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Boundary-Value Problems for Integrable Equations
346 Boundary-Value Problems for Integrable Equations
Boundary-Value Problems for Integrable Equations B Pelloni, University of Reading, UK ª 2006 Elsevier Ltd. All rights reserved.
Introduction Integrable equations are a special class of nonlinear equations arising in the modeling of a wide variety of physical phenomena. It has been argued that integrable PDEs are in a certain, specific sense ‘‘universal’’ models for physical phenomena involving weak nonlinearity. Indeed, integrable equations are obtained by a procedure involving rescaling and an asymptotic expansion from very large classes of nonlinear evolution equations, which preserves integrability while retaining in the limit weakly nonlinear effects. For this reason, integrable equations are a very important class of PDEs. Important examples are the nonlinear Schro¨dinger (NLS) equation iqt þ qxx 2jqj2 q ¼ 0;
¼ 1
½1
the Korteweg–deVries (KdV) equation qt þ qx qxxx þ 6qqx ¼ 0
½2
the modified KdV (mKdV) equation qt qxxx 6q2 qx ¼ 0;
¼ 1
½3
and the sine-Gordon (SG) equation in light-cone or laboratory coordinates qxt þ sin q ¼ 0
or
qtt qxx þ sin q ¼ 0
½4
A general method for solving the initial-value problem for integrable equations in one space dimension was discovered in 1967, when in a pioneering and much celebrated work (Gardner et al. 1967), the initial-value problems for KdV with decaying initial condition was completely solved. Soon afterwards, it was understood that this method, now known as the ‘‘inverse scattering transform,’’ is of more general applicability. Indeed, it can be applied to those nonlinear equations that can be written as the compatibility condition of a pair of linear eigenvalue equations. The method of solution for the Cauchy problem essentially relies on the possibility of expressing the equation through this pair, now called a Lax pair after the work of Lax (1968), who first clarified the connection. Zakharov and Shabat (1972) constructed such a pair for the NLS equation, and in subsequent years the Lax pairs associated with all important integrable equations in one and two spatial variables were constructed. These include the NLS, sG, mKdV,
Davey–Stewartson I and II, and Kamdotsev– Petviashvili I and II equations. There is no universally accepted definition of an integrable PDE, but on account of the above results, the existence of a Lax pair can be taken as the defining property of such equations. In the course of the 1970s, the inverse scattering transform was applied to solve the initial-value (Cauchy) problem for many integrable equations. In principle, there is no obstruction to solving analytically the initial-value problem by the inverse scattering transform as soon as a Lax pair is constructed for the equation, and appropriate decaying initial conditions are prescribed. The solution is then characterized in terms of a certain integral equation. This approach is equivalent to associating with the initial-value problem a classical problem in complex analysis, namely a matrix Riemann–Hilbert problem, defined in the complex spectral space. This point of view is currently taken by many authors as it provides a unifying and very flexible framework for the analysis. After the success of the inverse scattering transform in solving the Cauchy problem, it was natural to attempt to generalize the approach to boundaryvalue problems. To describe the difficulties involved in this generalization, consider the case of evolution equations in one space and one time dimensions. The independent variables can be denoted by (x, t), with t > 0 representing time. While the initial-value problem is posed on the full real line, hence for x 2 (1, 1), the simplest boundary-value problem is posed on a half-line, for x 2 (0, 1). In addition to initial conditions for initial time t = 0, it is necessary to prescribe conditions at the boundary x = 0. The number of conditions that must be prescribed to obtain a problem which admits a unique solution depends on the particular equation, but for evolution equation it is roughly equal to half the number of x-derivatives involved in the equation. For example, for the NLS equation, a well-posed problem is defined as soon as one boundary condition at x = 0 is prescribed; hence a typical boundary-value problem for this equation is obtained, for example, when q(x, 0) = q0 (x) and q(0, t) = g0 (t) are prescribed and compatible, so that q0 (0) = g0 (0). It follows that, while qxx (0, t) can be computed from the equation, qx (0, t) is not immediately known. An even more difficult situation arises for the KdV equation [2] (with the þ sign), for which a well-posed problem is again defined as soon as one boundary condition is prescribed, so that there are two unknown boundary values.
Boundary-Value Problems for Integrable Equations
Because of this simple fact, a straightforward application of the ideas of the inverse scattering transform immediately encounters one crucial difficulty. This transform method yields an integral representation of the solution which involves not only the given boundary conditions f (t), but also the other ‘‘unknown’’ boundary values – in our example for the NLS equation, the function qx (0, t). The problem of characterizing these unknown boundary values has impeded progress in this direction for over thirty years. On account of their physical significance, various boundary-value problems for the KdV equation have been considered, and classical PDE techniques (not specific to integrable models) have been used to establish existence and uniqueness results (Bona et al. 2001, Colin and Ghidaglia 2001, Colliander and Kenig 2001). These approaches, and in particular the approach of Colliander and Kenig, are quite general and possibly of wide applicability, and give global existence results in wide functional classes. However, they do not rely on integrability properties. Indeed, none of these results use the integrable structure of the equation in any fundamental or systematic way. However, the fact that these equations are integrable on the full line implies very special properties that should be exploited in the analysis and it is natural to try to generalize the inverse scattering transform approach. Such a generalization is sometimes directly possible. For example, it has been used for studying the problem on the half-line for the hyperbolic version of the sG equation [4a] which does not involve unknown boundary values (Fokas 2000, Pelloni). It has also been used to study some specific boundaryvalue problems for the NLS equation, for example, for homogeneous Dirichlet or Neumann conditions, when it is possible to use even or odd extensions of the problem to the full line (Ablowitz and Segur 1974), or more recently in Degasperis et al. (2001). In the latter case, however, the unknown boundary values are characterized through an integral Fredholm equation, which does not admit a unique solution. Some special cases of boundary-value problems for the KdV equation (Adler et al. 1997, Habibullin 1999) and elliptic sG (Sklyanin 1987) have also been studied via the inverse scattering transform. However all the examples considered are nongeneric, and it has recently been shown (Fokas, in press) that the boundary conditions chosen fall in the special class of the so-called ‘‘linearizable’’ boundary conditions, for which the problem can be solved as if it were posed on the full line. One cannot hope to use similar methods to solve the problem with generic boundary conditions.
