General soliton matrices in the Riemann–Hilbert problem for integrable nonlinear equations Valery S. Shchesnovich and Jianke Yang Citation: Journal of Mathematical Physics 44, 4604 (2003); doi: 10.1063/1.1605821 View online: http://dx.doi.org/10.1063/1.1605821 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/44/10?ver=pdfcov Published by the AIP Publishing This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 186.217.234.225 On: Tue, 14 Jan 2014 15:18:36 http://scitation.aip.org/content/aip/journal/jmp?ver=pdfcov http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/1722768765/x01/AIP-PT/JMP_CoverPg_1113/aipToCAlerts_Large.png/5532386d4f314a53757a6b4144615953?x http://scitation.aip.org/search?value1=Valery+S.+Shchesnovich&option1=author http://scitation.aip.org/search?value1=Jianke+Yang&option1=author http://scitation.aip.org/content/aip/journal/jmp?ver=pdfcov http://dx.doi.org/10.1063/1.1605821 http://scitation.aip.org/content/aip/journal/jmp/44/10?ver=pdfcov http://scitation.aip.org/content/aip?ver=pdfcov rtially s, by grable ces os of re of the n ctral os of n so- se JOURNAL OF MATHEMATICAL PHYSICS VOLUME 44, NUMBER 10 OCTOBER 2003 This article is copyrig General soliton matrices in the Riemann–Hilbert problem for integrable nonlinear equations Valery S. Shchesnovicha) and Jianke Yangb) Department of Mathematics and Statistics, University of Vermont, Burlington, Vermont 05401 ~Received 25 May 2003; accepted 5 June 2003! We derive the soliton matrices corresponding to an arbitrary number of higher- order normal zeros for the matrix Riemann–Hilbert problem of arbitrary matrix dimension, thus giving the complete solution to the problem of higher-order soli- tons. Our soliton matrices explicitly give all higher-order multisoliton solutions to the nonlinear partial differential equations integrable through the matrix Riemann– Hilbert problem. We have applied these general results to the three-wave interac- tion system, and derived new classes of higher-order soliton and two-soliton solu- tions, in complement to those from our previous publication@Stud. Appl. Math. 110, 297 ~2003!#, where only the elementary higher-order zeros were considered. The higher-order solitons corresponding to nonelementary zeros generically de- scribe the simultaneous breakup of a pumping wave (u3) into the other two com- ponents (u1 andu2) and merger ofu1 andu2 waves into the pumpingu3 wave. The two-soliton solutions corresponding to two simple zeros generically describe the breakup of the pumpingu3 wave into theu1 andu2 components, and the reverse process. In the nongeneric cases, these two-soliton solutions could describe the elastic interaction of theu1 and u2 waves, thus reproducing previous results ob- tained by Zakharov and Manakov@Zh. Éksp. Teor. Fiz.69, 1654~1975!# and Kaup @Stud. Appl. Math.55, 9 ~1976!#. © 2003 American Institute of Physics. @DOI: 10.1063/1.1605821# I. INTRODUCTION The importance of integrable nonlinear partial differential equations~PDEs! in 111 dimen- sions in applications to nonlinear physics can hardly be overestimated. Their importance pa stems from the fact that it is always possible to obtain certain explicit solutions, called soliton some algebraic procedure. At present, there is a wide range of literature concerning inte nonlinear PDEs and their soliton solutions~see, for instance, Refs. 1–4 and the referen therein!. The reader familiar with the inverse scattering transform method knows that it is zer the Riemann–Hilbert problem~or poles of the reflection coefficients in the previous nomenclatu! that give rise to the soliton solutions. These solutions are usually derived by using one several well-known techniques, such as the dressing method,1,5,6 the Riemann–Hilbert problem approach,2,3 and the Hirota method~see Ref. 1!. In the first two methods, the pure soliton solutio is obtained by considering the asymptotic form of a rational matrix function of the spe parameter, called the soliton matrix in the following. It is known that the generic case of zer the matrix Riemann–Hilbert problem is the case of simple zeros7–12 ~see also Ref. 13!. A