Mathematical Models and Methods in Applied Sciencesfc World Scienti�c Publishing CompanyNUMERICAL STUDY OF ELLIPTIC-HYPERBOLICDAVEY-STEWARTSON SYSTEM :DROMIONS SIMULATION AND BLOW-UPC. Bessey, C.H. Bruneauyy : Math�ematiques Appliqu�ees de BordeauxUniversit�e Bordeaux I33405 Talence Cedex.

This paper is devoted to the numerical approximation of the elliptic-hyperbolic form ofthe Davey-Stewartson equations. A well suited �nite di�erences scheme that preservesthe energy is derived. This scheme is tested to compute the famous dromion 1 � 1 anddromion 2 � 2 solutions. The accuracy of Crank-Nicolson scheme is discussed and it isshown that it induces a phase error. Then, the qualitative behavior of the solutions isstudied ; in particular the in uence of the initial datum and of the various parametersis pointed out. Finally, numerical experiments show the existence of blow-up solutions1. IntroductionThe aim of this paper is to study numerically the behavior of the solutions ofDavey-Stewartson system(DS) � iut + �uxx + uyy = �juj2u+ bu'x;'xx +m'yy = �(juj2)x;where the constants �, �, b, m and � are real. This system describes the evolution ofwater surface waves in presence of gravity and capillarity 1,2. Following Ghidaglia-Saut 3, we classify these systems as elliptic-elliptic (E-E), elliptic-hyperbolic (E-H),hyperbolic-elliptic (H-E) and hyperbolic-hyperbolic (H-H) according to the sign of(�;m) : (+;+), (+;�), (�;+) and (�;�). In this paper, we restrict ourselves tothe (E-H) case. As described in 4, the (E-H) mode needs appropriate boundaryconditions. Following 4, we takelim�!�1'(x; y; t) = '1(�; t);lim�!�1'(x; y; t) = '2(�; t);where � = cx � y, � = cx + y represent the characteristic variables, m = �c2 and'1 and '2 are given functions. 1

Few mathematical results are known in this case. Hayashi and Hirata 5 showglobal in time existence and uniqueness of a solution u 2 L1local(R; H3;0 \H0;3) withan hypothesis of regularity on initial data and boundary conditions, withHm;l = ff 2 L2; jj(1� @2x1 � @2x2)m=2(1 + x21 + x22)l=2f jj <1g:Moreover, they �nd a decay ratejju(t)jjL1 � C(1 + jtj) (jju0jjH3;0 + jju0jjH0;3 ):More recently, always under assumption of small initial datum, Hayashi 6 provesthe local existence and uniqueness of a solution in C([0; T ]; H1+�;0 \H0;1+�).On the other hand, for some values of the coe�cients in (DS), the system isintegrable by inverse scattering. A well-known example called DSI in the literature(see 7;8) is elliptic-hyperbolic and can be written as :(DSI) � iut + 1=2(uxx + uyy) = ��juj2u+ u'x;'xx � 'yy = 2�(juj2)x:For (DSI), the mathematical results are very complete (see Fokas-Sung 9) and more-over there exists a class of localized exact solutions called Dromions (see 8;10).From the numerical point of view, there are still very few results available since(DS) system has been derived only twenty years ago and mathematical and numeri-cal studies have started in the 90ies. To our knowledge, the only numerical work onDS (E-H) is the one of White and Weideman 11 who use a split step spectral methodon (DSI). Our goal is to derive a robust numerical scheme easy to implement andable to capture well the behavior of the solutions of DS (E-H) for various sets ofparameters. In addition, we want to compute directly both u and ' associated toDirichlet boundary conditions. We decide to split the two equations of (DS) systemby relaxing the u'x term. Then, we solve the nonlinear Schr�odinger type equationby means of Crank-Nicolson scheme. As the second equation is hyperbolic, we writeit in characteristic variables to make it easy to solve by a �nite di�erence scheme.To validate the scheme, we test it successfully by computing the well-knowndromions. However, Crank-Nicolson scheme induces a phase error that is quanti�edon the linear Schr�odinger equation. Then, we study the in uence of the initial datumand the various parameters on the behavior of the solutions.Finally, as blow-up phenomena can occur in some dispersive equations, we lookfor them in the focusing case (� = �1) when b or � are small enough. The resultsshow that a blow-up occurs even for a large set of values of b.This paper is organized as follow :In section 2, we recall some facts on the dromion solutions to DSI.In section 3, we present our �nite di�erence schemeIn section 4, we present some numerical experiments on dromions.In section 5, we present some simulations that lead to think that the solution ofDS (E-H) blows up in �nite time for some values of the coe�cients.2