347
Recently, Fokas (2000) introduced a general methodology to extend the ideas of the inverse scattering transform to boundary-value problems. This methodology provides the tools to analyze boundary-value problems for integrable equations to a considerable degree of generality. We note as a side remark that linear PDEs are trivially integrable, in the sense of admitting a Lax pair (in this case the Lax pair can be found algorithmically, while the construction of the Lax pair associated with a nonlinear equation is by no means trivial). As a consequence of this remark, the extension of the inverse scattering transform also provides a method for solving boundary-value problems for a large variety of linear PDEs of mathematical physics. What follows is a general description of the approach of Fokas, considering, for the sake of concreteness, the case of an integrable PDE in the two variables (x,t) which vary in the domain D (typically, for an evolution problem D = (0, 1) (0, T)). We assume that q(x, t) denotes the unique solution of a boundary-value problem posed for such an equation. The method consists of the following steps. 1. Write the PDE as the compatibility condition of a Lax pair. This is a pair of linear ODEs for the function = (x, t, k) involving the solution q(x, t) of the PDE, the derivatives of this solution, and a complex parameter k, called the spectral parameter. This can be done algorithmically for linear PDEs, and in this case (x, t,k) is a scalar function. For nonlinear integrable PDEs, (x, t, k) is in general a matrix-valued function. The equivalence of the PDE with a Lax pair can be reformulated in the language of differential forms, and in this language it is easier to describe the methodology in general. Assume then that (x, t, k) is a differential 1-form expressed in terms of a function q(x, t) and its derivatives, and of a complex variable k, and one which is characterized by the property that d = 0 if and only if q(x, t) satisfies the given PDE. The closure of the form yields the two important consequences 2(a) and 2(b) below. 2. (a) Since the domain D under consideration is simply connected, the closed form is also exact; hence, it is possible to find the particular, 0-form (x, t,k), solving d = . In particular, (x, t, k) can be chosen to be sectionally bounded with respect to k by solving either a Riemann–Hilbert problem or a d-bar problem in the complex spectral k plane, and the solution (x, t, k) is then expressed in terms of certain ‘‘spectral functions’’ depending on all the boundary values
348 Boundary-Value Problems for Integrable Equations
of the solution q(x, t) of the PDE. The function q(x, t) can then be expressed in terms of (x, t, k). (b) The integral of along the boundary of the domain D vanishes. This yields an integral constraint between all boundary values of the solution of the PDE, which becomes an algebraic constraint for the spectral functions. The resulting algebraic identity is called the ‘‘global relation.’’ 3. The last step is the analysis the k-invariance properties of the global relation. This analysis yields the characterization of the spectral functions in terms only of the given boundary conditions. The crucial and most difficult step in the solution process is the characterization described above. The analysis required depends on the type of problem under consideration. For nonlinear integrable evolution PDEs posed on the half-line x > 0, in general the characterization mentioned in step (3) involves solving a system of nonlinear Volterra integral equations. This is an important difference from the case of the Cauchy problem, where the solution is given by a single integral equation where all the terms are explicitly known. The method outlined above has been applied successfully to solve a variety of boundary-value problems for linear and integrable nonlinear PDEs. For concreteness, here the focus is on the important case of integrable evolution PDEs in one space, which illustrates clearly the generalities of this method.
Integrable Evolution Equations in One Space Dimension The crucial property of integrable PDEs which is used in the inverse scattering transform approach to solve the initial-value problem is the fact that they can be written as the compatibility of a Lax pair. Many integrable evolution equations of physical significance (such as NLS, KdV, sG, and mKdV) admit a Lax pair of the form x þ if1 ðkÞ3 ¼ Qðx; t; kÞ ~ t; kÞ t þ if2 ðkÞ3 ¼ Qðx;
½5
where (x, t, k) is a 2 2 matrix, 3 = diag(1, 1), fi (k), i = 1, 2, are analytic functions of the complex ~ are analytic functions of k, parameter k, and Q, Q of the function q(x, t) (and of its complex conjugate q(x, t) for complex-valued problems) and of its derivatives. For example, the NLS equation [1] is equivalent to the compatibility condition of the pair
x þ ik3 ¼ Q; 2
Q¼
0
q
q
0
½6 2
t þ 2ik 3 ¼ ð2kQ iQx 3 ijqj 3 Þ The first step towards a systematic new approach to solving boundary-value problem was the work of Fokas and Its, who associated the boundary-value problem for NLS on the half-line to a single Riemann–Hilbert problem determined by both equations in the Lax pair. The jump determining this Riemann–Hilbert problem has an explicit exponential dependence on both x and t. This differs from the classical inverse scattering approach, in which the x-part of the Lax pair is used to determine an x-transform with t-dependent scattering data, and the t-part of the Lax pair is then exploited to find the time evolution of these data. The work of Fokas and Its led to the understanding that both equations in the Lax pair [6] must be considered in order to construct a spectral transform appropriate to solve boundary-value problems. Fokas (2000) reviews his systematic way to solve these problems by performing the simultaneous spectral analysis of both equations in the Lax pair. The transform thus obtained, which is a nonlinearization of the Fourier transform, precisely generalizes the inverse scattering transform. This simultaneous analysis also leads naturally to the identification of the ‘‘global relation’’ which holds between initial and boundary data, and which plays an essential role in deriving an expression for the solution of the problem which does not involve unknown boundary values. The Riemann–Hilbert problem with explicit (x, t) dependence, the global relation, and the invariance properties of the latter with respect to the spectral parameter are the fundamental ingredients of this systematic approach to solve boundary-value problems for integrable equations. The steps involved in this method are summarized in the introduction. While steps (1) and (2) can be described generally, and, once the Lax pair is identified, can be performed algorithmically (at least under the assumption that the solution of the PDE exists), the last step is the most difficult part of the analysis, and it needs to be considered separately for each given problem. However, it is this step that yields the effective characterization of the solution. The results obtained for the particular case of eqn [1] are reviewed in detail in the next section, as they provide an important example, which can be generalized without any conceptual difficulty to eqns [2]–[4].