single simple zero produces a one-soliton solution. Several distinct zeros will produce multisolito lutions, which describe the interaction~scattering! of individual solitons. As far as the generic ca is concerned, there is no problem in the derivation of the corresponding soliton solutions. However, in the nongeneric cases, when at least one higher-order~i.e., multiple! zero is a!Also at: Instituto de Fisica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sa˜o Paulo, Brazil; electronic mail: valery@ift.unesp.br b!Electronic mail: jyang@emba.uvm.edu 46040022-2488/2003/44(10)/4604/36/$20.00 © 2003 American Institute of Physics hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 186.217.234.225 On: Tue, 14 Jan 2014 15:18:36 must h zeros f the nds to y everal multi- atrix the g will sion, trices nsider- rature ear hout mined exten- fs. 17 essing were vious igher- l. tance, . Such Natu- bitrary d. PDEs nding eneral. ltisoli- eneric ns in r corre- or ction higher- simulta- 4605J. Math. Phys., Vol. 44, No. 10, October 2003 General soliton matrices This article is copyrig present in the Riemann–Hilbert problem, the situation is not so definite. Higher-order zeros be considered separately, as, in general, the soliton solutions which correspond to suc cannot be derived from the known generic multisoliton solutions by coalescing some o distinct simple zeros. This is clear from the fact that a higher-order zero generally correspo a higher-order pole in the soliton matrix~or its inverse!, which cannot be obtained in a regular wa by coalescing simple poles in the generic multisoliton matrix. The procedure of coalescing s distinct simple zeros produces only higher-order zeros with equal algebraic and geometric plicities ~the geometric multiplicity is defined as the dimension of the kernel of the soliton m evaluated at the zero!, which is just the trivial case of higher-order zeros. For instance, if algebraic multiplicity is equal to or greater than the matrix dimension, then such coalescin produce a higher-order zero with the geometric multiplicity no less than the matrix dimen which could only correspond to the zero solution instead of solitons. Thus the soliton ma corresponding to higher-order zeros of the Riemann–Hilbert problem require a separate co ation. Soliton solutions corresponding to higher-order zeros have been investigated in the lite before, mainly for the 232-dimensional spectral problem. A soliton solution to the nonlin Schrödinger~NLS! equation corresponding to a double zero was first given in Ref. 14 but wit much analysis. The double- and triple-zero soliton solutions to the KdV equation were exa in Ref. 15 and the general multiple-zero soliton solution to the sine-Gordon equation was sively studied in Ref. 16 using the associated Gelfand–Levitan–Marchenko equation. In Re and 18, higher-order soliton solutions to the NLS equation were studied by employing the dr method. In Refs. 19–21, higher order solitons in the Kadomtsev–Petviashvili I equation derived by the direct method and the inverse scattering method. Finally, in our pre publication22 we have derived soliton matrices corresponding to a singleelementaryhigher-order zero—a zero which has the geometric multiplicity equal to 1. Our studies give the general h order soliton solutions for the integrable PDEs associated with the 232 matrix Riemann–Hilbert problem with a single higher-order zero. Indeed, any zero of the 232-dimensional Riemann– Hilbert problem is elementary since a nonzero 232 matrix can have only one vector in its kerne However, the previous investigations left some of the key questions unanswered. For ins the general soliton matrix corresponding to a single nonelementary zero remained unknown zeros arise when the matrix dimension of the Riemann–Hilbert problem is greater than 2. rally then, the ultimate question—the most general soliton matrices corresponding to an ar number of higher-order zeros in the generalN3N Riemann–Hilbert problem, was not addresse Because of these unresolved issues, the most general soliton and multisoliton solutions to integrable through theN3N Riemann–Hilbert problem~such as the NLS equation,23 the three- wave interaction system,2,24–27and the Manakov equations28! have not been derived yet. In this paper we derive the complete solution to the problem of soliton matrices correspo to an arbitrary number of higher-order normal zeros for the generalN3N matrix Riemann– Hilbert problem. These normal zeros are defined in Definition 1, and are nonelementary in g They include almost all physically important integrable PDEs where the involution property@see Eq. ~4!