2. Exact solutions of DSIUsing inverse scattering methods, Fokas and Santini 8;10 show for DSI that thereexists localized coherent structures which are governed by the non trivial boundaries'1 and '2, and call them \dromions" since they travel on the tracks (in ancientgreek \dromos") generated by the boundaries and are driven by them. They arelocalized traveling solutions which decay exponentially in both � and �, and caninteract upon the movement. Contrary to solitons, they do not preserve their formupon interaction and can exchange energy. To obtain them, Fokas and Santini write(DSI) in the form 8>>><>>>: iut +�u+ u[U1 + U2] = 0;'� = �U1 + �2 juj2;'� = �U2 + �2 juj2;where U1 = ��2 Z ��1(juj2)�d�0 + u1(�; t);U2 = ��2 Z ��1(juj2)�d�0 + u2(�; t):Therefore, u1(�; t) corresponds to �'2� (�; t) and u2(�; t) to �'1�(�; t). As usualby inverse scattering techniques, the analytic form of dromions solutions is veryhard to obtain. However, the (M;N) dromion solution, describing the interactionof N �M localized lumps, takes the following formu(�; �; t) = 2XtZY (2.1)with X = (Cx + I)�1V is a vector of size N;Y = (Cy + I)�1W is a vector of size M;Z = (A� �I)�1� and � a N �M matrix;A = �(Cy + I)�1[(Cx + I)�1�]t is a N �N matrix;where the superscript t denotes the transpose of a matrix. The N �N matrix Cxis given by (Cx)jk = mjmk�j + �k exp [�(�j + �k)(� � i(�j � �k)t)];and the M �M matrix Cy by(Cy)jk = lj lk�j + �k exp [�(�j + �k)(� � i(�j � �k)t)]:Finally, (V )j = lj exp [��j(� � i�jt)]; 1 � j � N;(W )j = mj exp [��j(� � i�jt)]; 1 � j �M;3

where �j ; �j ; lj ; mj and � 2 C and Re(�i);Re(�i) 2 R+ . Besides, the boundaryconditions are u1(�; t) = 2@�(Y tW ); (2.2)u2(�; t) = 2@�(XtV ); (2.3)which can be written asu1(�; t) = �2@� MXk=1mk exp [��k(� + i�kt)]Yk(�; t);u2(�; t) = �2@� LXj=1 lj exp [��j(� + i�jt)]Xj(�; t):Then, as t ! �1, u1(�; t) (resp. u2(�; t)) consists of M (resp. L) solitons eachtraveling with velocity �2Im(�k) (resp. �2Im(�j)). Moreover, always as t! �1,u(�; �; t) consists of M times N widely separated lumps, named �kj , k = 1::M; j =1::N , each traveling with velocity (�2Im(�k);�2Im(�j)). In the special case of�kj = 0 for k 6= j, the number of lumps is min (M;N).To illustrate these formula, we give here the (1�1) dromion expression. Let � =�R+ i�I , � = �R+ i�I , �̂ = �+2�It, �̂ = �+2�I t, �� = 1�R ln j�jp2�R , �� = 1�R ln j�jp2�R ,Ru = �R(�̂� ��) +�R(�̂� ��) and Iu = �(�I �̂+�I �̂) + (j�j2+ j�j2)t+ arg(lm), thenu1(�; t) = 2�R2cosh (�R(�̂ � ��))2 ; (2.4)u2(�; t) = 2�R2cosh (�R(�̂ � ��))2 ; (2.5)and u = 4�p�R�R exp f�Ru + iIug(1 + exp (�2�R(�̂ � ��)))(1 + exp (�2�R(�̂ � ��))) + j�j2 : (2.6)The dromion solutions are not the only explicit solutions of DSI. Indeed, recently,Hietarinta and Hirota 12 and Jaulent et al. 13 obtained a broader class of dromionsolutions in terms of Wronskian determinants. Finally, Gilson and Nimmo 14 con-sidered an alternative direct approach which uses a formulation of the solutions asgrammian determinants to obtain a much broader class of solutions (plane-wavesolitons, dromions, solito�s).3. Numerical schemeIn this section, we introduce the numerical method used to compute solutions ofthe slightly modi�ed (DS) (E-H) system to see the in uence of each derivative term4

