Ablowitz M J and Clarkson P A, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... the three wave equations (see David and Holm [5] for details) Some general references to literature on the integrable three wave equations is found in Whitham [6] Ablowitz and Haberman [7] Kaup [8, 9, 10] Zakharov and Manakov [11] Ablowitz and Segur [2] Newell [12] and Ablowitz and Clarkson [13]. The integrable Hamiltonian structure of the three wave equations is of course well known; we explore it from a somewhat novel point of view in what follows. As we will show, these equations possess a Lie Poisson structure in addition to the canonical Hamiltonian structure. One of the three wave ....

....Using n 1 independent symmetries the n wave system is ultimately reduced to quadratures. Solutions of the three wave system analyzed here are also traveling wave or stationary solutions of an integrable partial di#erential equation (for solution of the partial di#erential equation see Refs. [1, 7, 8, 9, 10, 11, 12, 13] ) In this sense the integrable structure outlined above generalizes to the structure of the partial di#erential equation. More generally, each integrable system of ordinary di#erential equations is associated with a hierarchy of evolution equations through (6.5) 6.6) by letting # # #x, ....

M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, (Cambridge University Press, Cambridge, 1991).

....coincide, i.e. # = # = # , we have dx #. Thus # is an eigenvalue of (5) in L (R) The eigenfunction # with the normalization condition: 11) dx = 1, is called a bound state. If it is possible to prove that there is only a finite number of negative eigenvalues #n (see [AC91]) Define (k n = # #n ) For each kn , there exists a constant c n such that the eigenfunction #n (x) c n e knx as x # and satisfies (11) 3.2. Continuous spectra. If # 0 the solutions of (5) don t vanish at but we can prove a result similar to proposition 2: #. Then, for each ....

M. Ablowitz and P. Clarkson. Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, 1991.

....Index Terms Estimation, modulation, multiplexing, nonlinear circuits, signal detection, solitons. I. INTRODUCTION S OLITONS are stable, mode like solutions to a special class of nonlinear wave equations that can be solved analytically using a technique known as inverse scattering [1]. The inverse scattering transform can be interpreted as a nonlinear Fourier analysis for these systems, which decompose wave dynamics into a superposition of normal modes. These normal modes are solitons, and their particle like properties have been observed in a variety of natural phenomena ....

....the individual solitary waves approach one another, they begin to interact nonlinearly. However, after passing through one another, they regain their shape and speed with only a slight positional shift [17] There are many physical systems that support soliton solutions in a wide range of media [1], 3] 5] 9] These can be distributed systems with dynamics described by partial differential equations and whose solitons propagate through a Fig. 2. Two solitary wave solutions to the Toda lattice. bulk medium such as water, optical fiber, or plasma. They can also be lumped or cascade ....

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M. Ablowitz and A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering. Cambridge, U.K.: Cambridge Univ. Press, 1991.

....with a ot, e third power law for self focusing collapse In order to estimate the size of the neighborhood of t0 where collapse is arrested a more careful analysis is required. Let B = iA so that B, B 2B a (38) The behavior of the solutions of (38) is characterized by the following result [1, 12]. Ay olutio of (38) atifyig lira B( 0 i aymptotic to kAi( for omc k. If Ikl 1 then a To apply this result we express k in terms of the parameters of the problem and note that A should agree with (34) in the domain ( fi0 6, when it is given by (32) Thus, k lA Ai(o) o B kAi( ....

M. Ablowitz and P. Clarkson. Solirons, Nonlinear Evolution Equations and Inverse Scatter- ing. Cambridge University Press, 1991.

....parts of the original dependent variable. The last and most complicated example involves a system of vector equations that needs to be split into equations for its scalar components in order to compute its Lie symmetries. 25 13.5.1. THE HARRY DYM EQUATION Consider the Harry Dym equation [1], u t u u xxx = 0: 13.17) Clearly, this is one equation with two independent variables and one dependent variable. The assignments of the variables are as follows: x 7 x[1] t 7 x[2] u 7 u[1] 13.18) This permits us to rewrite the equation (13.17) in a form accepted by the program ....

