Umeki, Makoto

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Title Generalized homoclinic solutions and numerical periodicity ofthe NLS equationAuthor(s)Umeki, MakotoCitation数理解析研究所講究録 (1996), 974: 21-29Issue Date1996-11URL http://hdl.handle.net/2433/60773RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityNLSGeneralized homoclinic solutions and numerical periodicity of the NLS equationMakoto UmekiDepartment of Physics, University of TokyoAbstractA generalization is given of the Ablowitz &Herbst’s exact solu-tions of the nonlinear Schrdinger (NLS) equation, which are tempo-rally homoclinic and periodic in space. It is found that, in numericalsimulations of the integrable difference scheme of the NLS equation,periodic and quasi-periodic motions are generated instead of $\mathrm{h}\mathrm{o}\mathrm{m}(\succ$clinic orbits. The periods are proportional to $-\ln\epsilon/\Omega_{i}$ , where $\epsilon$ isthe magnitude of numerical errors and $\Omega_{i}$ is the growth rate of theunstable modes.Homoclinicity in dynamical systems plays a crucial role in understandinga mechanism of creation of chaos. In low-dimensional systems, it is known asthe Melnikov’s theorem that a perturbation often destroys the unperturbedhomoclinic orbits in integrable systems and create the Smale’s horseshoemapping in the vicinity of them. In this sense, homoclinicity may be regardedas the fouffi route to chaos, in addition to period-doubling, quasi-periodicityand intermittency. In contrast with low-dimensional systems, the homoclinicstructures in higher or infinite dimensions have been studied very recently[1, 3, 4, 6] and the study should be applied to physical phenomena includingparametrically forced water waves [7].On the other hand, the nonlinear Schr\"odinger equation, which is one ofthe most-understood integrable equations by soliton theory, is still attractivefor study if we consider its periodic boundary problem. In a remarkable paper[1] Ablowitz and Herbst first showed that the focusing nonlinear Schr\"odingerequation$iu_{t}+u_{xx}+2\sigma|u|^{2}u=0$ , (1)with $\sigma=1$ possesses analytically expressed exact solutions which are homo-clinic to the periodic solution $u=a$ $\exp 2ia^{2}t$ . The $N$-homoclinic solutionwas derived through the transform $xarrow ix,$ $tarrow-t$ of 2N-dark-hole soli-ton solutions of the defocusing nonlinear Schr\"odinger equation (Eq. (1)974 1996 21-29 21with $\sigma=-1$ ) with the evenness condition $u(x, t)=u(-x, t)$ , which canbe derived directly by Hirota’s bilinear method. However, we show that theevenness condition is just sufficient but not necessary.The $M-\mathrm{d}\mathrm{a}\Gamma \mathrm{k}$-hole soliton solution of the defocusing NLS equation is givenby$u(x,t)=a\exp(-2ia^{2}t)g(x,t)/f(x, t)$ , (2)where$f(x,t)= \sum_{=\tilde{\mu}0,1}\exp[\sum_{j>k}^{M}\alpha_{jk\mu_{j}}\mu_{k}+\sum_{j}M=1\mu_{j}\eta_{j}]$ , (3)$g(x, t)= \sum_{\overline{\mu}=0,1}\exp[\sum_{j>k}^{M}\alpha jk\mu j\mu k+\sum_{1j=}\mu j(\eta_{j}+2iM\phi_{j})]$ , (4)and$\exp(\alpha_{jk})=[\frac{\sin(\phi_{j}-\phi_{k})/2}{\sin(\phi_{j}+\phi_{k})/2}]^{2}$ (5)$\eta_{j}=p_{j}(x-X_{j})-\Omega_{j}t+\gamma_{j}$ (6)$p_{j}=2a\sin\phi_{j}$ (7)$\Omega_{j}=\pm p_{j}\sqrt{4a^{2}-p_{j}^{2}}$ , (8)$\phi_{j},$$\gamma_{j}$ and $x_{j}$ are real constants, $\sum_{\overline{\mu}=0,1}$ is the summation over all possiblecombination of $\mu_{j}=0$ and 1 for $j=1,$ $\cdots,$ $M$ and $\sum_{j>k}^{M}$ denotes the sum-mation over all possible