on the behavior of the solution8>>>>><>>>>>: iut + �uxx + uyy = �juj2u+ bu'x; (a)'xx � c2'yy = �(juj2)x; (b)lim�!�1'(x; y; t) = '1(�; t);lim�!�1'(x; y; t) = '2(�; t);u(t = 0; x; y) = u0(x; y): (3.7)with �, > 0. Approximating this system, we face several di�culties ; namely, thesize of the domain, the coupling of equations and the hyperbolic type of the secondequation. Then, we have two possibilities for the numerical treatment of (3.7). Onone hand, we can use the structure of the system as in 11 and write (3.7) asiut + �uxx + uyy = �juj2u+ buVwhere V = �4c Z ��1(juj2)�d�0 + Z ��1(juj2)�d�0!+ '1� + '2�however, without the use of spectral methods, the integrals are di�cult to computeand, with such a formulation, it is not possible to compute '. On the other hand,we can approximate separately both equations of (3.7) which is more appropriatefor �nite di�erence schemes.Moreover, we want to preserve the energyM(u) = ZR2 juj2dxdy (3.8)3.1. Semi-discretization in timeThe main idea is to use the Crank-Nicolson scheme proposed by Delfour-Fortin-Payre 15 for the nonlinear Schr�odinger equation (NLS) :(NLS) iut +�u = �juj2u ; x 2 Rd ; t > 0;u(x; t = 0) = u0(x) ; x 2 Rd :This scheme is studied in 16;17;18. It is fully implicit and exactly preserves bothinvariants of (NLS). It takes the semi-discretized formiun+1 � un�t +��un+1 + un2 � = �� jun+1j2 + junj22 � un+1 + un2where un is the approximation of u at time tn = n�t. In fact, this equation iswritten at time tn+ 12 = (n + 1=2)�t. In order to use this scheme for DS (E-H), wejust have to add the discretization of the term u'x. We write it as�un+1 + un2 �'n+ 12x5

and replace the operator � by D = �@xx + @yy. Now, the principle of relaxationis to write (3.7 b) at the di�erent time tn = n�t, and proceeding in the same way,we obtain L('n+ 12 + 'n� 122 ) = �(junj2)xwhere L = @xx � c2@yy. We take as initial condition u0 = u0. Finally, we getiun+1 � un�t +D�un+1 + un2 � = �� jun+1j2+junj22 ��un+1+un2 �+b�un+1+un2 �'n+ 12x (3.9)L 'n+ 12 + 'n� 122 ! = �(junj2)x: (3.10)Then, there is conservation of the energy (3.8). Indeed, multiplying (3.9) byun+1 + un, integrating in space and taking the imaginary part, we getZR2 jun+1j2dxdy = ZR2 junj2dxdy: (3.11)Moreover, as the functions '1 and '2 are de�ned on the characteristic variables, wehave to rewrite (3.9) and (3.10) in the (�; �) plane8>>>>><>>>>>: S('n+ 12 + 'n� 122 ) = �4cT (junj2) (a)iun+1 � un�t + (c2� + )�(un+1 + un2 ) + 2(c2� � )S(un+1 + un2 ) = (b)(� jun+1j2 + junj22 + bcT ('n+ 12 ))(un+1 + un2 ) (3.12)where S = @�� and T = @� + @�.Remark 0.1 In this paper, we do not prove neither existence of a solution northe convergence of the scheme. However, the same scheme is used in 19 for the(NLS), DS (E-E) and DS (E-H) equations and results of existence and convergenceof solutions are proved.3.2. Full discretizationFor the spatial approximation, we restrict the in�nite domain to a large enoughbounded one and take homogeneous Dirichlet conditions on the boundary. Weconsider the approximated domain is large enough when the value of the initialdatum at the boundary is less than 10�6. The numerical tests show than this isenough to avoid a damage of the qualitative behavior of the solution.6

