KdV extensions with Painlevé property

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1 JOURNAL OF MATHEMATICAL PHYSICS VOLUME 39, NUMBER 4 APRIL 1998 KdV etensions with Painlevé propert Sen-ue Lou a) CCAST (World Laborator), P. O. Bo 8730, Beijing , People s Republic of China and Institute of Mathematical Phsics, Ningbo Universit, Ningbo 31511, People s Republic of China b Received 4 September 1996; accepted for publication August 1997 B means of the conformal invariance Möbious transformation invariance, the well known KdV equation is etended to i a 11-dimensional space-time smmetric form; ii two 1-dimensional space isotropic forms; and iii general 31-dimensional and N-dimensional forms. The etensions are proven to be integrable under the meaning that the possess the Painlevé propert American Institute of Phsics. S I. INTRODUCTION The soliton theor has attracted much attention from a rather diverse group of scientists e.g., both phsicists and mathematicians because it is widel applied in man phsicall significant fields such as plasma phsics, fluids, optics, relative field theor, astrophsics and geophsics, etc. and demonstrates a deep and fundamental relationship with both modern and core Mathematics e.g., algebraic and differential geometr, analsis, group theor and chaos. However, there is still a long list of important open problems in soliton theor. 1 Here are three of them: i ii iii Because of the real phsical space is 31-dimensional, one hopes to find some 31dimensional integrable models. Unfortunatel, ecept for the -dimensional self-dual Yang Mills SDYM equations, there is no one known integrable model in more than 1-dimensions. Usuall, the phsical space is isotropic. That is to sa the phsical laws should be invariant under the echange of space variables. However, ecept for the Dave Stewartson DS and the Nizhnik Novikov Veselov NNV equations, almost all the known 1dimensional integrable models are non-invariant under the echange of the space variables. If we consider relativistic phsics, the models should be also invariant under the echange of the space-time. Nevertheless, most of the famous integrable models are non-smmetric with respect to space-time. In this paper, we would like to treat these problems b means of the Painlevé analsis. The Painlevé tests, as formulated b Ablowitz, Ramani and Segur ARS and Weiss, Tabor and Carnevale WTC, 3 have been proven to be useful criteria for the identification of completel integrable models. In the net section, according to the fact that the conformal invariance in solution space Möbious transformations of fields plas important roles, we propose a 11-dimensional integrable model with space-time echange smmetr under the meaning that the model can be changed to a variant form with the Painlevé propert. Using the same idea, two space isotropic 1-dimensional models are studied in section III. In section IV, a general 31-dimensional Schwartz KdV etension is proved to be integrable under the meaning that a variant of the model possesses the Painlevé propert. Section V is devoted to discussion of the integrabilit of an N-dimensional etension. The last section is a summar and discussion. a Electronic mail: slou@fudan.ac.cn b Mailing address /98/39(4)/11/10/$ American Institute of Phsics Downloaded 8 Sep 001 to Redistribution subject to AIP license or copright, see

2 J. Math. Phs., Vol. 39, No. 4, April 1998 Sen-ue Lou 113 II. A SPACE-TIME SYMMETRIC 11 -DIMENSIONAL KdV EXTENSION It is known that the smmetr stud is one of the important works both in phsics and in mathematics, especiall in the stud of the soliton sstems because ever integrable model possesses infinitel man smmetries. 4 6 Recentl we found that the conformal invariance Möbious transformation invariance of the Schwartz form is quite important. 7 For instance, for the KdV equation U t U 6UU 0, 1 three sets of infinitel man nonlocal smmetries and one set of local smmetries and the Darbour transformation can be derived from the conformal invariance of the Schwartz KdV equation t ;0, where ;( / ) 1 ( / ) is the Schwartz derivative, while the KdV equation 1 and its Schwartz form are related b 8 U This fact hints to us that if we want to construct an integrable model, the conformal invariant equations ma be good candidates. In this paper we would like to construct some KdV tpe of etensions b using this idea. First, because of the relativit requirement, we etend to a space time smmetric form t ;c 1 t ;t 0, 3 where we have inserted a constant c 1 in such that 3 can be reduced to the usual Schwartz KdV when c 1 0. Equation 3 is smmetric for the echange of the space-time t for c 1 1. The model is obviousl conformal invariant in the solution space. In other words, 3 is invariant under the Möbious transformation ab, adbc. 4 cd To check whether the model 3 is integrable or not, we make some transformations first. Using the transformation ep F, 5 eq. 3 is changed to The further transformations F t F; 1 F F c 1 F F;t 1 F t F t 0. 6 F u, F t v 7 make 6 become an equation sstem possessing the Painlevé propert, uvvu c 1 uv tt 3c 1 u v t v u u v u c 1 v uvv c 1 u 0, u t v, 8 9 where eq. 9 comes from the compatibilit condition of the transformations 7. Downloaded 8 Sep 001 to Redistribution subject to AIP license or copright, see