pairs chosen from $M$ elements. Here the definitionof the phase $\gamma_{j}$ is different ffom that in [1] since we introduced $x_{j}$ . In orderto obtain general homoclinic solutions, we put$x=iX,$ $t=-T$ and $x_{j}=iX_{j}$ . (9)We denote, under the above transform, the relation between the solutions $u$and $U$ of the defocusing and focusing NLS equations, respectively, as$u(x, t;x_{j})=U(x, T;x_{j})$ . (10)The remaining condition that we should impose is the real-valuedness of$f(x, t)$ . This cannot be not satisfied if $M$ is odd, but if $M$ is even, it can besatisfied by letting $p_{2j-1}=-p2j,$ $\gamma 2j-1=\gamma_{2j},$ $\Omega_{2j-1}=\Omega_{2j},$ $\phi_{2j-}1=\phi_{2j}+\pi$and $X_{2j-1}=X_{2j}$ for $j=1,$ $\cdots,$ $N$ , where $M=2N$. This solution gives22not only periodic solutions in space but also quasi-periodic solutions if $p_{j}$ areincommensurate.The solution for $N=1$ is given explicitly by$U(X,T)=a \mathrm{e}\frac{1+2\cos\varphi \mathrm{e}^{\theta+i}\emptyset+A_{12}22\mathrm{e}\theta+4i\emptyset}{1+2\cos\varphi \mathrm{e}^{\theta}+A12\mathrm{e}2\theta}2ia^{2}T$, (11)where $\theta=\Omega T+\gamma,$ $\varphi=p(X-X_{1}),$ $p=2a\sin\phi$ and $A_{12}=\cos^{-2}\phi_{\mathrm{J}}$ Theasymptotic behaviors of $U(X, T)$ as $Tarrow\pm\infty$ are respectively given by$U_{-}=a\mathrm{e}^{2iaT}2[1+4i\sin\phi \mathrm{e}^{\theta}\cos\varphi]+i\emptyset$ , (12)$U_{+}=a\mathrm{e}^{2ia^{2}}[T+4i\phi 1-4iA^{-1}12\sin\phi \mathrm{e}-\theta-i\emptyset\cos\varphi]$ . (13)There is no essential difference between [1] and the present solution for $N=\dot{1}$ ,since the solution is invariant under the translation. If $N=2$ , however, weobtain a family of new solutions$U(X, T;X_{j})=a\exp(2ia2T)G(x, T)/F(X, T)$ (14)where$G(X, T)$ $=$ $1+2\mathrm{e}^{\theta_{1+}}2i\phi 1\cos\varphi_{1}+A_{12}\mathrm{e}^{2\theta_{1}4}+i\phi 1+2\mathrm{e}^{\theta 2i}\mathrm{c}\mathrm{o}3+\emptyset 3\mathrm{s}\varphi_{3}$$+A_{\mathrm{s}}4\mathrm{e}+2\theta_{3+}4i\phi s2\mathrm{e}2\theta_{1+3+}\theta i(\emptyset 1+\phi 3)\{A_{13}\cos(\varphi_{1}+\varphi_{3})$$+A_{2\mathrm{s}^{\mathrm{c}}}\mathrm{o}\mathrm{s}(\varphi 1-\varphi_{3})\}+2A_{13}A2\mathrm{s}A\mathrm{s}4\mathrm{e}^{\theta 2\theta}1+3+i(2\emptyset 1+4\emptyset 3)\mathrm{c}:0.\mathrm{s}\varphi_{1}$$+2A_{12}A13A23\mathrm{e}2\theta_{1}+\theta 3+i(4\phi_{1+}2\phi \mathrm{s})\cos\varphi_{\mathrm{s}}$$+A_{12}A_{1}^{2}3A^{2}23A34\mathrm{e}^{2()+4}\theta_{1}+\theta_{3}i(\phi 1+\phi_{3})$ (15)and$F(X, T)=G(X, T)$ with $\phi_{i}=0$ for all $i$ , (16)where $\theta_{i}=\Omega_{i}T+\gamma_{i}$ and $\varphi_{i}=p_{i}$ (X-Xi). We see that there exists necessarilya conjugate counterpart of every complex term in (4) under the transform (9)for arbitrary $N$ , which implies that the real valuedness of $f$ is satisfied. Notethat there is a possibility of the application of this solution to the motion ofa knotted closed filament.Next, we show how these homoclinic solutions are affected by numericalerrors in simulations. In order to imitate homoclinic orbits numerically as23precise as possible and examine the effect of errors, we perform a followingnumerical experiment.1) We adapt the Ablowitz-Ladik system [2]$iu_{nt}+(u_{n+1^{-}}2u_{n}+u_{n-1})/h^{2}+|u_{n}|^{2}(u_{n+1}+u_{n-1})=0$, (17)which is the integrable finite-difference scheme and does not show any chaoticmotions. The number of spatial discretization $N_{p}$ is taken as 64.2) We use three numerical codes written in single, double and quadrupleprecisions, which have about 7, 16 and 33 effective decimal digits of floatingpoint numbers. The truncation error is reduced to the level of the round-offerror. For this purpose, we choose as the scheme of time integration the$\mathrm{f}_{\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{t}}.