第六屆非線(xiàn)性數(shù)學(xué)物理國(guó)際會(huì)議暨全國(guó)第十三屆可積系統(tǒng)學(xué)術(shù)研討會(huì)1generalized perturbation nm-foldand multi rogue wave structures for some nonlinear systems

1、1Generalized perturbation (n;M)-fold Darboux transformations and multi-rogue wave structures for some nonlinear wave systems Xiaoyong WenDepartment of Mathematics, School of Applied Science, Beijing Information Science and Technology University (北京信息科技大學(xué) 理學(xué)院 數(shù)學(xué)系) Institute of Systems Science, AMSS,

2、Chinese Academy of Sciences(中國(guó)科學(xué)院 數(shù)學(xué)與系統(tǒng)科學(xué)研究院 系統(tǒng)所)(Work done together with Zhenya Yan, Yunqing Yang)X.Y. Wen, Y.Q. Yang, Z. Y. Yan, Phys. Rev. E 92 (2015) 0129172Outline Introduction 2. Generalized perturbation (n, M)-fold Darboux transformationsModified self-steepening NLS equationGerjikov-Ivanov (G

3、I) equation Conclusions and questions3 1. Introduction Rogue waves are always isolated huge waves with two to three times amplitude higher than its surrounding waves in the ocean. Rogue waves are also known as killer waves, huge waves, frank waves, giant waves and so on. Waves that appear from nowhe

4、re and disappear without a trace. Rogue waves ware firstly proposed in the study of ocean waves and gradually extended to other fields, such as Bose- Einstein condensates PRA,2009, optics fiberKibler et al. 2010; Solli et al. Nature,2007, Matter Bludov et al. PRA , 2009, finance Yan, PLA, 2011, etc.

5、4 1. Introduction The simplest rogue wave in this equation was given by Peregrine (1983). Mathematically the simplest model for describing rogue waves is the NLS equation:5 1. Introduction Mathematically rogue wave is the rational form solution for some nonlinear wave equations. Its remarkable chara

6、cter is localized in both space and time.6Main methods and techniques used: Darboux transformation Bilinear methods the focusing and defocusing AblowitzLadik equations (Ohta & Yang, 2014),the Davey-Stewartson equations (Ohta & Yang, 2012, 2013) KP-I，2-d Toda equation（Hu etc, 2015 ） Melnikov equation

7、 (Mu, and Qin,2014) , etc. the focusing NLS equation via generalized Darboux transformation (Guo, Ling, Liu, PRE 2012) (DNLS, Liu Stud. Appl. Math. 2012) (also Chen yong and Wang xin, Zhaqilao, Zhang haiqiang, etc) the determinant representation of the N-fold DT with iteration DNLSI (He,Xu,JPA 2011)

8、 DNSIII (He,Xu, JMP 2012) Hirota eqation (Akhmediev, He, Xu, PRA 2010), etc. 1. Introduction7 Wronskian determinant the focusing NLS equation and the KP-I equation (Dubard and Matveev Nat. Hazards Earth Syst. Sci., 2011) 1. Introduction Similarity transformation Gross-Pitaevskii (Yan, Dai et al. 201

9、3).8In fact, the Darboux transformation (DT) usually exhibits three different main types of DTs in integrable nonlinear wave systems: Darboux transformation without the Darboux matrix (Akhmediev,1988; Guo, Ling, Liu, 2012 ); Darboux transformation with the iteration Darboux matrix (Matveev,1991;Guo,

10、 Ling, Liu, 2012; He,et al. 2011,2012 ) Darboux transformation with the N-order Darboux matrix whose elements being the polynomials of the spectral parameters (Gu,1995,2005; Fan 2004) 1. IntroductionNext, we will present an approach to construct the generalized perturbation (n,M)-fold DT of integrab

11、le nonlinear wave equations based on Type (iii) DT.9 1. Introduction A procedure to construct generalized perturbation (n,M)-fold DT: Step 1: constructing N-fold DTStep 2: constructing generalized perturbation (1,N-1)-fold DT Step 3: constructing generalized perturbation (n,M)-fold DTFor examples: M

