Bäcklund Transformations for the Boussinesq Equation and Merging Solitons
BB¨acklund Transformations for the Boussinesq Equationand Merging Solitons
Alexander G. RasinDepartment of Mathematics,Ariel University, Ariel 40700, IsraelE-mail: [email protected] SchiffDepartment of Mathematics,Bar-Ilan University, Ramat Gan, 52900, IsraelE-mail: schiff@math.biu.ac.ilJanuary 3, 2017
Abstract
The B¨acklund transformation (BT) for the “good” Boussinesq equation and itssuperposition principles are presented and applied. Unlike many other standard in-tegrable equations, the Boussinesq equation does not have a strictly algebraic super-position principle for 2 BTs, but it does for 3. We present associated lattice systems.Applying the BT to the trivial solution generates standard solitons but also what wecall “merging solitons” — solutions in which two solitary waves (with related speeds)merge into a single one. We use the superposition principles to generate a variety ofinteresting solutions, including superpositions of a merging soliton with 1 or 2 regularsolitons, and solutions that develop a singularity in finite time which then disappearsat some later finite time. We prove a Wronskian formula for the solutions obtainedby applying a general sequence of BTs on the trivial solution. Finally, we show howto obtain the standard conserved quantities of the Boussinesq equation from the BT,and how the hierarchy of local symmetries follows in a simple manner from the super-position principle for 3 BTs. a r X i v : . [ n li n . S I] J a n Introduction
In this paper we explore the B¨acklund transformation (BT) of the Boussinesq equation(BEq) U tt − βU xx + U xxxx − U ) xx = 0 (1)where β is a positive constant. The BEq is one of the oldest of the classical integrablenonlinear partial differential equations (PDE) [10, 9], and its BT was given in bilinear formby Hirota and Satsuma [23] and in standard form by Chen [11]. who also gave a superpositionprinciple (see also [24, 41, 45]). However, certain aspects seem not to have been discussed.There is a second superposition principle, and using this it is possible to give a superpositionprinciple for 3 BTs that is algebraic (as opposed to the superposition principle of [11] thatinvolves derivatives). In addition, there does not seem to be a systematic study of solutionsgenerated by the BT, and this involves several surprises, as we shall shortly explain.Our original motivation for looking at the BT of the BEq was connected with latticeversions of the equation. In recent years there has been substantial interest in integrablelattice equations, and one of the origins of these is as superposition principles of BTs ofintegrable PDE (for example, the Q4 equation in the ABS classification [3] was originallydiscovered by Adler as the superposition principle for the Krichever-Novikov equation [2]).Discrete versions of the BEq have been given by Nijhoff et al. [33] as a scalar equation on alarge stencil (see also [32]), and by Tongas and Nijhoff [39] as a system of equations for 3 fieldson a rectangular plaquette. These, along with related “modified” and “Schwarzian” systems,have attracted much attention recently [5, 31, 17, 18, 19, 20, 43, 6, 42, 44]. Having fullyunderstood the superposition principle for the continuum BEq, we present two associatedlattice systems. One is a system of 2 equations for 2 fields on a rectangular plaquette (likethe discrete modified and Schwarzian BEqs, as introduced in [32, 5]), the other is a systemof 2 equations for a single field on a cube.However, it seems there is much to be learnt from simply applying the BT. Note that in(1) we have written the “good” version of the BEq, in which the signs of the U tt and U xxxx terms are the same. For the “bad” version, in which the signs are opposite, the N -solitonsolutions of the BEq equation were given by Hirota [21], using his eponymous method. Forthe good BEq there is