\mathrm{h}$-order Runge-Kutta method in the former two codes, and the sixth-order eight-stage method by $\mathrm{H}\mathrm{u}\mathrm{t}\mathrm{a}[5]$ in the quadruple case. The temporalsteplength $\Delta t$ is taken as $\Delta t=10^{-2},10^{-}3$ and $10^{-4}$ so that the local trunca-tion error is roughly estimated as $10^{-8},10^{-}15$ and $10^{-28}$ , which are on a levelof the round-off errors. Then, we denote the total size of the local numericalerror by $\epsilon_{n}$3) The initial conditions are given according to the asymptotic form $Tarrow-\infty$of the solution, (12) for $N=1$ . They are expressed by the sum of the fixedpoint $a$ $\exp(2ia^{2}\tau)$ plus a small disturbance which denotes unstable modes.The magnitude of the disturbance is $\epsilon_{n}$ . This may be considered as one ofthe best implementations of homoclinic orbits in a numerical sense.First, a 1-homoclinic solution is examined. We put $L=2\sqrt{2}\pi$ and theinitial condition is$U(X, T=0)=0.5[1-\epsilon(1+i)\cos(pX)]$ (18)with $p=2\pi/L$ and $\epsilon=\epsilon_{i}=\epsilon_{n}$ . The $\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{c}\succ \mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}$ behavior of $|U|$ inFigure 1 shows not homoclinic but periodic motion in time. We observethe sequence of appearance of bumps, whose positions are fixed or changedalternatively. It should be noted that the periods of three cases are different.The lefi side of Figure 2 shows the temporal evolution of the amplitude$A=\sqrt{|\tilde{U}_{1}|^{2}+|\tilde{U}-1|}$ where the Fourier coefficients $\tilde{U}_{j}$ is defined by$\tilde{U}_{j}=\frac{1}{N_{p}}\sum_{l=1}U(x_{\iota})N_{\mathrm{p}}\mathrm{e}-ikjX_{l}$ , $X_{l}=lL/N_{p}$ , $k_{j}=2\pi j/L$ (19)24The right side of Figure 2 shows the temporal plots of $({\rm Re}(V), {\rm Im}(V))$ at $X=$$0$ , where a new variable $V=U\exp(-2ia^{2}\tau)$ is introduced. This periodicmotion and its period can be explained as follows.The initial condition is expressed as$V=0.5+\epsilon_{i}\exp(i\varphi i)\cos(pX)$ . (20)Comparing (12) with (18), we may estimate $\epsilon_{i}\approx\exp(\Omega T_{i}+\gamma)$ and$T_{i}=(\ln\epsilon_{i}-\gamma)/\Omega$ (21)Similarly, we can estimate the final time $T_{f}$ of the single bump. The finalstate may be approximated as$V=\exp(4i\phi)[0.5+\epsilon_{f}\exp(i\varphi_{f})\cos(px)]$ . (22)Letting $\epsilon_{f}\approx\exp[-(\Omega T_{f}+\gamma)]$ , we have$T_{f}=-(\ln\epsilon_{f}+\gamma)/\Omega$ (23)Then, the lifetime of the numerical homoclinic solution is$T_{f}-T_{C}=-(\ln\epsilon i+\ln\epsilon_{n})/\Omega$ (24)Next, a -homoclinic solution is investigated by an initial condition$V=0.5\{1+\epsilon_{1}\exp(i\varphi i1)\cos[p_{1}(X-X1)]+\epsilon_{2}\exp(i\varphi i2)\cos[p_{2}(x_{-}X_{2})]\},$ (25)with $p_{1}=2\pi/L=p_{2}/2,$ $X_{1}=0,$ $L=4\sqrt{2}\pi$ and $\epsilon_{1}=\epsilon_{2}=10^{-15},$ $X_{2}=0$for one case and $\epsilon_{1}=\epsilon_{2}=10^{-8},$ $X_{2}=\pi/2$ for the other. A doubly-periodicmotion is observed numerically instead of the homoclinic solution. Followingthe above argument on the temporal period, we may estimate the periodsbetween bumps