability for perturbations around the solution. Consider a perturbed solution of the form (39) u(x, t) = u 0 (x, t)(1 + ε(x, t)), where u 0 is the plane-wave solution ae i (kx ωt) and ε 1. To the leading order, we get (40) iε t + i kε x + ε xx = a (ε +sε) + a (ε +sε) xx. Let us expand ε in Fourier series (with ξ n = n for ε periodic on [0,π) (41) ε(x, t) = ε n (t)exp(iξ n x). n= The equation of ε can then be written as a system of ODEs ( ) ( ) d εn εn (4) = G n, dt ŝε n ŝε n where (43) G n = i ( kξn ξ n a + a ξ n a + a ξ n a a ξ n kξ n + ξ n + a a ξ n The eigenvalues of G n, λ n is given by (44) λ n = i kξ n ± ξ n ξ n a + a ξ n The solution becomes unstable if one of the eigenvalues has a positive real eigenvalue or equivalently (45) a ξ n a ξ n > 0 A sufficient condition for stability is (46) a /. )

15 STRANG SPLITTING FOR QLS 15 Note that the stability threshold / agrees with the numerical observations in Section. 4.. Linear Stability Analysis for the Strang splitting algorithm. To study the stability of general initial data, where explicit solution is not available, we linearize around a solution to (1). Locally, the linear instability is essentially equivalent to the plane wave instability observed in Section., and analyzed in the previous subsection. The key in the observation is that the instability occurs for all k. This has been done in [3, Equation (8), where one observes that perturbation around a solution u = w + z leads to (47) z t = a [M w z +G w z + H w z + f (t), z(0) = g, where for w = w 1 + i w we observe M w = [ w1 w w 1 1 w 1 w 1 w, which has determinant 1 w. The matrix functions G w and H w come from the linearization and will be expressed in full in (48) below. Using this linearization and a Frechet based iteration argument in the space H, the authors then show local well-posedness for small data solutions to equation of the form (1). To understand the instability of the numerical scheme, we linearize the discretized Strang- Splitting algorithm and show that the linear instability threshold in the continuous problem exists in the discretized version as well. Letting u = w + z for some solution w of (4) and generally write h(x, t) = h 1 (x, t) + i h (x, t) for any complex function h, we have that the linearized continuous PDE (47) takes the form (see also [3) [ [ z1 w1 w (w = 1) [ [ z1 w z (1 w t 1 ) w + + w 1 w w1 + w (48) 1w z [3w1 + w w 1w [ [ 4w w 1 4w w z1 + [4w 1 w 1 [4w 1 w z [ w w 1 j =1 + (w j w j )) + w w [ j =1 (w j w j ) + w 1 w 1 w 1 w [ z1 z [ z1 z. We then have [ w1 w w M w = 1 1 w1 w 1 w [ 4w w1 T 4w w T, G w = 4w 1 w1 T 4w 1 w T, H w = [ w + w 1 w + w w 1 H 1 H 1 [w 1 w + w 1 w where H 1 = w 1 + w + w 1 w 1 + w 1 w 1 + 4w w + w w, H 1 = 3w 1 + w + 4w 1 w 1 + w 1 w 1 + w w + w w + 4w 1 w 1.

16 16 JIANFENG LU AND J.L. MARZUOLA form We wish to compare the linearization of the full PDE to the discretized linearization of the (49) z n+1/ = e i τ z n ; z + n+1/ = exp( iτ ( w n+1/ + ( w n+1/ ) ))( ) I d iτ H w z n+1/ n+1/ ; z n+1 = e i τ z + n+1/, where taking w n+1/ = w 1 + i w, we have [ w1 H w = w w [ w n+1/ w1 w + + w 1 w w1 + w 1w [3w1 + w w 1w [ 4w w 1 4w n w + [4w 1 w 1 [4w 1 w [ w w 1 j =1 + (w j w j ) + w w [ j =1 (w j w j ) + w 1 w 1 [w 1 w. To address the linear stability of the Strang-Splitting scheme, consider a linearly unstable mode corresponding to (48) such that z is regular. The linearized splitting scheme to the leading order works as (50) ( ( )) z n+1/ = I d + 1 τ z n ; τ 0 ( z + w1 n+1/ (I = w n w d + τ )) n w1 z n+1/ n w 1 w ; n ( ( )) z n+1 = I d + 1 τ z + n+1/ τ 0, and hence ( w1 w (w (51) z n+1 = z n + τ 1) ) (1 w1 ) w z n + O (τ ). 1w As a result, if w 1, w were constant (as we can assume only locally with any accuracy) and w 1 > 0, we observe that each Fourier mode z n+1,k can be approximated by the linearized dynamical system (5) z n+1,k = z n,k + τ ( w1 w k (w ) 1)k (w 1 1)k w 1 w k which has eigenvalues 1 ± τk w 1 and hence would clearly grow exponentially for w > /. For sufficiently large k, we observe that all the Fourier modes of w1, w are small perturbations and hence exponential growth will occur just as in the backwards heat flow generated from the continuous approximation. Indeed, for τ sufficiently small, we have that the linearized Strang-Splitting flow well approximates the unstable backwards heat flow and hence displays linear instability. However, of course the nonlinear effects are ignored in this computation. z n,k