a subtlety in applying the Hirota method, and there are a variety ofinteresting, non-standard, soliton-type solutions as discovered by Manoranjan et.al. [30, 29]and Bogdanov and Zakharov [7], citing unpublished work of Orlov. (Similar phenomenawere observed by Hietarinta and Zhang [19] in their study of solitons in a modified discreteBEq.) We show that applying the BT to the trivial solution can generate standard solitons,2ut also what we call “merging solitons” — solutions in which two solitary waves (withrelated speeds) merge into a single one. The superposition principle enables us to superposea merging soliton with a standard 1-soliton or a standard 2-soliton. We have not succeededso far to obtain a nonsingular solution involving the superposition of 2 or more mergingsolitons, but from the superposition of 3 merging solitons we find a solution which initiallydescribes 6 solitary waves, becomes singular in finite time, but then becomes regular again,leaving 3 solitary waves. The possibility of finite time singularities forming in the BEq iswell-known, originating, we believe, in [25].We also use the BT to prove a Wronskian formula for the general soliton solution ofBEq, a generalization of the formula given in [34] for the bad BEq.Finally, we show how to use the BT of the BEq to generate its conservation laws andsymmetries. The idea of using a BT to generate conservation laws of an integrable PDE isvery old, see for example [40]. In [37] we showed how the superposition principles of BTs ofa number of integrable PDEs can be used to generate their symmetries. This works for theBEq, but it is necessary to use the superposition principle for 3 BTs. This is a consequenceof the fact that the BEq is associated with the Lie group SL (3), while equations such asKorteweg-de Vries, Sine-Gordon and Camassa-Holm are associated with SL (2). We showhow to use the 3 BT superposition principle to obtain the local symmetries of the BEq, andalso obtain the recursion operator and some nonlocal symmetries.This paper is structured as follows: In Section 2 we give the BT of the BEq and itssuperposition principles. In Section 3 we discuss associated lattice equations. In Section4 we describe solutions of the BEq generated by the BT. In Section 5 we use the BT togenerate the symmetries and conservation of the BEq. In Section 6 we conculde and indicateareas for further study. We work with the potential BEq in the form f t = (cid:0) f x − f − h (cid:1) x , (2) h t = (cid:0) f xx − h x + f − f f x (cid:1) x + 2 f h x . (3)3his arises from the consistency of the Lax pair Y x = AY, Y t = BY where A = f f x − f − h λ + f xx + f − f f x − h x + 2 f h h − f ,B = − h f λ + f h − h x f x − f − fB λ + f xx + f − f f x − h x + f h h + f − f x ,B = − f xxx + f x + f f xx − f f x − h + h ( f x − f ) . The reason to take the equation in this apparently complicated form is that it simplifies theaction of the BT, as we shall see shortly. It is easy to check that (2)-(3) imply f tt = − f xxxx + 4 f x f xx (4)which is the standard form of the potential BEq. If we define w = h + f − f x then theabove system simplifies to f t = ( − w − f x ) x ,w t = w xx + f xxx − f x . So if u = f x , v = w x then u t = ( − v − u x ) x , (5) v t = (cid:0) v x + u xx − u (cid:1) x . (6)This is the two component form of the BEq that we use (it is maybe more standard toreplace the field v by ˜ v = − v − u x to simplify the first equation, but we find our formmarginally more convenient). By eliminating v we obtain the scalar form of the BEq: u tt = − u xxxx + (2 u ) xx . (7)Finally, if we write u = U + β we recover the familiar form (1). However, we will work withthe form (7), and just remember, when looking at explicit solutions, that we are interestedin solutions with u → β at spatial infinity. We will focus on the case β >