....components in order to compute its Lie symmetries. 25 13.5.1. THE HARRY DYM EQUATION Consider the Harry Dym equation [1] u t u u xxx = 0: 13.17) Clearly, this is one equation with two independent variables and one dependent variable. The assignments of the variables are as follows: x 7 x[1] ; t 7 x[2] u 7 u[1] 13.18) This permits us to rewrite the equation (13.17) in a form accepted by the program SYMMGRP. MAX; i.e. e1 : u[1, 0,1] u[1] 3 u[1, 3,0] For PDELIE and SYM DE the input form would be DIFF(U,T) U 3 DIFF(U,X,3) For SPDE and LIE the program accepts ....

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M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Notes on Mathematics 149 (Cambridge University Press, Cambridge, UK, 1991).

....to write the system in terms of the invariants of the group action and to perform a Gr obner basis type calculation on the invariantised system. The following simple example from the classical literature illustrates the idea in essence. The Halphen Darboux system of ordinary di erential equations ([1], p. 336) dw 1 dt = w 2 w 3 w 1 (w 2 w 3 ) dw 2 dt = w 1 w 3 w 2 (w 1 w 3 ) dw 3 dt = w 1 w 2 w 3 (w 1 w 2 ) 3) is invariant under permutations of the w i . One generating set of invariants of the permutation group is s 1 = w 1 w 2 w 3 s 2 = w 1 w 2 w 2 w 3 w 1 w 3 s 3 ....

.... In[u ,K] while the operator D j is calculated using the function f 7 Idiff(f,j) restart:with(Indiff) HNI ; IKolRitt ; IdSpoly ; Idi ; Idi parse ; Invariantize ; Iorthreduceall ; Ireduce; Ireduceall ; Kmat ] vars: x] ukns: u] GroupP: a,b,alpha,beta,delta] XiPhis: matrix( [1,0], 0,1] x, u(x) u(x) 0] 0,x] XiPhis : 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 1 0 0 1 x u(x) u(x) 0 0 x 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 The normalisation equations, Neqs, are the invariantised moving frame equations. The function HNI calculates the set ....

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M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering L.M.S. Lect. Notes Math., vol. 149 (C.U.P., Cambridge, 1991).

....freedom to identify the four points with 0, 1, #, t, where t is the cross ratio. However, as remarked by Yoshida in his wonderful book [61] it is not fair that only the fourth point is allowed to move freely . The more democratic version of the cross ratio used there is simply the point [ 1 , 2 , 3 ] on the line 1 2 3 = 0 in P 2 , with j invariant ( 2 1 2 2 2 3 ) 3 2 1 2 2 2 3 = 8x 3 c 2 . Note also that under the matrix Riccati equation, all four of the points move (relative to a D parallel trivialization) as does the democratic cross ratio, since x is not constant. I ....

.... to find r as a function of a projective coordinate t: up to projective transformation of t, it is the Schwarzian triangle function S(0, # 2, # 3; t) which is a modular function for SL(2, Z) This leads to a nice formula for a in terms of (not surprisingly) the discriminant modular form see [1] a(t) #(t) 6#(t) 9.2) I summarize this discussion by explaining how the solution of the matrix Riccati equation is related to these ideas. First note that the map sending a point of C to [# 1 , # 2 , # 3 ] is a local di#eomorphism, and so C can be identified locally with the ....

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M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge (1991).

.... to f(t) t = v t Gamma v xx Gamma v 2 x for some function f = f(t) Renaming v Gamma f u gives (14) and its conservation laws (15) To give a further example, we consider the Boussinesq equation describing surface water waves whose horizontal scale is much larger than the depth of the water [1], 14] u tt Gamma u xx 3uu xx 3u 2 x ffu xxxx = 0: 22) 6 Although already used in [2] 35] this example is shown again as it also serves to demonstrate an extension to non local conservation laws in section 5. 13 Calculating conservation laws, using (22) to substitute u xxxx , the ....