Boundary-Value Problems for Integrable Equations The NLS Equation
As already mentioned, the initial-value problem for NLS was solved, for decaying initial condition, by Zakharov and Shabat, and studied in depth by many others. However, by the mid-1990s only a handful of papers had been written on the solution of the boundary-value problem posed on the half-line, all on a specific example or aspect of the problem, or attempts at solving the problem using general PDE techniques. For this equation, the approach of Fokas yields the following results. Let the complex-valued function q(x, t) satisfy the NLS equation [1], for x > 0 and t > 0, for prescribed one initial and one boundary conditions. For the sake of concreteness, we select the specific initial and boundary conditions qðx; 0Þ ¼ q0 ðxÞ 2 SðRþ Þ qð0; tÞ ¼ g0 ðtÞ 2 SðRþ Þ q0 ð0Þ ¼ g0 ð0Þ
½7
where S denotes the space of Schwartz functions (similar results hold for different choices of boundary conditions, and less restrictive function classes). The solution of this initial boundary-value (IBV) problem can be constructed as follows (Fokas 2000, 2002; in press):
Given q0 (x) construct the spectral functions {a(k), b(k)}. These functions are defined by aðkÞ ¼ 2 ð0; kÞ;
bðkÞ ¼ 1 ð0; kÞ
where the vector (x, k) with components 1 (x, k) and 2 (x, k) is the following solution of the x-problem of the associated Lax pair evaluated at t = 0: x þ ik3 ¼ Qðx; 0; kÞ; 0 < x < 1; Im k 0 0 ðx; kÞ ¼ eikx þ oð1Þ as x ! 1 1 0 q0 ðxÞ Qðx; 0; kÞ ¼ q0 ðxÞ 0 (3 and Q(x, t, k) are defined after eqns [5] and [6], respectively). Given q0 (x) and g0 (t) characterize g1 (t) by the requirement that the spectral functions {A(t, k), B(t, k)} satisfy the global relation Bðt; kÞ RðkÞAðt; kÞ ¼ e4ik
2
t
cðt; kÞ aðkÞ
bðkÞ RðkÞ ¼ ; t 2 ½0; T; k 2 D aðkÞ
½8
349
where D denotes the first quadrant of the complex k-plane: D ¼ fkjRe k > 0; Im k > 0g denotes the closure of D, and c(t, k) is a D function of k analytic in D and of order O(1=k) as k ! 1. The spectral functions are defined by 2
Aðt; kÞ ¼ e2ik t 2 ðt; kÞ; 2ik2 t 1 ðt; kÞ Bðt; kÞ ¼ e
½9
where the vector (t, k) with components 1 and 2 is the following solution of the t-problem of the associated Lax pair evaluated at x = 0: ~ t þ 2ik2 3 ¼ Qð0; t; kÞ 0 < t < T;
k2C 0 ð0; kÞ ¼ 1 ~ Qð0; t; kÞ ¼
½10
jg0 ðtÞj2
2kg0 ðtÞ þ ig1 ðtÞ
2kg0 ðtÞ ig1 ðtÞ
jg0 ðtÞj2
!
Given a(k), b(k) and A(k), B(k), define a 2 2 matrix Riemann–Hilbert problem. This problem has the distinctive feature that its jump has explicit (x, t) dependence in the exponential form of exp {ikx þ 2ik2 t}. Determine q(x, t) in terms of the solution of this Riemann–Hilbert problem by using the fact that these functions are related by the Lax pair. Then the function q(x, t) solves the IBV problem [1]–[7] with q(x, 0) = q0 (x), q(0, t) = g0 (t), and q0x (0, t) = g1 (t). The above construction can be summarized in the following theorem (Fokas 2002): Theorem 1 Consider the boundary-value problem for the NLS equation [1] determined by the conditions [7]. Let a(k), b(k) be given by [8], and suppose that there exists a function g1 (t) such that if A(k), B(k) are defined by [9], then the global relation [8] holds. Let M(x, t, k) be the solution of the 2 2 Riemann–Hilbert problem with jump on the real and imaginary axes given by
M (x, t, k) = Mþ (x, t, k)J(x, t, k) with M = M in the second and fourth quadrants of C, M = Mþ in the first and third quadrants of C, and J(x, t, k) is defined 2 in terms of a, b, A, B and the exponential eikx2ik t : M = I þ O(1=k) as k ! 1 and has appropriate residue conditions if there are poles Then M(x,t,k) exists and is unique, and qðx; tÞ ¼ 2i lim ðkMðx; t; kÞÞ12 k!1
350 Boundary-Value Problems for Integrable Equations
The result above relies on characterizing the unknown boundary value g1 (t) a priori by requiring that the global relation hold. Recently, substantial progress has been made in this direction in the case of integrable nonlinear evolution equations, in particular of NLS. Namely Fokas (in press) contains an effective description of the map assigning to each given q(x, 0) = q0 (x) and g0 (t) = q(0, t) a unique value for qx (0, t) (called the Dirichlet to Neumann map) for the NLS, as well as for a version of the Korteweg– deVries and sG equations. We state below the relevant theorem for the case of the NLS equation.
representation has now been derived for all equations [1]–[3], see Fokas (in press). The analysis of the invariance properties of the global relation with respect to k also yields the characterization of all the boundary conditions for which the transform obtained to represent the solution linearizes. For these boundary conditions, called linearizable, the solution can be represented as effectively as for the Cauchy problem. For example, the linearizable boundary conditions for the NLS equation are given by any boundary values that satisfy
Theorem 2 Let q(x, t) satisfy the NLS equation on the half-line 0 < x < 1, t > 0 with the initial and boundary conditions [7]. Then g1 (t) := qx (0, t) is given by Z g0 ðtÞ 2 g1 ðtÞ¼ e2ik t ð2 ðt;kÞ2 ðt;kÞÞdk @D Z 4i 2 þ e2ik t kRðkÞ2 ðt; kÞdk @D Z 2i 2 þ e2ik t ðk½1 ðt;kÞ1 ðt;kÞ þig0 ðtÞÞdk @D
g0 ðtÞg1 ðtÞ g0 ðtÞg1 ðtÞ ¼ 0
with =(1 ,2 ) given by the solution of [10]. The Neumann datum g1 (t) is unique and exists globally in t. This result yields a rigorous proof of the global existence of the solution of boundary-value problems on the half-line for the NLS equation. Therefore, the assumption in Theorem 1 that a suitable function g1 (t) exists can be dropped.