12、odified self-steepening NLS equation and GI equation10 2. Generalized perturbation (n, M)-fold Darboux transformationModified self-steepening nonlinear Schrdinger equationwhere q q(x, t) is the slowly varying complex envelope of the wave, is a real constant and i is the imaginary number unit,(1)when

13、= 0, Eq.(1) reduces to the derivative NLS equation.is called the self-steepening term,(2)11 2. Generalized perturbation (n, M)-fold Darboux transformationthe linear iso-spectral problem (Lax pair) for Eq. (1)The compatibility condition between (3) and (4)gives rise to Eq. (1).(3)(4)12 2. Generalized

14、 perturbation (n, M)-fold Darboux transformationIn the following, we first proceed to establish the N-fold DT of Eq. (1) before we present a generalized perturbation (n, M)-fold DT. We introduce the following gauge transformation(5)under the transformation (5) , the Lax pair (3) and (4) can be trans

15、formed to the following same form Lax pair (6) and (7) :1.1 N-fold Darboux transformation13 2. Generalized perturbation (n, M)-fold Darboux transformationIn transformation (5), satisfies the Lax pair (3) and (4),(6)(7) is required to satisfy the Lax pair (6) and (7),T is a 22 Darboux matrix to be de

16、termined later.14 2. Generalized perturbation (n, M)-fold Darboux transformationAccording to the compatibility condition due to Eqs. (6) and (7) , we have(8)Hereby, we assume that the Darboux matrix T15 2. Generalized perturbation (n, M)-fold Darboux transformationthe complex functions and (j = 0, 1

17、, .,N 1) in T can be solved by the linear algebraic system , i.e.,(9)where (k = 1, 2, .,N) is a solution of the spectral problem (2) and (3) for the given spectral parameter and the initialsolution . (k = 1, 2,.,N) are the 4N roots of16 2. Generalized perturbation (n, M)-fold Darboux transformationS

18、ubstituting the Darboux matrix T into Eqs. (7) with (8) yields the following Theorem 1 for the N-fold DT of Eq. (1):(10)17 2. Generalized perturbation (n, M)-fold Darboux transformation18 2. Generalized perturbation (n, M)-fold Darboux transformationThe N-fold Darboux transformation with the initial

19、 solution (or is an initial plane wave solution) can be used to seek for multi-soliton solutions (or breather solutions) of Eq. (1). This is not our main aim. Our aim is to extend the N-fold DT to generate the generalized perturbation (n,M)-fold DT such that multi-rogue wave solutions of Eq. (1) are

20、 found in terms of determinants. Next, we first consider Generalized perturbation (1;N -1)-fold DT, then extend it to generate the generalized perturbation (n,M)-fold DT. 19 2. Generalized perturbation (n, M)-fold Darboux transformation1.2 Generalized perturbation (1;N -1)-fold DTHere we still consi

21、der the Darboux matrix T, but we only consider one spectral parameter not N distinct spectral parameters (k = 1, 2, .,N), in which the condition leads to the linear algebraic system including only two equations:(11)20 2. Generalized perturbation (n, M)-fold Darboux transformationFor (10) containing

22、2N unknown functions and (j =0, 1, .,N 1), we have two cases for parameter N: If N = 1, then we can determine only two complex functions and from Eqs. (10), in which we can not obtain the different functions and comparing from the above-mentioned 1-fold DT such that the new solutions can not be foun

23、d; If N 1, then we have 2(N 1) 0 free functions for and (j = 0, 1, .,N 1). This means that the number of the unknown variables and is larger than one of equations such that we have some free functions, which seems to be useful for the Darboux matrix T, but it may be difficult to show the invariant c

24、onditions (8).21 2. Generalized perturbation (n, M)-fold Darboux transformationTo generate new additional 2(N 1) equations from , we consider the Taylor expansionWe know thatand22 2. Generalized perturbation (n, M)-fold Darboux transformationwith . So we have 23 2. Generalized perturbation (n, M)-fo