Ablowitz, M.J., Clarkson, P.A. (1991) Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Math. Soc. Lec. Note Ser. 149. London: Cambridge University Press.

....space M c (n) together. 2) To solve this system of n variables, it suffices to solve the first flow equations of two variables. The literature in soliton theory is enormous, and we will only refer here to papers we use directly. There are many excellent survey articles and books; for example [2], 9] 28] 16] 29] where the reader can find more complete bibliographies. The relations among Backlund transformations, Poisson loop group actions, and the inverse scattering for the j th flow equation (j = Gamma1; 1; 2; will be studied in forthcoming joint papers with K. Uhlenbeck. ....

M. J. Ablowitz & P. A. Clarkson, Solitons, non-linear evolution equations and inverse scattering, Cambridge University Press, Cambridge, 1991.

....non linear partial di erential equations by Gardner, Green Kruskal and Miura and its connection with the singularity theory: Painlev e property, isomodromy deformations, etc. An idea of the amount of papers published in this eld is given by the 52 pages of references quoted in the monograph [1]. Although along of this paper essentially there are no new results, in section 4 I state a new result. It is a generalization of the so called Morales Ramis theorem about the non integrability of an analytical Hamiltonian system by means of the variational equations. This was conjectured by the ....

.... property From the H enon Heiles system studied above we observe the similarity between the variational method to prove non integrability used here and the Kovalevskaya Painlev e test to detect integrable systems : the existence of a logarithm is, in general, an obstruction to integrability, see [19, 65, 1] and references therein. Motivated by that, in this section I try to make some remarks about this connection and also to make some ideas on the meaning of the Painlev e property for the study of integrable systems. For this we need to precise the concept of integrability for complex analytical ....

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M.J.Ablowitz, P.A.Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991.

.... 1 2 ] 2 ; 72) where c is an arbitrary constant, and with = 1 p 6 x 5 6 t : Note that (72) does not follow from (70) in the limit for k 0: A solution similar to (72) was rst obtained by Ablowitz and Zeppetella [3] 5 Truncated Painlev e Expansion A PDE passes the Painlev e test [1, 9] if its solution u(x; t) expressed as a Laurent series in the complex plane, u(x; t) g (x; t) 1 X k=0 u k (x; t)g k (x; t) 73) has no worse singularities than movable poles. First, the negative integer and the coecient u 0 (x; t) are computed by substituting (73) into the given ....

M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge: Cambridge University Press, 1991.

....is given in Section 3. To illustrate the method we use the Toda lattice [19] a parameterized Toda lattice [8] and the discretized nonlinear Schr odinger (NLS) equation [20 22] In Section 4, we list results for an extended Lotka Volterra equation [23] a discretized modi ed KdV equation [24], a network equation [24] and some other lattices [2] The features, scope and limitations of our code di dens.m are described in Section 5, together with instructions for the user. In Section 6, we draw some conclusions. 2 De nitions 2.1 Conservation laws Consider a system of DDEs which is ....

....3. To illustrate the method we use the Toda lattice [19] a parameterized Toda lattice [8] and the discretized nonlinear Schr odinger (NLS) equation [20 22] In Section 4, we list results for an extended Lotka Volterra equation [23] a discretized modi ed KdV equation [24] a network equation [24], and some other lattices [2] The features, scope and limitations of our code di dens.m are described in Section 5, together with instructions for the user. In Section 6, we draw some conclusions. 2 De nitions 2.1 Conservation laws Consider a system of DDEs which is continuous in time, and ....

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M.J. Ablowitz and P.A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering (Cambridge University Press, Cambridge, U.K., 1991).