Generalizations and Summary of Results Results analogous to the ones presented in the previous section can be phrased exclusively in terms of integral equations rather than in terms of Riemann–Hilbert problems, as done for example in Khruslov and Kotlyarov (2003). This is the point of view of the school of Gelfand and Marchenko, and in this setting the functions are given in the so-called Gelfand–Levitan–Marchenko representation. Results on boundary-value problems for the NLS equation using this representation have been obtained only under additional assumptions on the unknown part of the boundary values. It was only after the idea that the x- and t-parts of the spectral equations should be treated simultaneously that this approach yielded complete results. However, the Gelfand–Levitan– Marchenko representation yields a crucial simplification for deriving the explicit form of the Dirichlet to Neumann map and proving Theorem 2. This
An example of boundary condition satisfying this constraint, encompassing also Dirichlet and Neumann homogeneous conditions, is q(0, t) qx (0, t) = 0, with a non-negative constant. As mentioned at the beginning of the previous section, the approach described in general can be used to obtain results similar to those given for the NLS equation for many other integrable evolution equations, in particular, mKdV (Boutet de Monvel et al. 2004), sG, and KdV (Fokas 2002). The results obtained are essentially the same as for NLS, starting from the general form [5] of the Lax pair, and include the derivation of the solution representation, the complete characterization of linearizable boundary conditions, and the analysis of the Dirichlet to Neumann map. The approach above can also be used for studying boundary-value problems posed on finite domains, for x 2 [0, 1]. This has been done for a model for transient simulated Raman scattering (Fokas and Menyuk 1999), for the sG equation in light-cone coordinates (Pelloni, in press), and for the NLS equation (Fokas and Its 2004). In this case also the method yields a representation of the solution which is suitable for asymptotic analysis. In this respect, the question of soliton generation from boundary data is of some importance, and has been recently considered by various authors (Fokas and Menyuk 1999, Boutet de Monvel and Kotlyarov 2003, Pelloni in press, Boutet de Monvel et al. 2004). The results are however still considered case by case, and there is no general framework for this problem identified yet. For problem on the half-line, solitons may be generated but not necessarily in correspondence to the singularities that generate soliton for the full line problem, even when the same singularities are present. For problems posed on finite domains, in some specific cases at least for the simulated Raman scattering, and the sG equations, it appears that the dominant asymptotic behavior is given by a similarity solution.
Braided and Modular Tensor Categories
In conclusion, the extension of the inverse scattering transform given by Fokas provides the tool for analyzing boundary-value problems specific to nonlinear integrable equations. This tool relies, in an essential way, on the integrability structure of the problem, and yields a full characterization of the solution as well as uniqueness and existence results. The solution representation thus obtained is not always fully explicit, but it is always suitable for asymptotic analysis using standard techniques such as the recent nonlinearization of the classical steepest descent method. See also: @ Approach to Integrable Systems; Integrable Discrete Systems; Integrable Systems and the Inverse Scattering Method; Integrable Systems: Overview; Nonlinear Schro¨dinger Equations; Riemann–Hilbert Methods in Integrable Systems; Separation of Variables for Differential Equations; Sine-Gordon Equation.
Further Reading Ablowitz MJ and Segur HJ (1974) The inverse scattering transform: semi-infinite interval. Journal of Mathematical Physics 16: 1054. Adler VE, Gurel B, Gurses M, and Habibullin IT (1997) Journal of Physics A 30: 3505. Bona J, Sun S, and Zhang BY (2001) A non-homogeneous boundary value problem for the Korteweg–deVries equation. Transactions of the American Mathematical Society 354: 427–490. Boutet de Monvel A, Fokas AS, and Shepelsky D (2004) The modified KdV equation on the half-line. Journal of the Institute of Mathematics of Jussieu 3: 139–164. Boutet de Monvel A and Kotlyarov VP (2003) Generation of asymptotic solitons of the nonlinear Schro¨dinger equation by boundary data. Journal of Mathematical Physics 44: 3185–3215. Colin T and Ghidaglia J-M (2001) An initial-boundary value problem for the Korteweg–deVries equation posed on a finite interval. Advanced Differential Equations 6(12): 1463–1492.
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Colliander JE and Kenig CE (2001) The generalized Korteweg– deVries equation on the half line (http://arxiv.org/abs/ math.AP/0111294). Degasperis A, Manakov S, and Santini PM (2001) The nonlinear Schro¨dinger equation on the half line. JETP Letters 74(10): 481–485. Fokas AS (2000) On the integrability of linear and nonlinear PDEs. Journal of Mathematical Physics 41: 4188. Fokas AS (2002) Integrable nonlinear evolution equations on the half line. Communications in Mathematical Physics 230: 1–39. Fokas AS (2005) A generalised Dirichlet to Neumann map for certain nonlinear evolution PDEs. Communications on Pure and Applied Mathematics 58: 639–670. Fokas AS and Its AR (2004) The nonlinear Schro¨dinger equation on the interval. Journal of Physics A: Mathematical and General 37: 6091–6114. Fokas AS and Menyuk CR (1999) Integrability and self-similarity in transient stimulated Raman scattering. Journal of Nonlinear Science 9: 1–31. Gardner GS, Greene JM, Kruskal MD, and Miura RM (1967) Method for solving the Korteweg–de Vries equation. Physical Review Letters 19: 1095. Habibullin IT (1999) KdV equation on a half-line with the zero boundary condition. Theoretical and Mathematical Fizika 119: 397. Khruslov E and Kotlyarov VP (2003) Generation of asymptotic solitons in an integrable model of stimulated Raman scattering by periodic boundary data. Mat. Fiz. Anal. Geom. 10(3): 366–384. Lax PD (1968) Integrals of nonlinear equations of evolution and solitary waves. Communications in Pure and Applied Mathematics 21: 467–490. Pelloni B (2005) The asymptotic behaviour of the solution of boundary value problems for the Sine–Gordon equation on a finite interval. Journal of Nonlinear Mathematical Physics 12: 518–529. Sklyanin EK (1987) Boundary conditions for integrable equations. Functional Analysis and its Applications 21: 86–87. Zakharov VE and Shabat AB (1972) An exact theory of twodimensional self-focusing and one-dimensional automodulation of waves in a nonlinear medium. Soviet Physics – JEPT 34: 62–78.
Braided and Modular Tensor Categories V Lyubashenko, Institute of Mathematics, Kyiv, Ukraine ª 2006 Elsevier Ltd. All rights reserved.
Introduction Tensor or monoidal categories are encountered in various branches of modern mathematical physics. First examples came without mentioning the name of a monoidal category as categories of modules over a group or a Lie algebra. The operation of a monoidal product in this case is the usual tensor product X C Y of modules (representations) X and Y. These categories are symmetric: the modules X Y and Y X are
isomorphic; moreover, the permutation isomorphism (the twist) c : X Y 7! Y X, x y ! y x, is involutive, c2 = idX Y . Next examples of monoidal categories were given by categories of representations of supergroups or Lie superalgebras. They are also symmetric: now the symmetry (Koszul’s rule) c : X Y ! Y X, x y 7! (1)deg x deg y y x, is the twist with a sign, which depends on the degree (or parity) deg x of elements x 2 X. The development of the theory of exactly solvable models in statistical mechanics led Drinfeld (1987) to the notion of quantum groups – Hopf algebras H with additional structures (quasitriangular Hopf algebras). H-Modules also form a monoidal category; however, it is not symmetric, but only braided.