# holds. The corresponding soliton solutions can be termed as the higher-order mu tons, to reflect the fact that these solutions do not belong to the class of the previous g multisoliton solutions. Our results give a complete classification of all possible soliton solutio the integrable PDEs associated with theN3N Riemann–Hilbert problem. In other words, ou soliton matrices contain the most general forms of reflection-less~soliton! potentials in the N-dimensional Zakharov–Shabat spectral operator. For these general soliton potentials, the sponding discrete and continuous eigenfunctions of theN-dimensional Zakharov–Shabat operat naturally follow from our soliton matrices. As an example, we consider the three-wave intera system, and derive single-soliton solutions corresponding to a nonelementary zero, and order two-soliton solutions. These solutions generate many new processes such as the neous breakup of a pumping wave (u3) into the other two components (u1 andu2) and merger of u1 andu2 waves into the pumpingu3 wave, i.e.,u11u21u3↔u11u21u3 . They also reproduce previous solitons in Refs. 2, 22, 26, 27 as special cases. hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 186.217.234.225 On: Tue, 14 Jan 2014 15:18:36 ed in oliton eros are op- are e the n– c te o- ute ate ation t ation. under invo- lica- - f a mplex ntities 4606 J. Math. Phys., Vol. 44, No. 10, October 2003 V. S. Shchesnovich and J. Yang This article is copyrig The paper is organized as follows. A summary on the Riemann–Hilbert problem is plac Sec. II. Section III is the central section of the paper. There we present the theory of s matrices corresponding to several higher-order zeros under the assumption that these z normal ~see Definition 1!, which include the physically important cases with the involution pr erty @see Eq.~4!#. Applications of these general results to the three-wave interaction system contained in Sec. IV. Finally, in the Appendix we briefly treat the more general case wher zeros are abnormal. II. THE RIEMANN–HILBERT PROBLEM APPROACH The integrable nonlinear PDEs in 111 dimensions are associated with the matrix Rieman Hilbert problem~consult, for instance, Refs. 1–12, 29–32!. The matrix Riemann–Hilbert problem ~below we work in the space ofN3N matrices! is the problem of finding the holomorphi factorization, denoted below byF1(k) and F2 21(k), in the complex plane of a nondegenera matrix functionG(k) given on an oriented curveg: F2 21~k,x,t !F1~k,x,t !5G~k,x,t ![E~k,x,t !G~k,0,0!E21~k,x,t !, kPg, ~1! where E~k,x,t ![exp@2L~k!x2V~k!t#. Here the matrix functionsF1(k) and F2 21(k) are holomorphic in the two complementary d mains of the complexk-plane:C1 to the left andC2 to the right from the curveg, respectively. The matricesL(k) andV(k) are called the dispersion laws. Usually the dispersion laws comm with each other, e.g., given by diagonal matrices. We will consider this case@precisely in this case E(k,x,t) is given by the above formula#. The Riemann–Hilbert problem requires an appropri normalization condition. Usually the curveg contains the infinite pointk5` of the complex plane and the normalization condition is formulated as F6~k,x,t !