....in the user guides [7, 8] Using the algorithmic language REFAL, Topunov [92] developed a software package for symmetry analysis that contains subroutines to reduce determining systems in passive form. 12 W. Hereman 4 Examples 4. 1 The Dym Kruskal equation Consider the Dym Kruskal equation [1], u t u 3 u xxx = 0: 5) Clearly, this is one equation with two independent variables and one dependent variable. The assignments of the variables are as follows: x 7 x[1] t 7 x[2] u 7 u[1] 6) Then, symmetry software such as SPDE, LIE, PDELIE, DIMSYM, SYM DE or SYMMGRP.MAX, ....

....systems in passive form. 12 W. Hereman 4 Examples 4.1 The Dym Kruskal equation Consider the Dym Kruskal equation [1] u t u 3 u xxx = 0: 5) Clearly, this is one equation with two independent variables and one dependent variable. The assignments of the variables are as follows: x 7 x[1] ; t 7 x[2] u 7 u[1] 6) Then, symmetry software such as SPDE, LIE, PDELIE, DIMSYM, SYM DE or SYMMGRP.MAX, automatically compute the determining equations for the coecients eta[1] x ; eta[2] t , and phi[1] u of the vector eld = x x t t u u : ....

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M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Notes on Mathematics 149, Cambridge University Press, Cambridge, UK (1991).

....of the KdV equation Consider the integrable discretization of the KdV equation: u n = u n (u n 1 u n 1 ) 10) which is known as the Kac Van Moerbeke equation or a special form of the Volterra system. It arises in the study of Langmuir oscillations in plasmas, and in population dynamics [16 18]. Notice that (10) is invariant under the scaling symmetry (t; u n ) t; 1 u n ) Hence, u n corresponds to one derivative with respect to t; i.e. u n d dt : All terms in (10) have the same rank if w(u n ) 1 = 2 w(u n ) thus, w(u n ) 1; which agrees with the scaling symmetry. Let ....

M.J. Ablowitz and P.A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering (Cambridge University Press, Cambridge, U.K., 1991). 12

....algorithm, involve one or two dependent variables. For simplicity of notation, the components of u will be denoted by u; v; instead of u 1 ; u 2 , etc. 2. 2 Algorithm To illustrate our algorithm, we consider the Korteweg de Vries (KdV) equation u t = 6uu x u 3x (5) from soliton theory [1, 42]. This ubiquitous evolution equation is long known to have in nitely many symmetries [14] Key to our method is the observation that (5) is invariant under the dilation symmetry (or scaling) t; x; u) 3 t; 1 x; 2 u) 6) where is an arbitrary parameter. The result of this ....

....are able to apply our algorithm to a larger class of polynomial PDE systems. Consider the wave equation, u tt u 2x 3uu 2x 3u 2 x u 4x = 0; 15) constant) which was proposed by Boussinesq to describe surface water waves whose horizontal scale is much larger than the depth of the water [1]. To apply our algorithm, we must rst rewrite (15) as a rst order system, u t = v x ; v t = u x 3uu x u 3x ; 16) where v is an auxiliary dependent variable. It is easy to verify that the terms u x and u 3x in the second equation obstruct uniformity in rank. To circumvent the problem we ....

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M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991.

.... Hamiltonian structures, and exact solutions of nonlinear ordinary and partial di erential equations (ODEs and PDEs) and di erential di erence equations (DDEs) has been the topic of major research projects in the dynamical systems [1] analytical mechanics [2,3] and the soliton communities [1,2,4 6]. In the context of ODEs, complete integrability is synonymous with solvability in terms of elementary or special functions (including the Painlev e transcendents) In contrast, the connotation complete integrability of PDEs is still subject to debate and clari cation [7] A summary of the ....