Boundary-Value Problems for Integrable Equations B Pelloni, University of Reading, UK ª 2006 Elsevier Ltd. All rights reserved.
Introduction Integrable equations are a special class of nonlinear equations arising in the modeling of a wide variety of physical phenomena. It has been argued that integrable PDEs are in a certain, specific sense ‘‘universal’’ models for physical phenomena involving weak nonlinearity. Indeed, integrable equations are obtained by a procedure involving rescaling and an asymptotic expansion from very large classes of nonlinear evolution equations, which preserves integrability while retaining in the limit weakly nonlinear effects. For this reason, integrable equations are a very important class of PDEs. Important examples are the nonlinear Schro¨dinger (NLS) equation iqt þ qxx 2jqj2 q ¼ 0;
¼ 1
½1
the Korteweg–deVries (KdV) equation qt þ qx qxxx þ 6qqx ¼ 0
½2
the modified KdV (mKdV) equation qt qxxx 6q2 qx ¼ 0;
¼ 1
½3
and the sine-Gordon (SG) equation in light-cone or laboratory coordinates qxt þ sin q ¼ 0
or
qtt qxx þ sin q ¼ 0
½4
A general method for solving the initial-value problem for integrable equations in one space dimension was discovered in 1967, when in a pioneering and much celebrated work (Gardner et al. 1967), the initial-value problems for KdV with decaying initial condition was completely solved. Soon afterwards, it was understood that this method, now known as the ‘‘inverse scattering transform,’’ is of more general applicability. Indeed, it can be applied to those nonlinear equations that can be written as the compatibility condition of a pair of linear eigenvalue equations. The method of solution for the Cauchy problem essentially relies on the possibility of expressing the equation through this pair, now called a Lax pair after the work of Lax (1968), who first clarified the connection. Zakharov and Shabat (1972) constructed such a pair for the NLS equation, and in subsequent years the Lax pairs associated with all important integrable equations in one and two spatial variables were constructed. These include the NLS, sG, mKdV,
Davey–Stewartson I and II, and Kamdotsev– Petviashvili I and II equations. There is no universally accepted definition of an integrable PDE, but on account of the above results, the existence of a Lax pair can be taken as the defining property of such equations. In the course of the 1970s, the inverse scattering transform was applied to solve the initial-value (Cauchy) problem for many integrable equations. In principle, there is no obstruction to solving analytically the initial-value problem by the inverse scattering transform as soon as a Lax pair is constructed for the equation, and appropriate decaying initial conditions are prescribed. The solution is then characterized in terms of a certain integral equation. This approach is equivalent to associating with the initial-value problem a classical problem in complex analysis, namely a matrix Riemann–Hilbert problem, defined in the complex spectral space. This point of view is currently taken by many authors as it provides a unifying and very flexible framework for the analysis. After the success of the inverse scattering transform in solving the Cauchy problem, it was natural to attempt to generalize the approach to boundaryvalue problems. To describe the difficulties involved in this generalization, consider the case of evolution equations in one space and one time dimensions. The independent variables can be denoted by (x, t), with t > 0 representing time. While the initial-value problem is posed on the full real line, hence for x 2 (1, 1), the simplest boundary-value problem is posed on a half-line, for x 2 (0, 1). In addition to initial conditions for initial time t = 0, it is necessary to prescribe conditions at the boundary x = 0. The number of conditions that must be prescribed to obtain a problem which admits a unique solution depends on the particular equation, but for evolution equation it is roughly equal to half the number of x-derivatives involved in the equation. For example, for the NLS equation, a well-posed problem is defined as soon as one boundary condition at x = 0 is prescribed; hence a typical boundary-value problem for this equation is obtained, for example, when q(x, 0) = q0 (x) and q(0, t) = g0 (t) are prescribed and compatible, so that q0 (0) = g0 (0). It follows that, while qxx (0, t) can be computed from the equation, qx (0, t) is not immediately known. An even more difficult situation arises for the KdV equation [2] (with the þ sign), for which a well-posed problem is again defined as soon as one boundary condition is prescribed, so that there are two unknown boundary values.
Boundary-Value Problems for Integrable Equations
Because of this simple fact, a straightforward application of the ideas of the inverse scattering transform immediately encounters one crucial difficulty. This transform method yields an integral representation of the solution which involves not only the given boundary conditions f (t), but also the other ‘‘unknown’’ boundary values – in our example for the NLS equation, the function qx (0, t). The problem of characterizing these unknown boundary values has impeded progress in this direction for over thirty years. On account of their physical significance, various boundary-value problems for the KdV equation have been considered, and classical PDE techniques (not specific to integrable models) have been used to establish existence and uniqueness results (Bona et al. 2001, Colin and Ghidaglia 2001, Colliander and Kenig 2001). These approaches, and in particular the approach of Colliander and Kenig, are quite general and possibly of wide applicability, and give global existence results in wide functional classes. However, they do not rely on integrability properties. Indeed, none of these results use the integrable structure of the equation in any fundamental or systematic way. However, the fact that these equations are integrable on the full line implies very special properties that should be exploited in the analysis and it is natural to try to generalize the inverse scattering transform approach. Such a generalization is sometimes directly possible. For example, it has been used for studying the problem on the half-line for the hyperbolic version of the sG equation [4a] which does not involve unknown boundary values (Fokas 2000, Pelloni). It has also been used to study some specific boundaryvalue problems for the NLS equation, for example, for homogeneous Dirichlet or Neumann conditions, when it is possible to use even or odd extensions of the problem to the full line (Ablowitz and Segur 1974), or more recently in Degasperis et al. (2001). In the latter case, however, the unknown boundary values are characterized through an integral Fredholm equation, which does not admit a unique solution. Some special cases of boundary-value problems for the KdV equation (Adler et al. 1997, Habibullin 1999) and elliptic sG (Sklyanin 1987) have also been studied via the inverse scattering transform. However all the examples considered are nongeneric, and it has recently been shown (Fokas, in press) that the boundary conditions chosen fall in the special class of the so-called ‘‘linearizable’’ boundary conditions, for which the problem can be solved as if it were posed on the full line. One cannot hope to use similar methods to solve the problem with generic boundary conditions.