y

0

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x

y

ξ

η

A B

C

x

Fig. 1. DomainThe non trivial boundary conditions on the mean ow ' are well adapted tosuch a domain. Indeed, if we represent it on �gure (1), we just have to prescribethem on (AB) and (AC). Obviously, we will impose that lim�!�1'1(�; t) =lim�!�1'2(�; t) = 0, so that, the point A will have to be chosen su�cientlyfar from the origin. We mesh the bounded domain with a (J � 1) � (K � 1)grid. We will denote by �x and �y the space steps, and unl the value at the point(x0 + (j � 1)�x; y0 + (k � 1)�y), (j = 1::J; k = 1::K), l = J(k � 1) + j, and 'n+ 12lthe value of ' at the same point. Always using centered �nite di�erences, the fullydiscrete scheme reads8>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

14�x�y 26664 ('n+12l+1+J�'n+12l+1�J�'n+12l�1+J+'n+12l�1�J )2 +('n� 12l+1+J�'n� 12l+1�J�'n� 12l�1+J+'n� 12l�1�J )2 37775= �4c [ junl+1j2 � junl�1j22�x + junl+J j2 � junl�J j22�y ]iun+1l � unl�t + (c2� + )[(un+1l+1 � 2un+1l + un+1l�1 + unl+1 � 2unl + unl�12�x2 )+(un+1l+J � 2un+1l + un+1l�J + unl+J � 2unl + unl�J2�y2 )]+2(c2� � )4�x�y 264 un+1l+1+J�un+1l+1�J�un+1l�1+J2 +un+1l�1�J+unl+1+J�unl+1�J�unl�1+J+unl�1�J2 375= [�( jun+1l j2 + junl j22 ) + bc('n+ 12l+1 � 'n+ 12l�12�x + 'n+ 12l+J � 'n+ 12l�J2�x )](un+1l + unl2 )7

We can now see that the boundary conditions are well adapted. Indeed, as'n+ 12l+1+J = 'n+ 12l+1�J + 'n+ 12l�1+J � 'n+ 12l�1�J + f('n� 12 ; un; �x; �y)we see clearly on �gure (2) that 'n+ 12l+1+J is the only unknown for the hyperbolicdependence

l

l+1-J

l+1+J

l-1-J

l-1+J

known row or column

known data

unknown data

Fig. 2. Meshequation on ' at point l. All other terms are known from the boundary conditionsor from the previous step in an explicit way going along the x-direction and thenalong the y-direction or conversely.Then, the �rst step is to compute ' 12 with S' 12 = �4cT (ju0j2). At each timestep, we solve alternatively the hyperbolic equation (3.12-(a))S'n+ 12 = �S'n� 12 + �2cT (junj2)and the equation (3.12-(b)) using a standard iteration procedure based on a �xed-point algorithm. So, we get successive approximations of un+1 by solving linearsystems until we estimate that there is enough precision.4. Numerical experiments4.1. Computation of dromion solutionsAs we mentioned in section 2, there are explicit solutions of the sub case (DSI). So,we �rst test our scheme on (DSI) system and dromion solutions. We start with the1-1 dromion solutionu(�; �; t) = 4i exp (�(� + � + 4t)� i(� + �))(1 + exp (�2� � 4t))(1 + exp (�2� � 4t)) + 1corresponding to � = �1, � = � = l = m = 1 + i and � = 1 in (2.6). It representsa soliton moving with speed 2p2 in the negative direction on the line � = �. The8

non trivial boundary conditions associated with ' are'1(�; t) = �2 tanh (� + 2t) +A'2(�; t) = �2 tanh (� + 2t) +Bwhere A = B = �2 in order that lim�!�1'1(�; t) = lim�!�1'2(�; t) = 0. More-over, as'(�; �; t) = �2 Z ��1 juj2(�; s; t)ds+ Z ��1 juj2(s; �; t)ds!+ '1(�; t) + '2(�; t)an easy computation gives'(�; �; t) = �4 11 + exp (2� + 4t) + 11 + exp (2� + 4t)(1 + exp (�2� � 4t))(1 + exp (�2� � 4t)) + 1 + '1(�; t) + '2(�; t)We take the initial datum at t = �3. The domain is the square [�12; 12]� [�12; 12]with a 128� 128 mesh, �t = 10�3 and t 2 [�3; 3]. On the �gures (3),(4), we plotthe module of the theoretical and the numerical solution at t = 0.