17 STRANG SPLITTING FOR QLS 17 To make this more precise, we observe from [36 for the full PDE model (3) that we can construct initial data for (4) having has L norm larger than / but sufficiently localized in frequency such that the solution exists locally in time. A simple example of a solution with a global existence time and initial L norm larger than / is an exact plane wave solution for the periodic problem. However, perturbations of such exact solutions are still linearly unstable as calculated in Section 4.1. The backward heat equation that represents the linearization of the continuous model exists locally in time when considering frequency localized data, however, this time scale will depend on the frequency cut-off and potentially be quite short given the nonlinear interactions. Using Theorem 1, we observe that in the semi-discretized equation choosing τ 1 sufficiently small compared to the scale of local existence for (4), the numerical solution computed using the Strang-Splitting scheme is O (τ ). Of course, from (51), on a single time-step, the Strang-Splitting solution has a polynomial instability. However, the linearized equations in (48) give exponential growth dynamics for the full solution on the time scale of local existence. Since on the scale of existence, the numerical solution remains O (τ ), the linear instability is inherited by the numerical solution over repeated iterations of the time step. 5. REGULARITY OF THE TIME EVOLUTION In the analysis of the convergence of the Strang splitting scheme, we have used the analyticity of the linear Schrödinger evolution. This is in general however not true for the quasilinear Schrödinger evolution. The Strang splitting scheme actually regularizes the time flow of the original PDE. In this section, we give some further discussion for the regularity of the time evolution. Let us show that the solution map of the quasilinear Schrödinger evolution is continuous in time. The continuity partially hinges upon the proof of uniqueness for the evolution of (1). In particular, take two solutions to (1), say u 1 and u. Setting v = u 1 u and linearizing (1), we have an equation of the form { i vt + g j k (u) j k v +V v +W v = 0, v(0, x) = u 1 (0) u (0) with V = V (u 1, u 1,u, u ), W = h(u 1, u 1,u, u ) + g (u 1,u ) u 1 for functions V, h and g related to Taylor expanding the metric and the nonlinearity. Then, to solve this linear equation, we use [36, Proposition 5.1 (see also [35, Proposition 5.) to show that the weak Lipschitz bound (53) v L H σ v(0) H σ holds for any 0 σ s 1 via energy estimates on the linearized equation, where we recall that the initial condition lies in H s for s > d+5. In the well-posedness result for the linearized version of (1), see Proposition 5.1 of [36 for instance, we have at most continuity of the solution map with respect to time in the H s norm however. The key ideas to the proof follow from the theory of frequency envelopes as discussed in both [35, 36, Sections and 5, where it is proven that the size of a dyadic frequency component of

18 18 JIANFENG LU AND J.L. MARZUOLA the solution to (1) in a natural energy space is bounded by a uniform constant times the corresponding dyadic frequency component of the initial data in H s. To be more precise, we shall use a Littlewood-Paley decomposition of the spatial frequencies, S i (D) = 1, i=0 where S i localizes to frequency ξ [ i 1, i+1 for i > 0 and to frequencies ξ for i = 0. By a frequency envelope, we recall from [35, Section.4 we mean that given a translation invariant space U such that u U S k u U, a frequency envelope for u in U is a positive sequence a j so that k=0 (54) S j u U a j u U, a j 1. We say that a frequency envelope is admissible if a 0 1 and it is slowly varying, a j δ j k a k, j,k 0, 0 < δ 1. An admissible frequency envelope always exists, say by (55) a j = δj + u 1 U max δ j k S k u U. k Abusing notation and avoiding for simplicity the atomic space formulations in [35, 36, we rely upon a uniform bound over the evolution such that effectively (56) u L H s u 0 H s. We note that the L H s norm appears the estimate here is due to the cubic interactions in the nonlinearity and the compactness of our domain, otherwise one must enforce further summability as in [35. A key estimate is the following proposition. Proposition (Proposition 5.3, [35; Proposition 5.4, [36). Let u be a small data solution to (1), which satisfies (56). Let {a j } be an admissible frequency envelope for the initial data u 0 in H s. Then {a j } is also a frequency envelope for u in L H s. Once we have Proposition, the continuity of the solution map can be established as is Section 5.7 of [35. Namely, we consider a sequence of initial data {u n 0 } u 0 in H s. Frequency envelope bounds can then be chosen such that there exists a uniform N ɛ for which a (n) N ɛ ɛ for all n, which gives a uniform upper bound by Proposition on the high frequencies of each corresponding solution u (n) to (1) with initial data u (n) 0 in the L H s norm. Separating into low and high frequencies, using the smallness of the high frequencies and the uniform convergence in weaker Sobolev norms provided by (53), the result follows. Let us emphasize however that we generally gain no more than continuity of the solution map from such arguments. Hence, in order to accurately compare the flow of the full solution map

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