as$T_{\mathrm{P}^{i}}=-(\ln\epsilon i+\ln\epsilon_{f})/\Omega_{i}$ (26)Then, the ratio of the two periods is determined by the ratio of the lineargrowth rates as$\nu=\frac{T_{p2}}{T_{p1}}=\frac{\Omega_{1}}{\Omega_{2}}$ . (27)25The present example gives $\Omega_{1}=\sqrt{7}/8,$ $\Omega_{2}=1/2$ and $\nu--0.661$ and thenumerically obtained values are $T_{p1}=84,$ $T_{p2}=57$ and $\nu=0.679$ . Therefore,the theoretical prediction is confirmed by this numerical simulation.A generalized homoclinic solution of the NLS equation is given and it isshown that it appears as periodic and quasi-periodic motions in numericalsimulations with minimized numerical errors. The dependence of the periodon the magnitude of errors may not be restricted to this example but isexpected to be applicable to systems having homoclinic structures and smallperturbations or noise.I thank Professor M. J. Ablowitz for his two seminars given in Profes-sor Wadati’s group in Department of Physics, University of Tokyo, whichmotivated me to study this subject.References[1] Ablowitz, M. J. and Herbst, B. M., SIAM J. Appl. Math. 50, 339 (1990).[2] Ablowitz, M. J. and Ladik, J. F., Stud. in Appl. Math. 55, 213 (1976).[3] Ablowitz, M. J., Schober, C. and Herbst, B. M., Phys. Rev. Lett. 71,2683 (1993).[4] Calini, A., Ercolani, N. M., McLaughlin, D. W. and Schober, C. M.,Physica D89,227 (1996).[5] Huta, A., Acta Fac. Rerum Natur. Univ. Comenian. Math. 2, 21 (1957).;Lambert, J. D. Computational Methods in Ordinary,$D$ifferential Equations(John Wiley&Sons, 1973)[6] McLaughlin, D. W. and Schober C. M., Physica D, 57,447 (1992).[7] Umeki, M., J. Phys. Soc. Jpn. 60,146 (1991).26Figure 1. Numerical simulations of the 1-homoclinic solutionwith (a) single, (b) double and (c) quadruple precisions.27$\mathrm{F}\mathrm{i}\mathrm{u}\mathrm{r}_{2}\mathrm{e}\mapsto_{||}^{2\mathrm{L}}|U_{1}+U-1|0^{\cdot}\mathrm{f}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{F}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{r}\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{f}\mathrm{t}\cdot \mathrm{T}\mathrm{e}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{V}\mathrm{o}1\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{o}\mathrm{f}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}_{\mathrm{S}}\tilde{U}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{d}\tilde{U}\mathrm{h}\mathrm{t}\mathrm{e}\mathrm{a}\mathrm{m}\mathrm{p}1\mathrm{i}\mathrm{t}\mathrm{u}\mathrm{d}\mathrm{e}A=-1$ .Right: Plots of $({\rm Re}(V), {\rm Im}(V))$ at $X=0$.Precisions are single $(\mathrm{a},\mathrm{b})$ , double $(\mathrm{c},\mathrm{d})$ and quadruple $(\mathrm{e},\mathrm{f})$ .28$T$Figure 3. Numerical simulations of the -homoclinic solution. Left: Tem-poral evolution of $|U|$ with double precision. Rght: Temporal evolution of$A_{1}=\sqrt{|\tilde{U}_{1}|^{2}+|\tilde{U}-1|}$ and $A_{\mathit{2}}=\sqrt{|\tilde{U}_{2}|^{2}+|\tilde{U}-\mathit{2}|}$. The initial condition is $U=$$0.5[1|\epsilon \mathrm{e}^{i\phi 0}1\cos p_{1}X+\epsilon \mathrm{e}^{i}\emptyset 02\cos p_{1}(X-x\mathit{2})],$ $L=4\sqrt{2}\pi,p_{1}=\mathrm{p}_{2}/2=2\pi/L$,and (a-c) $\epsilon=10^{-\mathrm{l}_{\backslash }})r,d\mathrm{v}_{2}=0$, (d-r) $\epsilon=10^{-\hslash},\mathfrak{l}^{\gamma}2\Delta \mathrm{Y}2=\pi/2$.29

NLS方程式のホモクリニック解と数値的周期性(計算流体力学に関わる数理的諸問題)

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NLS方程式のホモクリニック解と数値的周期性(計算流体力学に関わる数理的諸問題)

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