3 114 J. Math. Phs., Vol. 39, No. 4, April 1998 Sen-ue Lou To prove the Painlevé propert of the sstem 8 and 9, we can use the standard WTC approach. 3 According to the WTC paper, we sa a model has the Painlevé propert if its solutions are single valued about an arbitrar movable singular manifold which is given b ( 1,,..., n,t)0. For the sstem 8 and 9, we can epand u and v as u j0 u j j, v j0 v j j, 10 with u j,v j being arbitrar functions of (,t) and u 0 0,v 0 0. B using leading order analsis i.e., substituting uu 0, vv 0, into 8, 9, we have 1, 1, u 0, v 0 u 0 t /. 11 Now substituting 10 with 11 into 8 and 9, we can get the recursion relations on the functions u j and v j : j1 j1u 0 u j c 1 v 0 v j f u i,v i,i j1, 1 j1v j u 0 u j v 0 u j1,t v j1,, 13 where f is a complicated function of u i,v i,i j1 and the derivatives of the singularit manifold. From 1 and 13, the resonance values of j are given b j1u 0 j1v 0 c 1 j 1v 0 j 1u 0 j1 j1 u 0 c 1 v 0 0, 14 i.e., j1,1,1. 15 The resonance at j1 corresponds to the arbitrar singularit manifold. At two other resonances, j1, there are two compatibilit conditions u 0 v 0 c 1 tt u 0 u 0 c 1 v 0t v 0 t0, u 0t v 0 0, that need to be satisfied. It is clear that 16 and 17 are satisfied identicall due to 11. The Painlevé propert of the sstem 8 and 9 is proven. So the equation sstem 8 and 9 is integrable under the meaning that it possesses the Painlevé propert. Now we turn to stud 1-dimensional eamples. III. TWO SPACE ISOTROPIC 1 -DIMENSIONAL EXTENSIONS The well known KdV equation has some different etensions in 1-dimensions, sa, the Kadomtsev Petviashvili KP equation, 9 u t u 6uu 3u 0, 18 the Boiti Leon Manna Pempinelli BLMP equation 10 u t u 3u 1 u 0, 19 and the Nizhnik Novikov Veselov NNV equation, 11 u t u u 3u 1 u 3u 1 u 0, 0 Downloaded 8 Sep 001 to Redistribution subject to AIP license or copright, see

4 J. Math. Phs., Vol. 39, No. 4, April 1998 Sen-ue Lou 115 etc. Actuall, the NNV equation is onl a combination of two BLMP equations because of u 3(u 1 u ) and u 3(u 1 u ) being commutative. Correspondingl, the etensions of the Schwartz KdV to Schwartz KP and Schwartz BLMP equations have the forms 1 and t ; 3 3 0, t 3; ; 0, 1 respectivel. In this section, we would like to propose two new tpes of 1-dimensional Schwartz KdV etensions with the space echange,, invariance. The first one is t ; t ;0, 3 which will be reduced back to the Schwartz KdV for. The second etension is t ; 3 3 a t ; 3 3 0, 4 which will be reduced back to the Schwartz KP 1 for a0 and to the Schwartz KdV for. Similar to the 11-dimensional case, making the transformation ep F 5 for both 3 and 4, we obtain F t F F; F t F F;F F 0 6 for 3 and F t F; 3F F F 3 F F a F t F; 3F F F 3 F F F F af F 0 7 for 4. Furthermore, making the transformations eq. 6 becomes F u, F v, F t w, 8 uvwuv3v u u v uvvu uv u v u v 0, 9 Downloaded 8 Sep 001 to Redistribution subject to AIP license or copright, see