347
Recently, Fokas (2000) introduced a general methodology to extend the ideas of the inverse scattering transform to boundary-value problems. This methodology provides the tools to analyze boundary-value problems for integrable equations to a considerable degree of generality. We note as a side remark that linear PDEs are trivially integrable, in the sense of admitting a Lax pair (in this case the Lax pair can be found algorithmically, while the construction of the Lax pair associated with a nonlinear equation is by no means trivial). As a consequence of this remark, the extension of the inverse scattering transform also provides a method for solving boundary-value problems for a large variety of linear PDEs of mathematical physics. What follows is a general description of the approach of Fokas, considering, for the sake of concreteness, the case of an integrable PDE in the two variables (x,t) which vary in the domain D (typically, for an evolution problem D = (0, 1) (0, T)). We assume that q(x, t) denotes the unique solution of a boundary-value problem posed for such an equation. The method consists of the following steps. 1. Write the PDE as the compatibility condition of a Lax pair. This is a pair of linear ODEs for the function = (x, t, k) involving the solution q(x, t) of the PDE, the derivatives of this solution, and a complex parameter k, called the spectral parameter. This can be done algorithmically for linear PDEs, and in this case (x, t,k) is a scalar function. For nonlinear integrable PDEs, (x, t, k) is in general a matrix-valued function. The equivalence of the PDE with a Lax pair can be reformulated in the language of differential forms, and in this language it is easier to describe the methodology in general. Assume then that (x, t, k) is a differential 1-form expressed in terms of a function q(x, t) and its derivatives, and of a complex variable k, and one which is characterized by the property that d = 0 if and only if q(x, t) satisfies the given PDE. The closure of the form yields the two important consequences 2(a) and 2(b) below. 2. (a) Since the domain D under consideration is simply connected, the closed form is also exact; hence, it is possible to find the particular, 0-form (x, t,k), solving d = . In particular, (x, t, k) can be chosen to be sectionally bounded with respect to k by solving either a Riemann–Hilbert problem or a d-bar problem in the complex spectral k plane, and the solution (x, t, k) is then expressed in terms of certain ‘‘spectral functions’’ depending on all the boundary values
348 Boundary-Value Problems for Integrable Equations
of the solution q(x, t) of the PDE. The function q(x, t) can then be expressed in terms of (x, t, k). (b) The integral of along the boundary of the domain D vanishes. This yields an integral constraint between all boundary values of the solution of the PDE, which becomes an algebraic constraint for the spectral functions. The resulting algebraic identity is called the ‘‘global relation.’’ 3. The last step is the analysis the k-invariance properties of the global relation. This analysis yields the characterization of the spectral functions in terms only of the given boundary conditions. The crucial and most difficult step in the solution process is the characterization described above. The analysis required depends on the type of problem under consideration. For nonlinear integrable evolution PDEs posed on the half-line x > 0, in general the characterization mentioned in step (3) involves solving a system of nonlinear Volterra integral equations. This is an important difference from the case of the Cauchy problem, where the solution is given by a single integral equation where all the terms are explicitly known. The method outlined above has been applied successfully to solve a variety of boundary-value problems for linear and integrable nonlinear PDEs. For concreteness, here the focus is on the important case of integrable evolution PDEs in one space, which illustrates clearly the generalities of this method.
Integrable Evolution Equations in One Space Dimension The crucial property of integrable PDEs which is used in the inverse scattering transform approach to solve the initial-value problem is the fact that they can be written as the compatibility of a Lax pair. Many integrable evolution equations of physical significance (such as NLS, KdV, sG, and mKdV) admit a Lax pair of the form x þ if1 ðkÞ3 ¼ Qðx; t; kÞ ~ t; kÞ t þ if2 ðkÞ3 ¼ Qðx;
½5
where (x, t, k) is a 2 2 matrix, 3 = diag(1, 1), fi (k), i = 1, 2, are analytic functions of the complex ~ are analytic functions of k, parameter k, and Q, Q of the function q(x, t) (and of its complex conjugate q(x, t) for complex-valued problems) and of its derivatives. For example, the NLS equation [1] is equivalent to the compatibility condition of the pair
x þ ik3 ¼ Q; 2
Q¼
0
q
q
0
½6 2
t þ 2ik 3 ¼ ð2kQ iQx 3 ijqj 3 Þ The first step towards a systematic new approach to solving boundary-value problem was the work of Fokas and Its, who associated the boundary-value problem for NLS on the half-line to a single Riemann–Hilbert problem determined by both equations in the Lax pair. The jump determining this Riemann–Hilbert problem has an explicit exponential dependence on both x and t. This differs from the classical inverse scattering approach, in which the x-part of the Lax pair is used to determine an x-transform with t-dependent scattering data, and the t-part of the Lax pair is then exploited to find the time evolution of these data. The work of Fokas and Its led to the understanding that both equations in the Lax pair [6] must be considered in order to construct a spectral transform appropriate to solve boundary-value problems. Fokas (2000) reviews his systematic way to solve these problems by performing the simultaneous spectral analysis of both equations in the Lax pair. The transform thus obtained, which is a nonlinearization of the Fourier transform, precisely generalizes the inverse scattering transform. This simultaneous analysis also leads naturally to the identification of the ‘‘global relation’’ which holds between initial and boundary data, and which plays an essential role in deriving an expression for the solution of the problem which does not involve unknown boundary values. The Riemann–Hilbert problem with explicit (x, t) dependence, the global relation, and the invariance properties of the latter with respect to the spectral parameter are the fundamental ingredients of this systematic approach to solve boundary-value problems for integrable equations. The steps involved in this method are summarized in the introduction. While steps (1) and (2) can be described generally, and, once the Lax pair is identified, can be performed algorithmically (at least under the assumption that the solution of the PDE exists), the last step is the most difficult part of the analysis, and it needs to be considered separately for each given problem. However, it is this step that yields the effective characterization of the solution. The results obtained for the particular case of eqn [1] are reviewed in detail in the next section, as they provide an important example, which can be generalized without any conceptual difficulty to eqns [2]–[4].