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η ξFig. 4. Exact dromion 1-1We can not see any revelant di�erences. To emphasize it, we look at the evolutionof the contour of the localized solution for t = �3; 0; 3 on �gure (5).9

−10 −5 0 5 10

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ξFig. 5. Contour of numerical dromion 1-1The structure of the dromion is perfectly preserved. Moreover, we show theevolution of the L1-norm and see that it is well conserved on �gure (6).

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Fig. 6. L1 normNext, to understand better what driven by the tracks means, we plot ' for thesame values of t on (�g(7,8,9). 10

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ηFig. 9. ' at t=3In fact, if we compare position of u on �g(5) with cross section position of thecontour of ' on �g(10,11,12), we can see that u is exactly localized on the tracksleft by '.

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Fig. 10. ' at t=-3 −10 −5 0 5 10

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Fig. 11. ' at t=0 −10 −5 0 5 10

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Fig. 12. ' at t=3In order to analyze better the error we made, we plot on �gure (13, 14, 15)N1 = jjuexjj2 � jjunumjj2, N2 = jjuex � unumjj2 and N3 = jjjuexj � junumjjj2 whereuex is the exact solution and unum the numerical one.11

−3.0 −1.0 1.0 3.0−0.25

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Fig. 13. N1 = jjuexjj2 � jjunumjj2

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Fig. 14. N2 = jjuex � unumjj2 −3.0 −1.0 1.0 3.00.0

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Fig. 15. N3 = jjjuexj � junumjjj2The N1 quantity is less than 10�5 at any time as expected from (3.11). However,the phase error is quite high as revealed by N2.This phase error comes from the Crank-Nicolson scheme. Indeed, let us considerthe linear Schr�odinger equation(LS) � iut +�u = 0;u(0; x) = u0(x);and the semi-discrete Crank-Nicolson schemes8<: iun+1 � un�t +�un+1 + un2 = 0;u0(x) = u0(x): (4.13)12

The solution of (LS) is given byu(x; t) = S(t)u0(x)where dS(t)(�) = exp (�i�2t) and c(:) denotes the Fourier transform. Writing (4.13)as un+1 = (Op1)�1Op2unwith Op1 = ( i�t + �2 ) and Op2 = ( i�t � �2 ). We get[un+1 = exp (i�)cun = exp (i(n+ 1)�)cu0with � = ��2�t+ �6�t312 + o(�10�t5). Thus, if t = n�t,jjuex(t)� unjj2 = 2j sin ( t�t2�624 )j jju0jj2So, the error phase could grow up to 2jju0jj2, which is not negligible. On �gure (16),we plot the relative phase error (N2=jju0jj2) in one dimension for u0(x) = sin(6x),which is the eigen vector of Laplacian operator associated to the eigenvalue k2 = 36corresponding to �6 in the above formula. We take �t = 10�2, �x = 5:10�1 andx 2 [0; �].

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Fig. 16. Phase error for (LS) equationObviously, we present here the simple example to analyze the error phase ofCrank-Nicolson scheme. If we want to understand why N3 is bad too, we mustplot jjjuexj � junumjjj2 for a superposition of two solutions of (LS). The result foru0(x) = sin (6x) + sin (8x) is plotted on �g(17).13

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Fig. 17. N2=jju0jj2 and N3=jju0jj2 for (LS) equationWe see that the error for (LS) is quite high. In fact, juexj � junumj couples thephase as soon as the number of phase is more than two.A possible correction for (LS) consists in solvingiun+1 � un�t +�un+1 + un2 + �t212 �3un+1 + un2 = 0but, although this is better for the linear case, it is not so good for the nonlinearSchr�odinger equation.We continue our tests with the 2-2 dromion and a diagonal matrix �, whichcorresponds to the interaction of two lumps. We take �1 = 2 � 2i, �2 = 4 � 0:5i,l1 = 2 + 1, l2 = 1 + 2i, �1 = 1 � 2i, �2 = 3 � 0:5i, m1 = 1 + i, m2 = 2 + 3i,�11 = 1+ i, �22 = 2+3i and �12 = �21 = 0 in (2.1) to (2.3). The resulting algebraicequations are solved by making use of MapleV. Thus, we get the explicit expressionof u(�; �; t), u1(�; t) and u2(�; t) which are too large to be printed in this paper.Therefore, we can compare the numerical solution computed from u(�; �;�1) withthe exact solution above. The space domain is [�10; 10]� [�10; 10], with 257 pointsin every direction and �t = 10�3. We plot the contour of the exact solution and thenumerical one for t = �1; 0; 1 (�g (18)). 14