→I as k→`. ~2! This normalization condition is called the canonical normalization. Setting the normaliz condition to an arbitrary nondegenerate matrix functionS(x,t) leads to the gauge equivalen integrable nonlinear PDE, e.g., the Landau–Lifshitz equation in the case of the NLS equ3 Obviously, the new solutionF̂6(k,x,t) to the Riemann–Hilbert problem, normalized toS(x,t), is related to the canonical solution by the following transformation F̂6~k,x,t !5S~x,t !F~k,x,t !. ~3! Thus, without any loss of generality, we confine ourselves to the Riemann–Hilbert problem the canonical normalization. For physically applicable nonlinear PDEs the Riemann–Hilbert problem possesses the lution properties, which reduce the number of the dependent variables~complex fields!. The following involution property of the Riemann–Hilbert problem is the most common in app tions F1 † ~k!5F2 21~ k̄!, k̄5k* . ~4! Here the superscript ‘‘†’’ represents the Hermitian conjugate, and ‘‘* ’’ the complex conjugate. Examples include the NLS equation, the Manakov equations, and theN-wave system. The analy sis in this article includes this involution~4! as a special case. In this case, the overline o quantity represents its Hermitian conjugation in the case of vectors and matrices and the co conjugation in the case of scalar quantities. In other cases, the original and overlined qua may not be related. hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 186.217.234.225 On: Tue, 14 Jan 2014 15:18:36 l equa- hat the e s t ions, ros 4607J. Math. Phys., Vol. 44, No. 10, October 2003 General soliton matrices This article is copyrig To solve the Cauchy problem for the integrable nonlinear PDE posed on the whole axisx, one usually constructs the associated Riemann–Hilbert problem starting with the linear spectra tion ]xF~k,x,t !5F~k,x,t !L~k!1U~k,x,t !F~k,x,t !, ~5! whereas thet-dependence is given by a similar equation ] tF~k,x,t !5F~k,x,t !V~k!1V~k,x,t !F~k,x,t !. ~6! The nonlinear integrable PDE corresponds to the compatibility condition of the system~5! and~6!: ] tU2]xV1@U,V#50. ~7! The essence of the approach based on the Riemann–Hilbert problem lies in the fact t evolution governed by the complicated nonlinear PDE~7! is mapped to the evolution of th spectral data given by simpler equations such as~1! and ~20a!–~20b!. When the spectral data i known, the matricesU(k,x,t) andV(k,x,t) describing the evolution ofF6 can then be retrieved from the Riemann–Hilbert problem. In our case, the potentialsU(k,x,t) andV(k,x,t) are com- pletely determined by the~diagonal! dispersion lawsL(k) and V(k) and the Riemann–Hilber solutionF[F6(k,x,t). Indeed, let us assume that the dispersion laws are polynomial funct i.e., L~k!5( j 50 J1 Ajk j , V~k!5( j 50 J2 Bjk j . ~8! Then using similar arguments as in Ref. 32 we get U52P$FLF21%, V52P$FVF21%. ~9! Here the matrix functionF(k) is expanded into the asymptotic series, F~k!5I 1k21F (1)1k22F (2)1¯ , k→`, and the operatorP cuts out the polynomial asymptotics of its argument ask→`. An important property of matricesU andV is that Tr U~k,x,t !52Tr L~k!, ~10! Tr V~k,x,t !52Tr V~k!, which evidently follows from Eq.~9!. This property guarantees that the Riemann–Hilbert ze are (x,t) independent. Let us consider as an example the physically relevant three-wave interaction system.2,24,25,27 SetN53, L~k!5 ikA, A5S a1 0 0 0 a2 0 0 0 a3 D , V~k!5 ikB, B5S b1 0 0 0 b2 0 0 0 b3 D , ~11! whereaj andbj are real with the elements ofA being ordered:a1.a2.a3 . From Eq.~9! we get U52L~k!1 i @A,F (1)#, V52V~k!1 i @B,F (1)#. ~12! Setting hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 186.217.234.225 On: Tue, 14 Jan 2014 15:18:36 ree s s the ilbert ral data ta rs 4608 J. Math. Phys., Vol. 44, No. 10, October 2003 V. S. Shchesnovich and J. Yang This article is copyrig u15Aa12a2F12 (1) , u25Aa22a3F23 (1) , u35Aa12a3F13 (1) , ~13! assuming the involution~4!, and using Eq.~12! in ~7! we get the three-wave system: ] tu11v1]xu11 i«ū2u350, ~14a! ] tu21v2]xu21 i«ū1u350, ~14b! ] tu31v3]xu31 i«u1u250. ~14c! Here v15 b22b1 a12a2 , v25 b32b2 a22a3 , v35 b32b1 a12a3 , ~15! «5 a1b22a2b11a2b32a3b21a3b12a1b3 @~a12a2!~a22a3!~a12a3!#1/2 . ~16! The group velocities satisfy the following condition: v22v3 v12v3 52 a12a2 a22a3 ,0. ~17! The three-wave system~14! can be interpreted physically. It describes the interaction of th wave packets with complex envelopesu1 , u2 , andu3 in a medium with quadratic nonlinearity. In general, the Riemann–Hilbert problem~1!