25、ld Darboux transformationLet We choose s = 0, 1, ., N 1 to generate 2N algebraic equations(12)24Therefore we have obtained system (12) containing 2N algebraic equations with the 2N unknowns functions and (j = 0, 1, 2, .,N 1). When the eigenvalue is suitably chosen so that the determinant of the coef

26、ficients for system (11) is non-zero, hence the Darboux matrix T is uniquely determined by system (12). It can be shown that Theorem 1 still holds for the Darboux matrix T with ( j = 0, 1, .,N 1) being determined by system (12). Owing to new distinct functions obtained in the N-order Darboux matrix

27、T, so we can derive the new DT with the same eigenvalue . Here we call Eqs. (5) and (10) associated with new functions determined by system (12) as a generalized perturbation (1,N 1)-fold DT. 2. Generalized perturbation (n, M)-fold Darboux transformation25Notice that in the name of the generalized p

28、erturbation (1,N1)-fold DT, the number 1 means that we use the number of the distinct spectral parameters and N1 means that the order of the highest derivative of the vector eigenfunction .So we have the following Theorem 2 for the (1,N 1)-fold DT of Eq.(1): 2. Generalized perturbation (n, M)-fold D

29、arboux transformation26 2. Generalized perturbation (n, M)-fold Darboux transformation27 2. Generalized perturbation (n, M)-fold Darboux transformation28 2. Generalized perturbation (n, M)-fold Darboux transformation1.3 Generalized perturbation (n;M)-fold DTIn fact, we also further extend the above-

30、found generalized perturbation (1,N 1)-fold DT, in which we only use one spectral parameter and its th-order perturbation derivatives of and with . Here we further extend to use n distinct spectral parameters (i = 1, 2, ., n) and their corresponding highest order ( ) perturbation derivatives, where

31、these non-negative integers n, are required to satisfy with , where N is the same as one in the Darboux matrix T.29We consider the Darboux matrix T and the eigenfunctions (i = 1,2, . . . ,n) are the solutions of the linear spectral problem (3) and (4) for the spectral parameter and initial solution

32、of Eq. (1). Thus we haveLetwith i = 1,2,.,n and , we may obtain the following linear algebraic system with the 2N equations: 2. Generalized perturbation (n, M)-fold Darboux transformation30 2. Generalized perturbation (n, M)-fold Darboux transformation(13)where i = 1,2,.,n.31Therefore we have obtain

33、ed system (12) containing 2N algebraic equations with the 2N unknowns functions and (j = 0, 1, 2, .,N 1). When the eigenvalues are suitably chosen so that the determinant of the coefficients for system (12) is non-zero, hence the Darboux matrix T is uniquely determined by system (12).It can be shown

34、 that Theorem 1 still holds for the Darboux matrix T with ( j = 0, 1, .,N 1) being determined by system (12). Owing to new distinct functions obtained in the N-order Darboux matrix T, so we can derive the new DT with n different eigenvalue (i =1,2, .,n) . Here we call Eqs. (5) and (10) associated wi

35、th new functions determined by system (12) as a generalized perturbation (n, M)-fold DT. 2. Generalized perturbation (n, M)-fold Darboux transformation32Notice that in the name of the generalized perturbation (n, M)-fold DT, the number n means that we use the number of the distinct spectral paramete

36、rs and means that the order of the highest derivative of the vector eigenfunction .So we have the following Theorem 3 for the (n, M)-fold DT of Eq.(1): 2. Generalized perturbation (n, M)-fold Darboux transformation33 2. Generalized perturbation (n, M)-fold Darboux transformation34 2. Generalized per

37、turbation (n, M)-fold Darboux transformation35Remark: Notice that when n = 1 and , Theorem 3 reduces to Theorem 2; when n = N and , Theorem 3 reduces to Theorem 1. 2. Generalized perturbation (n, M)-fold Darboux transformation361.4 Multi-rogue wave solutions and parameters controlling 2. Generalized