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10Numerical solution, t=0

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Fig. 18. Contour of dromion 2-2The two solutions are very close to each other and we note only a small discrep-ancy for t = 1 behind the upper bump. We also draw the module of the solution(�g(19,20,21)) at the same times to show that it is really the interaction of twolocalized lumps.−10

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Fig. 19. Numerical dromion2-2 at t=-1 −10

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Fig. 20. Numerical dromion2-2 at t=0 −10

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Fig. 21. Numerical dromion2-2 at t=1We precise the movement of the tracks left by ' with the representation of thecontour of ' (�g(22,23,24)). 15

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Fig. 22. ' at t=-1 −10 −8 −6 −4 −2 0 2 4 6 8 10−10

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8

10

Fig. 23. ' at t=0 −10 −8 −6 −4 −2 0 2 4 6 8 10−10

−8

−6

−4

−2

0

2

4

6

8

10

Fig. 24. ' at t=1As the � matrix is diagonal, the number of lumps is only two instead of four.However, there are four cross-points on the tracks. Two of them localize the existinglumps and the two others give the location of the two missing ones as stated in 8.The phase error (�g(25)) is bigger than the one of dromion 1-1, but the dynamic ofmovement is more sophisticated.

−1.0 −0.5 0.0 0.5 1.0

time

0.0

0.5

1.0

1.5

2.0

Fig. 25. jjuex � unumjj24.2. Role of the initial datum and parametersNow that we have tested our scheme with exact solutions, we can examine theaction of DS on other initial data. For that, we put the gaussian datum u0 =4 exp [�(x2 + y2)] in DSI with '1 = '2 = 0. Like it is stated in Fokas-Santini 8, allinitial data with '1 = '2 = 0 should disperse at in�nity and this is exactly whatwe get (�g(26,27,28,29,30)). 16

−10

−5

0

5

10 −10−5

05

10

0

1

2

3

4

Fig. 26. Gaussian datum at t=0−10

−5

0

5

10 −10−5

05

10

0

1

2

3

4

Fig. 27. Gaussian datum at t=0.321

−10

−5

0

5

10 −10−5

05

10

0

1

2

3

4

Fig. 28. Gaussian datum at t=0.642−10

−5

0

5

10 −10−5

05

10

0

1

2

3

4

Fig. 29. Gaussian datum at t=0.963We see the e�ect of dispersion on the L1-norm which decreases with time(�g(31). 17

−10 −5 0 5 10

−10

−5

0

5

10

t=0.

−10 −5 0 5 10

−10

−5

0

5

10

t=0.321

−10 −5 0 5 10

−10

−5

0

5

10

t=0.642

−10 −5 0 5 10

−10

−5

0

5

10

t=0.963

Fig. 30. Contour of gaussian datum 0.0 0.3 0.5 0.8 1.00.0

1.0

2.0

3.0

4.0

5.0

Fig. 31. L1 norm of gaussian datumThen, we go on by changing some coe�cients while keeping the others to thevalues taken for DSI. The initial datum and the boundary conditions are the samethan those of dromion 1-1 test. It is di�cult to make an exhaustive review ofthe e�ects of each parameter due to their number. However, we can imagine thein uence of some coe�cients. For example, � and should manage the dispersion.Some tests show that � and do not have the same in uence. � acts directly onthe x-direction, whereas acts on the y-direction, but without the same strength.Indeed, x and y do not have the same role in DS. For example, we present here thetest for � = = 1. Then, the equations become

� iut +�u = juj2u+ u'x;'xx � 'yy = �2(juj2)x:We see that the initial lump disperses away much faster in the x-direction than inthe y-direction �g(32,33). 18