Boundary-Value Problems for Integrable Equations The NLS Equation
As already mentioned, the initial-value problem for NLS was solved, for decaying initial condition, by Zakharov and Shabat, and studied in depth by many others. However, by the mid-1990s only a handful of papers had been written on the solution of the boundary-value problem posed on the half-line, all on a specific example or aspect of the problem, or attempts at solving the problem using general PDE techniques. For this equation, the approach of Fokas yields the following results. Let the complex-valued function q(x, t) satisfy the NLS equation [1], for x > 0 and t > 0, for prescribed one initial and one boundary conditions. For the sake of concreteness, we select the specific initial and boundary conditions qðx; 0Þ ¼ q0 ðxÞ 2 SðRþ Þ qð0; tÞ ¼ g0 ðtÞ 2 SðRþ Þ q0 ð0Þ ¼ g0 ð0Þ
½7
where S denotes the space of Schwartz functions (similar results hold for different choices of boundary conditions, and less restrictive function classes). The solution of this initial boundary-value (IBV) problem can be constructed as follows (Fokas 2000, 2002; in press):
Given q0 (x) construct the spectral functions {a(k), b(k)}. These functions are defined by aðkÞ ¼ 2 ð0; kÞ;
bðkÞ ¼ 1 ð0; kÞ
where the vector (x, k) with components 1 (x, k) and 2 (x, k) is the following solution of the x-problem of the associated Lax pair evaluated at t = 0: x þ ik3 ¼ Qðx; 0; kÞ; 0 < x < 1; Im k 0 0 ðx; kÞ ¼ eikx þ oð1Þ as x ! 1 1 0 q0 ðxÞ Qðx; 0; kÞ ¼ q0 ðxÞ 0 (3 and Q(x, t, k) are defined after eqns [5] and [6], respectively). Given q0 (x) and g0 (t) characterize g1 (t) by the requirement that the spectral functions {A(t, k), B(t, k)} satisfy the global relation Bðt; kÞ RðkÞAðt; kÞ ¼ e4ik
2
t
cðt; kÞ aðkÞ
bðkÞ RðkÞ ¼ ; t 2 ½0; T; k 2 D aðkÞ
½8
349
where D denotes the first quadrant of the complex k-plane: D ¼ fkjRe k > 0; Im k > 0g denotes the closure of D, and c(t, k) is a D function of k analytic in D and of order O(1=k) as k ! 1. The spectral functions are defined by 2
Aðt; kÞ ¼ e2ik t 2 ðt; kÞ; 2ik2 t 1 ðt; kÞ Bðt; kÞ ¼ e
½9
where the vector (t, k) with components 1 and 2 is the following solution of the t-problem of the associated Lax pair evaluated at x = 0: ~ t þ 2ik2 3 ¼ Qð0; t; kÞ 0 < t < T;
k2C 0 ð0; kÞ ¼ 1 ~ Qð0; t; kÞ ¼
½10
jg0 ðtÞj2
2kg0 ðtÞ þ ig1 ðtÞ
2kg0 ðtÞ ig1 ðtÞ
jg0 ðtÞj2
!
Given a(k), b(k) and A(k), B(k), define a 2 2 matrix Riemann–Hilbert problem. This problem has the distinctive feature that its jump has explicit (x, t) dependence in the exponential form of exp {ikx þ 2ik2 t}. Determine q(x, t) in terms of the solution of this Riemann–Hilbert problem by using the fact that these functions are related by the Lax pair. Then the function q(x, t) solves the IBV problem [1]–[7] with q(x, 0) = q0 (x), q(0, t) = g0 (t), and q0x (0, t) = g1 (t). The above construction can be summarized in the following theorem (Fokas 2002): Theorem 1 Consider the boundary-value problem for the NLS equation [1] determined by the conditions [7]. Let a(k), b(k) be given by [8], and suppose that there exists a function g1 (t) such that if A(k), B(k) are defined by [9], then the global relation [8] holds. Let M(x, t, k) be the solution of the 2 2 Riemann–Hilbert problem with jump on the real and imaginary axes given by
M (x, t, k) = Mþ (x, t, k)J(x, t, k) with M = M in the second and fourth quadrants of C, M = Mþ in the first and third quadrants of C, and J(x, t, k) is defined 2 in terms of a, b, A, B and the exponential eikx2ik t : M = I þ O(1=k) as k ! 1 and has appropriate residue conditions if there are poles Then M(x,t,k) exists and is unique, and qðx; tÞ ¼ 2i lim ðkMðx; t; kÞÞ12 k!1
350 Boundary-Value Problems for Integrable Equations
The result above relies on characterizing the unknown boundary value g1 (t) a priori by requiring that the global relation hold. Recently, substantial progress has been made in this direction in the case of integrable nonlinear evolution equations, in particular of NLS. Namely Fokas (in press) contains an effective description of the map assigning to each given q(x, 0) = q0 (x) and g0 (t) = q(0, t) a unique value for qx (0, t) (called the Dirichlet to Neumann map) for the NLS, as well as for a version of the Korteweg– deVries and sG equations. We state below the relevant theorem for the case of the NLS equation.
representation has now been derived for all equations [1]–[3], see Fokas (in press). The analysis of the invariance properties of the global relation with respect to k also yields the characterization of all the boundary conditions for which the transform obtained to represent the solution linearizes. For these boundary conditions, called linearizable, the solution can be represented as effectively as for the Cauchy problem. For example, the linearizable boundary conditions for the NLS equation are given by any boundary values that satisfy
Theorem 2 Let q(x, t) satisfy the NLS equation on the half-line 0 < x < 1, t > 0 with the initial and boundary conditions [7]. Then g1 (t) := qx (0, t) is given by Z g0 ðtÞ 2 g1 ðtÞ¼ e2ik t ð2 ðt;kÞ2 ðt;kÞÞdk @D Z 4i 2 þ e2ik t kRðkÞ2 ðt; kÞdk @D Z 2i 2 þ e2ik t ðk½1 ðt;kÞ1 ðt;kÞ þig0 ðtÞÞdk @D
g0 ðtÞg1 ðtÞ g0 ðtÞg1 ðtÞ ¼ 0
with =(1 ,2 ) given by the solution of [10]. The Neumann datum g1 (t) is unique and exists globally in t. This result yields a rigorous proof of the global existence of the solution of boundary-value problems on the half-line for the NLS equation. Therefore, the assumption in Theorem 1 that a suitable function g1 (t) exists can be dropped.