–~2! has multiple solutions. Different solution are related to each other by the rational matrix functionsG(k) ~which also depend on the variable x and t):2–6,13 F̃6~k,x,t !5F6~k,x,t !G~k,x,t !. ~18! The rational matrixG(k) must satisfy the canonical normalization condition:G(k)→I for k→` and must have poles only inC2 @the inverse functionG21(k) then has poles inC1 only#. Such a rational matrixG(k) will be called the soliton matrix below, since it gives the soliton part of solution to the integrable nonlinear PDE. To specify a unique solution to the Riemann–Hilbert problem the set of the Riemann–H data must be given. These data are also called the spectral data. The full set of the spect comprises the matrixG(k,x,t) on the right-hand side of Eq.~1! and the appropriate discrete da related to the zeros of detF1(k) and detF2 21(k). In the case of involution~4!, the zeros of detF1(k) and detF2 21(k) appear in complex conjugate pairs,k̄ j5kj* . It is known7–12 ~see also Ref. 13! that in the generic case the spectral data include simple~distinct! zerosk1 ,...,kn of detF1(k) and k̄1 ,...,k̄n of detF2 21(k), in their holomorphicity domains, and the null vecto uv1&,...,uvn& and ^v̄1u,... ,̂ v̄nu from the respective kernels: F1~kj !uv j&50, ^v̄ j uF2 21~ k̄ j !50. ~19! Using the property~10! one can verify that the zeros do not depend on the variablesx andt. The (x,t) dependence of the null vectors can be easily derived by differentiation of~19! and use of the linear spectral equations~5! and ~6!. This dependence reads hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 186.217.234.225 On: Tue, 14 Jan 2014 15:18:36 he n e lvable of of t the alescing o this al non- liton it will s ert f c also recall t - 4609J. Math. Phys., Vol. 44, No. 10, October 2003 General soliton matrices This article is copyrig uv j&5exp$2L~kj !x2V~kj !t%uv0 j&, ~20a! ^v̄ j u5^v̄0 j uexp$L~ k̄ j !x1V~ k̄ j !t%, ~20b! whereuv0 j& and ^v̄0 j u are constant vectors. The vectors in Eqs.~20a! and ~20b! together with the zeros constitute the full set of t generic discrete data necessary to specify the soliton matrixG(k,x,t) and, hence, unique solutio to the Riemann–Hilbert problem~1!–~2!. Indeed, by constructing the soliton matrixG(k) such that the following matrix functions: f1~k!5F1~k!G21~k!, f2 21~k!5G~k!F2 21~k! ~21! are nondegenerate and holomorphic in the domainsC1 and C2 , respectively, we reduce th Riemann–Hilbert problem with zeros to another one without zeros and hence uniquely so ~for details see, for instance, Refs. 2–4, 13!. Below by matrixG(k) we will imply the matrix from equation~21! which reduces the Riemann–Hilbert problem~1!–~2! to the one without zeros. The corresponding solution to the integrable PDE~7! is obtained by using the asymptotic expansion the matrixF(k) ask→` in the linear equation~5!. In theN-wave interaction model it is given by formula ~12!. The pure soliton solutions are obtained by using the rational matrixF5G(k). The above set of discrete spectral data~19! holds only for the generic case where zeros detF1(k) and detF2 21(k) are simple. If these zeros are higher-order rather than simple, wha discrete spectral data should be and how they evolve withx andt is unknowna priori. Moreover, we have stressed in Sec. I that the case of higher-order zeros cannot be treated by co simple zeros, thus is highly nontrivial. In the next sections, we give the complete solution t problem. III. SOLITON MATRICES FOR GENERAL HIGHER-ORDER ZEROS In this section we derive the soliton matrices for an arbitrary matrix dimensionN and an arbitrary number of higher-order zeros under the assumption that these zeros are norm~see Definition 1!. Normal higher-order zeros are most common in practice. In general, they are elementary. Our approach is based on a generalization of the idea in our previous paper.22 A. Product representation of soliton matrices Our starting point to tackle this problem is to derive a product representation for so matrices. This product representation is not convenient for obtaining soliton solutions, but lead to the summation representation of soliton matrices, which is very useful. In treating the soliton matrix as a product of constituent matrices@called