−100

10

−10

0

100

0.5

1

t=−1.5

−100

10

−10

0

100

0.5

1

t=−1.072

−100

10

−10

0

100

0.5

1

t=−0.640

−100

10

−10

0

100

0.5

1

t=−0.216

Fig. 32. � = = 1

−10 −5 0 5 10

−10

−5

0

5

10

t=−0.216

−10 −5 0 5 10

−10

−5

0

5

10

t=−0.640

−10 −5 0 5 10

−10

−5

0

5

10

t=−1.072

−10 −5 0 5 10

−10

−5

0

5

10

t=−1.5

Fig. 33. � = = 119

Then, we choose to study the e�ects of the b parameter on the behavior ofdromion 1-1. We plug into (DSI) respectively b = 0:5 and b = 1:5. We note(�g(34,35,36,37)) that dromion 1-1 is not stable at all with respect to b.

−100

10

−10

0

100

0.5

1

t=1.5

−100

10

−10

0

100

0.5

1

t=0.640

−100

10

−10

0

100

0.5

1

t=−0.430

−100

10

−10

0

100

0.5

1

t=−1.5

Fig. 34. b = 0:5−10 −5 0 5 10

−10

−5

0

5

10

t=−1.5

−10 −5 0 5 10

−10

−5

0

5

10

t=−0.430

−10 −5 0 5 10

−10

−5

0

5

10

t=0.640

−10 −5 0 5 10

−10

−5

0

5

10

t=1.5

Fig. 35. b = 0:520

−100

10

−10

0

100

0.5

1

t=−1.5

−100

10

−10

0

100

0.5

1

t=−0.430

−100

10

−10

0

100

0.5

1

t=0.640

−100

10

−10

0

100

0.5

1

t=1.5

Fig. 36. b = 1:5

−10 −5 0 5 10

−10

−5

0

5

10

t=1.5

−10 −5 0 5 10

−10

−5

0

5

10

t=0.640

−10 −5 0 5 10

−10

−5

0

5

10

t=−0.430

−10 −5 0 5 10

−10

−5

0

5

10

t=−1.5

Fig. 37. b = 1:521

Other tests show that the results are about the same when modifying the valueof � and �. So, dromion 1-1 is not stable with respect to the coe�cients of (DSI).5. Blow-up of DS (E-H)It is well known that (NLS) admits blow-up solutions (see Ginibre-Velo 20, Kato21 or Glassey 22 for instance). Now, the dynamic of explosion is better understood(17,23). Moreover, as predicted by Ghidaglia and Saut 3, Papanicolaou-Sulem-Sulemand Wang 24 show numerically that the blow-up occurs in the elliptic-elliptic caseof (DS). Unfortunately, no results are known to validate or not the blow-up in the(E-H) mode.From now on, we set � = = 1=2, c = 1 and '1(�; t) = '2(�; t) = 0; 8t. Then,in the (�; �) plane, DS (E-H) becomes( iut +�u = �juj2u+ bu'�+�; (a)'�� = �4 (juj2� + juj2�): (b) (5.14)If we assume that � = �1, blow-up should arise when b goes to zero, as the �rstequation tends to the focusing (NLS) equation. In the same way, as � ! 0, (5.14 b)gives '(�; �; t) = 1(�; t)+ 2(�; t). Using the hypothesis '1(�; t) = '2(�; t) = 0; 8t,we get '(�; �; t) = r, r 2 R. Putting it on (5.14 a), we get exactly (NLS). Therefore,we will work �nally on the system( iut +�u = �juj2u+ bu'�+�;'�� = �4 (juj2� + juj2�): (5.15)In order to verify our statement, we take as initial condition the one used in17 and 23 for blow-up of (NLS). So, u0 = 4 exp [�(x2 + y2)]. In the last reference17, the presumed blow-up time for this initial datum is compute numerically andis t� = 0:1425. This reference time allows us to validate our suppositions. Forthe numerical tests, we take a 256 � 256 mesh, �t = 10�4 and the domain is[�4; 4]� [�4; 4].We begin by the � � test consisting to compute the approximate solutions ofequations (5.15), with the gaussian initial datum, for � ! 0 and b = 1. We plot on�g(38) sup�;� ju(�; �; t)j for di�erent values of �.22

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.160.0

25.0

50.0

75.0

100.0−0.1

0.010.1 0.2

σ= −0.3

σ=

t*

−0.2σ=

σ=σ=

σ=

Fig. 38. Evolution of blow-up time for b = 1 and di�erent values of �We see clearly that blow-up seems to arise and that the sign of � changes therelative position of the blow-up time for (5.15) compared to t�.Next, we go on with b! 0, setting � = �2. Now, we plot the results on �g (39)for a wider range of values of b.