5 116 J. Math. Phs., Vol. 39, No. 4, April 1998 Sen-ue Lou u t w, v t w after multipling it b u v while eq. 7 is changed to u v w vaw uvu auv 3vv 3auu 3v 5 u 3au 5 v uvwv u awu v 4v u u 4au v v 3u 3 v 3 3av 3 u 3 u 3 v 3 uu avv 0, u t w, v t w after multipling it b u 3 v 3. To prove the Painlevé propert of 9 31, we epand u,v,w as u j0 u j j, v j0 v j j, w j0 w j j 35 with 1, w 0 t, u 0 w 0 / t, v 0 w 0 / t, 36 where eq. 36 is given b leading order analsis. Substituting 35 with 36 into the sstem 9 31, we have j1 j1u 0 u j v 0 v j f u i,v i,w i,i j1, j1w j u 0 u j w 0 u j1,t w j1,, j1w j v 0 v j w 0 v j1,t w j1,, where f is a function of u i,v i,w i,i j1 and the derivatives of the singularit manifold. It is clear that the resonances occur at j1,1,1,1. From 37 39, we can get three resonance conditions which should be satisfied identicall and have the forms u 0 v 0 u 0 v 0 v 0 u 0 0, u 0t w 0 0, v 0t w Obviousl, 40 4 are true because of 36. Using the leading order analsis to the sstem 3 34 leads to the same Laurent series 35 with 36. Substituting 35 with 36 into 3 34 ields j1 j1 ju 0 3 v 0 3 u 0 u j av 0 v j f u i,v i,i j1, j1w j u 0 u j w 0 u j1,t w j1,, j1w j v 0 v j w 0 v j1,t w j1,, where f is again a function of u i,v i,w i,i j1 and the derivatives of. Now the resonances are located at Downloaded 8 Sep 001 to Redistribution subject to AIP license or copright, see

6 J. Math. Phs., Vol. 39, No. 4, April 1998 Sen-ue Lou 117 j1,1,1,1,, 46 and the corresponding compatibilit conditions are u 0 v 0 u 0 v 0 v 0 u 0 u 0 v 0 u 0 au 0 v 0 u 0 v 0 u 0 v 0 v 0 u 0 v 0 0, 47 u 0 v 3 0 u 0 u 0 3u 0 v 0 u 0 u 0 u 0 u 0 3v 0 av 0 u 3 0 v 0 v 0 3u 0 v 0 v 0 v 0 v 0 v 0 3u 0 0, u 0t w 0 0, v 0t w The compatibilit conditions 47 and 48 come from 43 for j1 and j, respectivel. It is also straightforward to see that the conditions are satisfied identicall because of eq. 36. From the above calculations, we know that both sstems 9 31 and 3 34 possess the Painlevé propert. Then two space smmetric 1-dimensional Schwartz KdV etensions 3 and 4 are integrable under the meaning that the can be changed to the variant forms with the Painlevé propert. The ecellent propert of the 11-dimensional and 1-dimensional Schwartz equations leads us to construct integrable models in higher dimensions. IV. A GENERALIZED 31 -DIMENSIONAL KdV EXTENSION As mentioned in the Introduction, because the real phsical space-time is 31-dimensional, we discuss a 31-dimensional etension of the Schwartz KdV equation before discussing more general higher dimensional etension. According to discussions of the last two sections, one ma guess that the following 31dimensional sstem, u u v w g a 1 u a v v a w zz 3 w a g tt 4 v w z g a 3 u 1 u a v a 3 w a 4 1 a g 1u a v a 3 w a 4 g a 5 u a u 6 g a u 7 v a v 8 u a 9 u w a w 10 u a v 11 w a w 1 v a g 13 w a w 14 g a g 15 v a v 16 0, g u t g, g t g 51 5 v t g, w t g z, with arbitrar constants a i, i1,,...,16, mapossess the Painlevé propert. The sstem can be simplified to a single Schwartz form a 1 ;a ;a 3 ;za 4 ;ta 5 t a 6 t a 7 a 8 a 9 z a 10 z a 11 z a 1 z a 13 t z a 14 z t a 15 t a 16 t 0, 55 Downloaded 8 Sep 001 to Redistribution subject to AIP license or copright, see