elementary matrice in Ref. 2, see formulas~24! and~27! below# one can consider each zero of the Riemann–Hilb problem separately. For instance, consider a pair of zerosk1 and k̄1 , respectively, ofF1(k) and F2 21(k) from Eq. ~1!, each having orderm: detF1~k!5~k2k1!mw~k!, detF2 21~k!5~k2 k̄1!mw̄~k!, ~22! wherew(k1)Þ0 andw̄( k̄1)Þ0. The geometric multiplicity ofk1 ( k̄1) is defined as the number o independent vectors in the kernel ofF1(k1) (F2 21( k̄1)), see~19!. In other words, the geometri multiplicity of k1 ( k̄1) is the dimension of the kernel space ofF1(k1) (F2 21( k̄1)). It can be easily shown that the order of a zero is always greater or equal to its geometric multiplicity. It is obvious that the geometric multiplicity of a zero is less than the matrix dimension. Let us how the soliton matrices are usually constructed~see, for instance, Refs. 2 and 13!. Starting from the solutionF6(k) to the Riemann–Hilbert problem~1!–~2!, one looks for the independen vectors in the kernels of the matricesF1(k1) andF2 21( k̄1). Assuming that the geometric multi plicities of k1 and k̄1 are the same and equal tor 1 , then we have hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 186.217.234.225 On: Tue, 14 Jan 2014 15:18:36 ix e e rices on e- s in lbert or ey are 4610 J. Math. Phys., Vol. 44, No. 10, October 2003 V. S. Shchesnovich and J. Yang This article is copyrig F1~k1!uv i1&50, ^v̄ i1uF2 21~ k̄1!50, i 51, . . . ,r 1 . ~23! Next, one constructs the constituent matrix x1~k!5I 2 k12 k̄1 k2 k̄1 P1 , ~24! where P15( i , j r 1 uv i1&~K21! i j ^v̄ j 1u, Ki j 5^v̄ i1uv j 1&. ~25! Here P1 is a projector matrix, i.e.,P1 25P1 . It can be shown that detx15(k2k1) r1/(k2k̄1) r1 ~note that the geometric multiplicityr 1 is equal to rankP1). If r 1,m then one considers the new matr functions F̃1~k!5F1~k!x1 21~k!, F̃2 21~k!5x1~k!F2 21~k!. By virtue of Eqs.~23!, the matricesF̃1(k) and F̃2 21(k) are also holomorphic in the respectiv half planes of the complex plane~see Lemma 1 in Ref. 22!. In addition,k1 and k̄1 are still zeros of detF̃1(k) and detF̃2 21(k). Assuming that the geometric multiplicities of zerosk1 andk̄1 in new matricesF̃1(k) and F̃2 21(k) are still the same and equal tor 2 , then the above steps can b repeated, and we can define matrixx2(k) analogous to Eq.~24!. In general, if the geometric multiplicities of zerosk1 and k̄1 in matrices F̃1~k!5F1~k!x1 21~k!¯x l 21 21 ~k!, F̃2 21~k!5x l 21~k!¯x1~k!F2 21~k! ~26! are the same and given byr l ( l 51,2,. . . ), then we can define a matrixx l similar to Eqs.~24! and ~25! but the independent vectorsuv i l & and^v̄ i l u ( i 51, . . . ,r l) are from the kernels ofF̃1(k1) and F̃2 21( k̄1) in Eq. ~26!. When this process is finished, one would get the constituent mat x1(k),..., x r(k) such thatr 11r 21¯1r n5m, and the product representation of the solit matrix G(k), G~k!5xn~k!¯x2~k!x1~k!. ~27! This product representation~27! is our starting point of this paper. In arriving at this repr sentation, our assumptions are that the zerosk1 and k̄1 have the same algebraic multiplicity@see Eq. ~22!#, and their geometric multiplicities in matricesF̃1(k) andF̃2 21(k) of Eq. ~26! are also the same for alll ’s. For convenience, we introduce the following definition. Definition 1: A pair of zeros k1 and k̄1 in the matrix Riemann–Hilbert problem is called normal if the zeros have the same algebraic multiplicity, and their geometric multiplicitie matricesF̃1(k) and F̃2 21(k) of Eq. (26) are also the same for all l’ s. In the text of this paper, we only consider normal zeros of the matrix Riemann–Hi problem. The case of abnormal zeros will be briefly discussed in the Appendix. Remark 1:Under the involution property~4!, all zeros are normal. Thus, our results f normal zeros cover almost all the physically important integrable PDEs. Remark 2:Normal zeros include the elementary zeros of Ref. 22 as special cases, but th nonelementary in general. It is an important fact~see Ref. 22, Lemma 2! that the sequence of ranks of the projectorsPl in the matrixG(k) given by Eq.