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.240.0

25.0

50.0

75.0

100.0

b=2

b=1

b=0.5

b=0.3

b=0.1

b=−0.1 b=−0.15

b=−0.25

b=−0.3

t*

Fig. 39. Evolution of blow-up time for � = �2 and di�erent values of bIf b is small enough, the results are the same than for the � � test. But we seein addition that a strong concentration and may be a blow-up occurs for values ofb in O(1) when b is positive. Therefore, this fact seems to indicate that DS (E-H)system has an inner blow-up mechanism. For negative values of b, a stabilization23

appears. The �rst value of b leading to this stabilization is hard to compute becauseof relative instability. We plot on �g (40) a more complete study on a 512 � 512�ner mesh for those values leading to stabilization.

0.00 0.05 0.10 0.15 0.20 0.250.0

50.0

100.0

150.0

200.0

b=−0.18

b=−0.19

b=−0.20

b=−0.21

b−0.22

b=−0.25

Fig. 40. Stabilization of blow-up for � = �2

We show now that the sign of � has an in uence on the position of the supposedblow-up time of DS with respect to the blow-up time of (NLS). The �g(39) and�g(41) illustrate this fact with � = �2 and � = +2.24

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.200.0

25.0

50.0

75.0

100.0b=−0.1

b=0.1

b=0.2

t*

Fig. 41. Evolution of blow-up time for � = +2 and varying bFinally, although we do not have a re�nement procedure, we show on variousmeshes the validity of the tests above as the L1-norm increases twofold each timethe mesh size is divided by two. So, we think that blow-up for DS (E-H) reallyexists. We plot on (�g(42)) sup�;� ju(�; �; t)j for di�erent mesh and juj (�g(43,44))att = 0:1704 for � = �2 and b = �0:15.

0.00 0.05 0.10 0.15 0.200.0

50.0

100.0

150.0

200.0

Mesh=128x128

Mesh=256x256

Mesh=512x512

Fig. 42. Evolution of blow-up time for b = �0:15 and di�erent grid mesh25

−2

−1

0

1

2

−2

−1

0

1

20

5

10

15

20

25

Fig. 43. juj at t=0 −2

−1

0

1

2

−2

−1

0

1

20

5

10

15

20

25

Fig. 44. juj at t=0.17046. ConclusionWe develop a new scheme in order to solve DS (E-H) systems. This tool allows toshow that dromions 1-1 are not stable with respect to coe�cients and that blow-upmechanism can exist for DS (E-H). We con�rm that from an initial datum with'1 = '2 = 0, the solution disperses away for (DSI). In addition, we prove thatCrank-Nicolson type schemes create a periodic phase error that can be quite bigfor some values of t. Unfortunately, we can not prove existence of solution and theconvergence of our semi-discrete scheme. However, the principle of relaxation isapplicable to a wide range of systems, and, in particular, our scheme is relativelyeasy to transpose to other versions of DS.AcknowledgmentThe authors thank T. Colin for valuable discussions and fruitful suggestions. Theyalso thank P. Fabrie for helping them to understand better the phase error createdby Crank-Nicolson scheme. In addition, the �rst author is very greatful to P. De-gond who kindly welcome him at the Math�ematiques pour l'Industrie et la Physiquelaboratory in Toulouse.1. Davey A. and Stewartson K. On three-dimensional packets of surface waves, Proc. R.Soc. Lond. A, vol 338, pages 101{110, 1974.2. Djordjevic V.D. and Redekopp L.G. On two-dimensional packets of capillary-gravitywaves, J. Fluid Mech., vol 79, pages 703{714, 1977.3. Ghidaglia J-M. and Saut J-C. On the initial value problem for the Davey-Stewartsonsystems, Nonlinearity 3, pages 475{506, 1990.4. Ablowitz M.J. and Segur H. On the evolution of packets of water waves, J. Fluid Mech.,vol 92, n0 (4), pages 691{715, 1979.5. Hayashi N. and Hirata H. Global existence and asymptotic behavior in time of small26

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