7 118 J. Math. Phs., Vol. 39, No. 4, April 1998 Sen-ue Lou with epf, uf, vf, wf z, and gf t. To check the Painlevé propert of the sstem 51 54, we use the leading order analsis first again. Substituting u,v,w,gu 0,v 0,w 0,g 0 56 to leads to 1, g 0 t, u 0 g 0 t, v 0 g 0 t, w 0 g 0 z t. 57 Then the Laurent epansions of u,v,w,g have the forms u j0 w j0 u j j1, w j j1, v j0 g j0 v j j1, g j j1. 58 Using 58 with 57 in 51 54, weget j1j1a 1 u 0 u j a v 0 v j a 3 w 0 w j a 4 g 0 g j fu i,v i,w i,g i,ij1, j1g j u 0 u j g 0 u j1,t g j1,, j1g j v 0 v j g 0 v j1,t g j1,, j1g j w 0 w j g 0 w j1,t g j1,z with the resonances j1,1,1,1. 63 The remaining discussions are similar to that of the last two sections. Four compatibilit conditions u 0 v 0 w 0 g 0 a 1 a a 3 zz a 4 tt a 1 w 0 v 0 g 0 u 0 a u 0 w 0 g 0 v 0 a 3 u 0 v 0 g 0 z w 0z a 4 u 0 v 0 w 0 t g 0t 0, u 0t g 0 0, v 0t g 0 0, and w 0t g 0z 0 67 are satisfied naturall thanks to 57. So the sstem possesses Painlevé propert and the generalized 31-dimensional Schwartz sstem 55 is integrable because it can be transformed to some variants with Painlevé propert. Some special cases of 55 are worth mentioning. i When we take the constants a i 0 ecept a 1 a 4 a 5 a 6 1, the model 55 reduces back to the 11-dimensional space-time smmetric model given in section II. ii If we take a 1 a a 5 a 15 1 and all other a i being zero, we get the 1-dimensional space isotropic model 3. iii If we take all the constants as one, then the model 55 becomes a 31-dimensional relativistic integrable model, i.e., the model is spacetime echange invariant. Downloaded 8 Sep 001 to Redistribution subject to AIP license or copright, see

8 J. Math. Phs., Vol. 39, No. 4, April 1998 Sen-ue Lou 119 Actuall, one can easil obtain various other tpes of 31-dimensional models b means of the conformal invariant quantities which are invariant under the Möbious transformation. However, we prefer to give a possible integrable generalization in arbitrar dimensions rather than to discuss this problem further. V. AN N-DIMENSIONAL KdV EXTENSION Generall, an N-dimensional conformal invariant model ma have the form G i ; i,,i, j,1,,...,n0, j 68 where G is an arbitrar functions of the conformal invariants ; i, i / j,(i,j,1,,...,n)0 and their derivatives. We believe that there are man tpes of special function G which ma be changed to a variant form with the Painlevé propert. In this section, we write down onl one special G and change it to a form with the Painlevé propert. Using the same idea as in lower dimensions discussed in sections II IV, we ma prove the following special case of 68, N k0 a k ; k g i j,i, j,1,,...,n0, 69 where g is an arbitrar polnomial functions of i / j (i, j,1,,...,n), ma be transformed to a form possessing Painlevé propert. Taking the transformations eq. 69 is changed to epf, u i F i, i1,,... N, 70 N k1 u k m k N k1 a k n; u kk k u k 3u kk u k 1 u k g u i,i,j1,,...,n0, u j 71 u 1i u ii, i1,,...,n, 7 N m where m j are positive integers and are fied such that ( k1 u k k )g(u i /u j, i, j1,,...,n) is onl a polnomial function of u i, i1,,...,n. To prove the Painlevé propert of 71 and 7, we epand u i as u i j0 u ij j i, i1,,...,n, 73 where i and u i0 are fied b leading order analsis. The result is i 1, u i0 i, i1,,...,n. 74 Substituting eq. 73 with 74 into 71 and 7, we obtain N j1 j1 a k u k0 u kj fu ki,k1,,...,n,ij1, k1 75 j1u 1 j u k0 u kj u 10 u k j1,t u 1 j1,k, k,3,...,n 76 with N1 resonances, Downloaded 8 Sep 001 to Redistribution subject to AIP license or copright, see