....property; i.e. its solutions have no movable singularities other than poles in the complex plane. A later version of the Painlev e test due to Weiss et al. 46] allows testing of the PDE directly, without recourse to the reduction(s) to an ODE. A PDE is said to have the Painlev e property [4] if all solutions in the complex plane are single valued around all movable singularities. In other words, all the solutions of the equation must be branchless around the singular points whose positions depend on the initial conditions. For ODEs, it suces to show that the general solution has no ....

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M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering. London Math. Soc. Lec. Note Ser. 149 (Cambridge University Press, London, 1991).

....systems of PDEs using only ODE solution techniques. Applied to symmetry problems, their algorithm will nd all polynomial rational solutions of the determining equations provided the symmetry group is nite dimensional. 6 Examples 6. 1 The Harry Dym Equation Consider the Harry Dym equation [1], u t u 3 u xxx = 0: 17) Clearly, this is one equation (m = 1) with two independent variables (p = 2) and one dependent variable (q = 1) The assignments of the variables are as follows: x 7 x[1] t 7 x[2] u 7 u[1] 18) This permits to rewrite the equation (17) in a form accepted by the ....

....is nite dimensional. 6 Examples 6. 1 The Harry Dym Equation Consider the Harry Dym equation [1] u t u 3 u xxx = 0: 17) Clearly, this is one equation (m = 1) with two independent variables (p = 2) and one dependent variable (q = 1) The assignments of the variables are as follows: x 7 x[1] ; t 7 x[2] u 7 u[1] 18) This permits to rewrite the equation (17) in a form accepted by the program SYMMGRP.MAX, i.e. e1 : u[1, 0,1] u[1] 3 u[1, 3,0] For PDELIE and SYM DE the input form would be DIFF(U,T) U 3 DIFF(U,X,3) For SPDE and LIE the program accepts ....

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M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering (Cambridge University Press, Cambridge, 1991).

....solutions will exist for such equations. The development and implementation of a comprehensive algorithm that could handle other types of bilinear representations [9] is in progress. 2. 2 Algorithm and Implementation Details about the method can be found in almost any book on soliton theory, [1, 2, 4] here we outline the procedure. Hirota s method requires: i) a clever change of dependent variable; ii) the introduction of a novel di erential operator; iii) a perturbation expansion to solve the resulting bilinear equation. Our leading example is the Korteweg de Vries equation, u t 6uu x ....

....5 3 The Painlev e Test 3.1 Purpose The MACSYMA program PAINLEVE SINGLE.MAX allows one to determine whether or not a given single nonlinear ODE or PDE with (real) polynomial terms ful lls the necessary conditions for having the Painlev e property. A PDE is said to possess the Painlev e property [1] if its solutions in the complex plane are single valued in the neighborhood of non characteristic, movable singular manifolds. For ODEs, this means that solutions should have no worse singularities than movable poles. Such equations are prime candidates for being completely integrable. 3.2 ....

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M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, London Mathematical Society Lecture Notes on Mathematics 149 (Cambridge University Press, Cambridge, UK, 1991).

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Ablowitz, M.J., and Clarkson, P.A., Solitons, Nonlinear Evolution Equations and the Inverse Scattering Transform, L.M.S. Lecture Notes in Mathematics, vol. 149, C.U.P., Cambridge, 1991.

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Ablowitz M J and Clarkson P A, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge University Press, Cambridge, 1991.

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M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge: Cambridge University Press, 1991.

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M.J. Ablowitz and P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering . Cambridge University Press, Cambridge (1991).

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Ablowitz, M.J., Clarkson, P.A. (1991). Solitons, Nonlinear Evolution Equations and Inverse Scattering . London Math. Soc. Lect. Note Ser. 149. Cambridge (U.K.): Cambridge University Press.

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Ablowitz, M.J., Clarkson, P.A. (1991). Solitons, Nonlinear Evolution Equations and Inverse Scattering . London Math. Soc. Lect. Note Ser. 149. London: Cambridge University Press.

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M.J. Ablowitz, P.A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge Univ. Press, Cambridge, 1991.

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