38、 perturbation (n, M)-fold Darboux transformationNext, we consider the seed plane wave solutionThe solutions of Lax pair (3) and (4) are In the following, we give some multi-rogue wave solutionsvia generalized perturbation (1,N 1)-fold DT.(14)37where (k = 1, 2 ,., N) are real free parameters and is a

39、 small parameter. 2. Generalized perturbation (n, M)-fold Darboux transformation38 2. Generalized perturbation (n, M)-fold Darboux transformationthen we expand (14) as two Taylor series at = 0. Here we choose , i.e., to simplify the calculation.We fix the spectral parameter withTherefore, We have39C

40、ase 1: when N=1, it is particularly worth pointing out that we re-derive the seed solutionIn the following, We consider the four cases: N =1,2,3,4. 2. Generalized perturbation (n, M)-fold Darboux transformationwhere (i= 2, 3) are omitted since they are complicated.40Case 2: N=2 2. Generalized pertur

41、bation (n, M)-fold Darboux transformation41 For parameters a = 1,c = 1, = 2: For parameters a = 1,c = 1, = 0: 2. Generalized perturbation (n, M)-fold Darboux transformation42 2. Generalized perturbation (n, M)-fold Darboux transformation43Case 3: N=3 2. Generalized perturbation (n, M)-fold Darboux t

42、ransformation44 2. Generalized perturbation (n, M)-fold Darboux transformation45 2. Generalized perturbation (n, M)-fold Darboux transformation46 2. Generalized perturbation (n, M)-fold Darboux transformation47Case 4: N=4 2. Generalized perturbation (n, M)-fold Darboux transformation48 2. Generalize

43、d perturbation (n, M)-fold Darboux transformation49 2. Generalized perturbation (n, M)-fold Darboux transformation50 2. Generalized perturbation (n, M)-fold Darboux transformation51 2. Generalized perturbation (n, M)-fold Darboux transformation521.5 Dynamical behaviors of multi-rogue wave solutions

44、2. Generalized perturbation (n, M)-fold Darboux transformationCase 1: First-order rogue wave 53 2. Generalized perturbation (n, M)-fold Darboux transformation54Case 2: Second-order rogue wave 2. Generalized perturbation (n, M)-fold Darboux transformation55 2. Generalized perturbation (n, M)-fold Dar

45、boux transformation56 2. Generalized perturbation (n, M)-fold Darboux transformation57Case 3: Third-order rogue wave 2. Generalized perturbation (n, M)-fold Darboux transformation58 2. Generalized perturbation (n, M)-fold Darboux transformation59 2. Generalized perturbation (n, M)-fold Darboux trans

46、formation60 2. Generalized perturbation (n, M)-fold Darboux transformation2. Gerjikov-Ivanov equationits Lax pair(14)(15)61 2. Generalized perturbation (n, M)-fold Darboux transformationRemark: The above Darboux matrix T is also applied to Chen-Li-Liu equation, Kundu equation, DNLS equation, etc.62

47、2. Generalized perturbation (n, M)-fold Darboux transformation63 2. Generalized perturbation (n, M)-fold Darboux transformation64 3. Conclusions and questions We have presented a generalized perturbation (n;M)-fold DT. Some Lax integrable nonlinear wave systems such as the NLS equation, KPI equation

48、, mKdV equation, nonlocal NLS equation, AB system, the NLS-type equation, KP-type equation and AKNS hierarchy can be solved by means of generalized perturbation (n;M)-fold DTs and multi-rogue wave solutions can be given in terms of fractional forms of determinants.2. We derive multi-rogue wave solut

49、ions via the generalized perturbation (1;N-1)-fold DT with one spectral parameter. However, if we choose two or more different spectral parameters, for example, if we choose two different spectral parameters and for the MNLS equation, then the higher-order rogue waves can be degraded to lower-order

50、rogue waves. It is still a problem on how to choose several different spectral parameters?653. The generalized perturbation (n;M)-fold DT can be extended to the discrete integrable nonlinear wave equations such as discrete NLS equation or the AblowitzLadik equations, but we dont derive discrete rogue