Generalizations and Summary of Results Results analogous to the ones presented in the previous section can be phrased exclusively in terms of integral equations rather than in terms of Riemann–Hilbert problems, as done for example in Khruslov and Kotlyarov (2003). This is the point of view of the school of Gelfand and Marchenko, and in this setting the functions are given in the so-called Gelfand–Levitan–Marchenko representation. Results on boundary-value problems for the NLS equation using this representation have been obtained only under additional assumptions on the unknown part of the boundary values. It was only after the idea that the x- and t-parts of the spectral equations should be treated simultaneously that this approach yielded complete results. However, the Gelfand–Levitan– Marchenko representation yields a crucial simplification for deriving the explicit form of the Dirichlet to Neumann map and proving Theorem 2. This
An example of boundary condition satisfying this constraint, encompassing also Dirichlet and Neumann homogeneous conditions, is q(0, t) qx (0, t) = 0, with a non-negative constant. As mentioned at the beginning of the previous section, the approach described in general can be used to obtain results similar to those given for the NLS equation for many other integrable evolution equations, in particular, mKdV (Boutet de Monvel et al. 2004), sG, and KdV (Fokas 2002). The results obtained are essentially the same as for NLS, starting from the general form [5] of the Lax pair, and include the derivation of the solution representation, the complete characterization of linearizable boundary conditions, and the analysis of the Dirichlet to Neumann map. The approach above can also be used for studying boundary-value problems posed on finite domains, for x 2 [0, 1]. This has been done for a model for transient simulated Raman scattering (Fokas and Menyuk 1999), for the sG equation in light-cone coordinates (Pelloni, in press), and for the NLS equation (Fokas and Its 2004). In this case also the method yields a representation of the solution which is suitable for asymptotic analysis. In this respect, the question of soliton generation from boundary data is of some importance, and has been recently considered by various authors (Fokas and Menyuk 1999, Boutet de Monvel and Kotlyarov 2003, Pelloni in press, Boutet de Monvel et al. 2004). The results are however still considered case by case, and there is no general framework for this problem identified yet. For problem on the half-line, solitons may be generated but not necessarily in correspondence to the singularities that generate soliton for the full line problem, even when the same singularities are present. For problems posed on finite domains, in some specific cases at least for the simulated Raman scattering, and the sG equations, it appears that the dominant asymptotic behavior is given by a similarity solution.
Braided and Modular Tensor Categories
In conclusion, the extension of the inverse scattering transform given by Fokas provides the tool for analyzing boundary-value problems specific to nonlinear integrable equations. This tool relies, in an essential way, on the integrability structure of the problem, and yields a full characterization of the solution as well as uniqueness and existence results. The solution representation thus obtained is not always fully explicit, but it is always suitable for asymptotic analysis using standard techniques such as the recent nonlinearization of the classical steepest descent method. See also: @ Approach to Integrable Systems; Integrable Discrete Systems; Integrable Systems and the Inverse Scattering Method; Integrable Systems: Overview; Nonlinear Schro¨dinger Equations; Riemann–Hilbert Methods in Integrable Systems; Separation of Variables for Differential Equations; Sine-Gordon Equation.
Further Reading Ablowitz MJ and Segur HJ (1974) The inverse scattering transform: semi-infinite interval. Journal of Mathematical Physics 16: 1054. Adler VE, Gurel B, Gurses M, and Habibullin IT (1997) Journal of Physics A 30: 3505. Bona J, Sun S, and Zhang BY (2001) A non-homogeneous boundary value problem for the Korteweg–deVries equation. Transactions of the American Mathematical Society 354: 427–490. Boutet de Monvel A, Fokas AS, and Shepelsky D (2004) The modified KdV equation on the half-line. Journal of the Institute of Mathematics of Jussieu 3: 139–164. Boutet de Monvel A and Kotlyarov VP (2003) Generation of asymptotic solitons of the nonlinear Schro¨dinger equation by boundary data. Journal of Mathematical Physics 44: 3185–3215. Colin T and Ghidaglia J-M (2001) An initial-boundary value problem for the Korteweg–deVries equation posed on a finite interval. Advanced Differential Equations 6(12): 1463–1492.
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Colliander JE and Kenig CE (2001) The generalized Korteweg– deVries equation on the half line (http://arxiv.org/abs/ math.AP/0111294). Degasperis A, Manakov S, and Santini PM (2001) The nonlinear Schro¨dinger equation on the half line. JETP Letters 74(10): 481–485. Fokas AS (2000) On the integrability of linear and nonlinear PDEs. Journal of Mathematical Physics 41: 4188. Fokas AS (2002) Integrable nonlinear evolution equations on the half line. Communications in Mathematical Physics 230: 1–39. Fokas AS (2005) A generalised Dirichlet to Neumann map for certain nonlinear evolution PDEs. Communications on Pure and Applied Mathematics 58: 639–670. Fokas AS and Its AR (2004) The nonlinear Schro¨dinger equation on the interval. Journal of Physics A: Mathematical and General 37: 6091–6114. Fokas AS and Menyuk CR (1999) Integrability and self-similarity in transient stimulated Raman scattering. Journal of Nonlinear Science 9: 1–31. Gardner GS, Greene JM, Kruskal MD, and Miura RM (1967) Method for solving the Korteweg–de Vries equation. Physical Review Letters 19: 1095. Habibullin IT (1999) KdV equation on a half-line with the zero boundary condition. Theoretical and Mathematical Fizika 119: 397. Khruslov E and Kotlyarov VP (2003) Generation of asymptotic solitons in an integrable model of stimulated Raman scattering by periodic boundary data. Mat. Fiz. Anal. Geom. 10(3): 366–384. Lax PD (1968) Integrals of nonlinear equations of evolution and solitary waves. Communications in Pure and Applied Mathematics 21: 467–490. Pelloni B (2005) The asymptotic behaviour of the solution of boundary value problems for the Sine–Gordon equation on a finite interval. Journal of Nonlinear Mathematical Physics 12: 518–529. Sklyanin EK (1987) Boundary conditions for integrable equations. Functional Analysis and its Applications 21: 86–87. Zakharov VE and Shabat AB (1972) An exact theory of twodimensional self-focusing and one-dimensional automodulation of waves in a nonlinear medium. Soviet Physics – JEPT 34: 62–78.
Braided and Modular Tensor Categories V Lyubashenko, Institute of Mathematics, Kyiv, Ukraine ª 2006 Elsevier Ltd. All rights reserved.
Introduction Tensor or monoidal categories are encountered in various branches of modern mathematical physics. First examples came without mentioning the name of a monoidal category as categories of modules over a group or a Lie algebra. The operation of a monoidal product in this case is the usual tensor product X C Y of modules (representations) X and Y. These categories are symmetric: the modules X Y and Y X are
isomorphic; moreover, the permutation isomorphism (the twist) c : X Y 7! Y X, x y ! y x, is involutive, c2 = idX Y . Next examples of monoidal categories were given by categories of representations of supergroups or Lie superalgebras. They are also symmetric: now the symmetry (Koszul’s rule) c : X Y ! Y X, x y 7! (1)deg x deg y y x, is the twist with a sign, which depends on the degree (or parity) deg x of elements x 2 X. The development of the theory of exactly solvable models in statistical mechanics led Drinfeld (1987) to the notion of quantum groups – Hopf algebras H with additional structures (quasitriangular Hopf algebras). H-Modules also form a monoidal category; however, it is not symmetric, but only braided.