~27!, i.e., built in the described way, is nonincreasing: hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 186.217.234.225 On: Tue, 14 Jan 2014 15:18:36 her- - nce of r-order ns f a umber eros, r of ith ranks, s ur 4611J. Math. Phys., Vol. 44, No. 10, October 2003 General soliton matrices This article is copyrig rankPns2>¯>sr 1 is defined as follows: sn[the index of the last positive integer in the array@r 1112n,r 2112n, . . . ,r n112n#. The sequence of integers$r n ,r n21 , . . . ,r 1% is then the rank sequence associated with the pai zeros(k1 ,k̄1) and the new sequence$s1 ,s2 , . . . ,sr 1 % is called the block sequence associated w this pair of zeros. Remark:It is easy to see that the sum of the block sequence is equal to the sum of all ( n51 r 1 sn5( l 51 n r l , with the sum being equal to the algebraic order of the Riemann–Hilbert zeros (k1 ,k̄1). For example, if the rank sequence is$3% @only one constituent matrix in~27!—trivial higher- order zero#, then the block sequence is$1,1,1%; if the rank sequence is$1,1,1,1% ~an elementary zero!, then the block sequence is$4%; if the rank sequence is$2,3,5,7%, then the block sequence i $4,4,3,2,2,1,1%. With these definitions the most general soliton matricesG(k) andG21(k) for a single pair of higher-order normal zeros (k1 ,k̄1) are given as follows. This result is a generalization of o previous result22 to nonelementary higher-order zeros. hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 186.217.234.225 On: Tue, 14 Jan 2014 15:18:36 s n ula 4612 J. Math. Phys., Vol. 44, No. 10, October 2003 V. S. Shchesnovich and J. Yang This article is copyrig Lemma 1: Consider a single pair of higher-order normal zeros(k1 ,k̄1) in the Riemann– Hilbert problem. Suppose their geometric multiplicity is r1 , and their block sequence i $s1 ,s2 , . . . ,sr 1 %. Then the soliton matricesG(k) and G21(k) can be written in the following summation forms: G~k!5I 1 ( n51 r 1 S̄n , G21~k!5I 1 ( n51 r 1 Sn . ~29! Here Sn and S̄n are the following block matrices, S̄n5( l 51 sn ( j 51 l uq̄ j (n)&^ p̄l 112 j (n) u ~k2 k̄1!sn112 l 5~ uq̄sn (n)&,...,uq̄1 (n)&D̄n~k!S ^ p̄1 (n)u ] ^ p̄sn (n)u D , ~30a! Sn5( l 51 sn ( j 51 l upl 112 j (n) &^qj (n)u ~k2k1!sn112 l 5~ up1 (n)&,...,upsn (n)&Dn~k!S ^qsn (n)u ] ^q1 (n)u D , ~30b! where Dn(k) and D̄n(k) are triangular Toeplitz matrices of the size sn3sn : D̄n~k!51 1 ~k2 k̄1! 0 ... 0 1 ~k2 k̄1!2 1 ~k2 k̄1! � ] ] � � 0 1 ~k2 k̄1!sn ... 1 ~k2 k̄1!2 1 ~k2 k̄1! 2 , ~31! Dn~k!5S 1 ~k2k1! 1 ~k2k1!2 ... 1 ~k2k1!sn 0 � � ] ] � 1 ~k2k1! 1 ~k2k1!2 0 ... 0 1 ~k2k1! D . The vectorsupi (n)&,^ p̄i (n)u,^qi (n)u,uq̄i (n)& ( i 51, . . . ,sn) are independent of k, and in the two sets $up1 (1)&, . . . ,up1 (r 1) &% and $^ p̄1 (1)u, . . . ,̂ p̄1 (r 1)u% the vectors are linearly independent. Remark 1:If r 151, the zerosk1 and k̄1 are elementary.22 In this case, the above solito matrices reduce to those in Ref. 22. Remark 2:The total number of allup& vectors or̂ p̄u vectors from alln blocks are equal to the algebraic order of the zerosk1 and k̄1 . Proof: The representation~29! can be proved by induction. Consider, for instance, the form for G(k). Obviously, this formula is valid forn51 in Eq.~27!, whereG(k) contains only a single hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 186.217.234.225 On: Tue, 14 Jan 2014 15:18:36 4613J. Math. Phys., Vol. 44, No. 10, October 2003 General soliton matrices This article is copyrig matrix x1(k). Now, suppose that this formula is valid forn.1. We need to show that it is valid for n11 as well. Indeed, denote the soliton matrices forn andn11 by G(k), andG̃(k), respec- tively, the rightmost multiplier inG̃(k) being x̃(k). Then we have G̃~k!5G~k!x̃~k!5S I 1 A1 k2 k̄1 1 A2 ~k2 k̄1!2 1¯1 An ~k2 k̄1!nD S I 1 R k2 k̄1 D 5I 1 Ã1 k2 k̄1 1 Ã2 ~k2 k̄1!2 1¯1 Ãn11 ~k2 k̄1!n11 , ~32! where R[~ k̄12k1!P̃5( l 51 r̃ uul&^ūl u. ~33! Here we have normalized the vectorsuul& and ^ūl u such that ^ūl uui&5~ k̄12k1!d l ,i , ~34! andr̃ 5rankR. In view of Eq.~28!, we know thatr̃>r 1 , wherer 1 is the geometric multiplicity of k1 andk̄1 in the soliton matricesG(k) andG21(k). The coefficients at the poles inG̃(k) are given by Ã15A11R, Ãj5Aj1Aj 21R, j 52, . . . ,n, Ãn115AnR. ~35! Consider first the coefficientsÃ2 to Ãn11 . The explicit form of the coefficientsAj can be obtained from Eqs.