9 10 J. Math. Phs., Vol. 39, No. 4, April 1998 Sen-ue Lou 77 where f is a ver complicated function of u ki, k1,,...,n,ij1. The resonances j1 corresponds to the arbitrar singular manifold. For other resonances, j1, after finishing some tedious calculations, we get N compatibilit conditions N k0 a k u k0,k k k u k0 k 0, 78 u k0,t u 10,k 0, k,3,...,n. 79 Obviousl the resonance conditions 78 and 79 are satisfied identicall because of eq. 74. Then equation 69 are integrable under the meaning that its variant 71 and 7 possesses the Painlevé propert. VI. SUMMARY AND DISCUSSION According to the fact that man integrable models can be transformed to the conformal invariant forms Schwartz forms and various integrable properties such as the Darbour transformation, infinitel man smmetries, etc., can be derived from the conformal invariance, we believe that one ma get various integrable models b etending the known integrable Schwartz equations. Starting from this idea, we have constructed some significant KdV etensions. Some of them are isotropic in space and some of them are relativistic i.e., space-time smmetric. Especiall the Schwartz KdV equation has been etended to real 31-dimensional phsical space. The Painlevé properties of these etensions are all proved b using the standard WTC approach. Though the etensions are all proven to be integrable under the meanings that the possess the Painlevé propert and the Schwartz forms with conformal invariance, there are still man important problems worth further stud: i ii iii Usuall, one would sa a model is integrable under some different meanings. For instance, a model is integrable under the meanings that it can be solved b the inverse scattering transformation, it possesses the Painlevé propert, it possesses La Pairs, it possesses bi-hamiltonian structure and infinitel man smmetries and conservation laws and so on. Are the models obtained here integrable under other meanings, sa, under the meaning that the can be solved b inverse scattering transformation? Though the models are obtained onl from the consideration of the integrabilit, we believe that the ma be found also from the real phsical sstems because the real phsical space-time is isotropic, relativistic and 31-dimensional. The most possible phsical sstems which can be used to derive these models are the SDYM equations because of the considerabl increasing the validit of Ward s conjecture 13,1 : man (and perhaps all?) of the ordinar and partial differential equations that are regarded as being integrable or solvable ma be obtained from the self-dual gauge field equations (or its generalizations) b reduction. Almost all the known integrable models, sa, the KdV, KP, DS, N-wave interaction equations, nonlinear Schrödinger, chiral field equation, Toda equations and sine-gordon equation, have been obtained from the reduction of the SDYM equations. 1 How do we get the model obtained in this paper from concrete phsical sstems, sa, SDYM equation? Generall, one can construct an N-dimensional integrable model which possesses the Painlevé propert quite freel b means of the conformal invariants. For instance, an arbitrar polnomial function of some conformal invariants has been included in our special eample 69. Though not all the equations in the Schwartz form are integrable can be changed to a form with the Painlevé propert, it is still significant to ask: What kind of Downloaded 8 Sep 001 to Redistribution subject to AIP license or copright, see

10 J. Math. Phs., Vol. 39, No. 4, April 1998 Sen-ue Lou 11 conformal invariant form function G in 68 ma be changed to a form with Painlevé propert? ACKNOWLEDGMENTS The work was supported b the National Nature Science Foundation of China and the Nature Science foundation of Zhejiang province in China. I thank Professors Q-p Liu, X-b Hu and G-j Ni for their helpful discussions. 1 M. J. Ablowitz and P. A. Clarkson, Solitons Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Societ Lecture Note Series 149 Cambridge U.P., Cambridge, M. J. Ablowitz, A. Ramani, and H. Segur, Lett. Nuovo Cimento 3, J. Weiss, M. Tabor, and G. Carnevale, J. Math. Phs. 4, P. J. Olver, Application of Lie Group to differential equations, Graduate tets in Mathematics Springer, Berlin, 1986; Q. W. Bluman and S. Kumei, Smmetries and differential equations Springer, Berlin, B. Fuchssteiner, Prog. Theor. Phs. 70, ; P. M. Santini and A. S. Fokas, Commun. Math. Phs. 115, ; A. S. Fokas and P. M. Santini, Commun. Math. Phs. 116, ; J. Math. Phs. 9, ; H.H. Chen, Y. C. Lee, and J. E. Lin, Phs. Lett. 91, ; Phsica D 6, ; D. David, N. Kamran, D. Levi, and P. Winternitz, J. Math. Phs. 7, S- Lou, Phs. Lett. B 30, ; Phs. Lett. A 175, 31993; 181, ; J. Math. Phs. 35, ; 35, ; Phs. Rev. Lett. 71, ; J. Phs. A 6, ; 7, ; J. Math. Phs. 35, S- Lou, J. Phs. A 30, ; 30, M. C. Nucci, J. Phs. A, B. B. Kadomtsev and V. I. Patviashvili, Sov. Phs. Dokl. 15, M. Boiti, J. Jp. Leon, M. Manna, and F. Pempinelli, Inverse Probl., L. P. Nizhnik, Sov. Phs. Dokl. 5, ; A. P. Veselov and S. P. Novikov, Sov. Math. Dolk. 30, ; 30, ; S. P. Novikov and A. P. Veselov, Phsica D 18, Q- Wu and S- Lou, Commun. Theor. Phs. 7, R. S. Ward, Philos. Trans. R. Soc. London, Ser. A 315, Downloaded 8 Sep 001 to Redistribution subject to AIP license or copright, see