~29!, ~30!, and~32! as Aj[ ( n51 r 1 Aj (n)5 ( n51 r 1 ( l 51 sn112 j uq̄l (n)&^ p̄sn122 j 2 l (n) u, ~36! where the inner sum is zero ifsn112 j <0. Substituting this expression into~35! and defining the following new vectors in each block: ^p! 1 (n)u5^ p̄1 (n)uR, ^p! j (n)u5^ p̄ j (n)uR1^ p̄ j 21 (n) u, j 52, . . . ,sn , ~37! @for blocks of size 1,sn51, the second formula in~37! is dropped#, we then put the coefficients Ã2 ,...,Ãn11 into the required form Ãj5 ( n51 r 1 ( l 51 s̃n112 j uq! l (n)&^p! s̃n122 j 2 l (n) u, j 52, . . . ,n11, where uq! l (n)&[uq̄l (n)&, l 51, . . . ,s̃n21, ands̃n5sn11, i.e., the size of eachn-block grows by one as we multiply byx̃(k) in formula~32!. Next, we consider the coefficientÃ1 . Defining the vector̂ p! s̃n (n)u[^ p̄sn (n)u and utilizing the definition ~37!, we can rewriteA1 (n) as hted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 186.217.234.225 On: Tue, 14 Jan 2014 15:18:36 s, at , of 4614 J. Math. Phys., Vol. 44, No. 10, October 2003 V. S. Shchesnovich and J. Yang This article is copyrig A1 (n)5 ( l 51 s̃n21 uq̄l (n)&^p! s̃n112 l (n) u2( l 52 sn uq̄l (n)&^ p̄sn122 l (n) uR. ~38! To setÃ15A11R into the required form Ã15 ( m5r 111 r̃ uq! 1 (m)&^p! 1 (m)u1 ( n51 r 1 ( l 51 s̃n uq! l (n)&^p! s̃n112 l (n) u, ~39! we must define exactly one new vectoruq! s̃n (n)& for eachn-block @in the second term of Eq.~39!# and r̃ 2r 1 new blocks of size 1 containing 2(r̃ 2r 1) new vectorsuq̄1 (m)& and ^p! 1 (m)u. Due to formulas ~35! and ~38!, the new vectors to be defined must satisfy the following equation: ( m5r 111 r̃ uq! 1 (m)&^p! 1 (m)u1 ( n51 r 1 uq! s̃n (n)&^ p̄1 (n)uR5R2 ( n51 r 1 ( l 52 sn uq̄l (n)&^ p̄sn122 l (n) uR, ~40! where the definition of̂p! 1 (n)u in Eq. ~37! has been utilized. Substituting the expression~33! for R into the above equation, we get ( m5r 111 r̃ uq! 1 (m)&^p! 1 (m)u5( l 51 r̃ uj l&^ūl u, ~41! where uj l&[S I 2 ( n51 r 1 ( l 52 sn uq̄l (n)&^ p̄sn122 l (n) u D uul&2 ( n51 r 1 uq! s̃n (n)&^ p̄1 (n)uul&, l 51, . . . ,r̃ . To show that Eq.~41! is solvable, we need to use an important fact that the matrix M5~Mn,l !, Mn,l5^ p̄1 (n)uul&, n51, . . . ,r 1 , l 51, . . . ,r̃ 1 , has rankr 1 . This fact can be proved by contradiction as follows. Suppose the matrixM has rank less thanr 1 . Then itsr 1 rows are linearly dependent. Thu there are such scalarsC1 ,C2 , . . . ,Cr 1 , not equal to zero simultaneously, that the vector ^hu[ ( n51 r 1 Cn^ p̄1 (n)u is orthogonal to alluul& ’s, i.e., ^huul&50, 1< l< r̃ . ~42! According to our induction assumption that soliton matrices involvingn multipliers in formula ~27! have the form~29!, we can easily show, by equating the coefficient at the highest polek 5 k̄1 in the left-hand side~lhs! of the identityG(k)G21(k)5I to zero, that̂ p̄1 (n)uG21( k̄1)50 for all 1m, ~93a! Bn (n,m)51 c1 c2 . . . cs m (n)21 cs m (n) 0 c1 c2 � cs m (n)21 ] 0 � � ] ] � � � c2 ] � � 0 c1 ] � � � 0 ] � � � ] 0 . . . . . . . . . 0 2 , ns2 (n)> . . . >sr 1 (n) (n) is the block sequence of zeros (kn ,k̄n) as in Lemma 2~see Definition 2!, andbj ,cj are arbitrary complex constants which are generally different in different submat Bn (n,m) . The invariance matrixB̄ has the form ofBT ~in general, with arbitrary elements unrelate to those ofB). The above forms~92! and~93! of the invariance matricesBn andB̄n follow immediately from the following argument. Consider, for instance, the matrixBn . The commutability relation with the part of the matrix D(k) corresponding to thenth pair of zeros, i.e.,D (n)(k) 5diag@D1 (n)(k), . . . ,Dr 1 (n) (n) (k)# where matricesDn (n)(k) are given by Eq.~48!, produces the following set of independent matrix equations: Dn (n)~k!Bn (n,m)5Bn (n,m)Dm (n)~k!, n,m51, . . . ,r 1 (n) . ~94! For n5m, the above equations are equivalent to the commutability conditions for the s elementary higher-order zero considered in Ref. 22, thus the form~93! for the diagonal blocks Bn (n,n) follows accordingly. Consider now the case whenn.m ~the other case can be consider similarly!. We have thensn (n)