0, in which case(1) is a linearly stable perturbed wave equation.It is straightforward to verify that the potential BEq in the f, h form (2)-(3) has a BT f → f new = f − s , h → h new = h − f x + f s (8)4here s satisfies the equations s xx = θ − ss x − s + 3 f x s + 3 f xx − f f x − h x , (9) s t = θ − ss x − s + 3 f x s + f xx − f f x − h x , (10)or, equivalently, s xx = θ − ss x − s + 3 us − v , (11) s t = θ − ss x − s + 3 us − u x − v . (12)This is a BT in the sense that if f, h satisfy the potential BEq system (2)-(3), then so do f new , h new given by (8). Furthermore, the equations for s , (11)-(12), are consistent if andonly u, v satsify the BEq system (5)-(6).Denote by f , h ( f , h ) the solution obtained from f, h using a BT with parameter θ ( θ ), and by f , h ( f , h ) the solution obtained from f , h ( f , h ) using a BT withparameter θ ( θ ). Assuming commutativity of BTs gives f = f , h = h . Eliminating s from 4 copies of (8) we have h = h − f x + f ( f − f ) ,h = h − f x + f ( f − f ) ,h = h − f x + f ( f − f ) ,h = h − f x + f ( f − f ) . Using commutativity and taking the obvious linear combination of these equations to elim-inate all h fields, we arrive at the superposition principle, as given by Chen [11] f x − f x + ( f + f − f − f )( f − f ) = 0 , (13)which can be solved for f : f = f + f − f − f x − f x f − f . (14)In place of giving a proof for commutativity, it is possible to directly verify that the newsolution given by (14) is a solution of the potential BEq (4).5ut in fact there is also a second superposition formula. Assuming the commutativityof 2 BTs, we have four versions of equation (9):( f − f ) xx = θ − f − f )( f − f ) x − ( f − f ) + 3 f x ( f − f ) + 3 f xx − f f x − h x , ( f − f ) xx = θ − f − f )( f − f ) x − ( f − f ) + 3 f x ( f − f ) + 3 f xx − f f x − h x , ( f − f ) xx = θ − f − f )( f − f ) x − ( f − f ) + 3 f x ( f − f ) + 3 f xx − f f x − h x , ( f − f ) xx = θ − f − f )( f − f ) x − ( f − f ) + 3 f x ( f − f ) + 3 f xx − f f x − h x , where h = h − f x + f ( f − f ), h = h − f x + f ( f − f ). Taking a suitable linear combinationof these equations eliminates the second derivatives of f, f , f , f and the function h , givingthe result0 = ( f − f ) (cid:0) f x + f x − f − f − f − f + 2 f ( f + f ) − f f + f ( f + f ) (cid:1) + θ − θ + ( f − f ) f x − ( f − f ) f x . Finally, adding f times equation (13) gives0 = ( f − f ) (cid:0) f x + f x − f − f − f − f + ( f + f )( f + f ) + f f − f f (cid:1) + θ − θ + f f x − f f x . (15)This can be solved for f x : f x = f + f + f + f − ( f + f )( f + f ) − f f + f f + θ − θ + f f x − f f x f − f − f x . (16)Equation (16) does not follow directly from (14). Differentiating the right hand side of (14)will include second derivatives of f , f . However note that if these are eliminated by usingthe first two of the 4 versions of (9) above, then (16) can be proved directly, without needingto assume commutativity. Writing u = f x in (13) and (15) we obtain the pair of quad-graph equations0 = u − u + ( f + f − f − f )( f − f ) , (17)0 = ( f − f ) (cid:0) u + u − f − f − f − f + ( f + f )( f + f ) + f f − f f (cid:1) + θ − θ + f u − f u . (18)6Here we are thinking of f, f , f , f as 4 values of the field f around the vertices of a rectan-gle. Other notations common in the literature are f, ˜ f , ˆ f , ˆ˜ f and f n,m , f n +1 ,m , f n,m +1 , f n +1 ,m +1 .)These equations are somewhat simplfied by introducing the field g = u − f :0 = g − g + ( f + f )( f − f ) . (19)0 = ( f − f ) ( g + g + ( f + f )( f + f ) + f f ) + θ − θ + f g − f g . (20)It is straightforward to check that these equations have the consistency around the cube(CAC) property [3], and also arise as the consistency conditions for the following Lax pair: Y = f − − ( f f + g ) f θ + f f + f g − f g g + g − f − ( f + f ) Y ,Y = f − − ( f f + g ) f θ + f f + f g − f g g + g − f − ( f + f ) Y .
The lattice potential Boussinesq system (19)-(20) should be compared with the latticeBoussinesq system of Tongas and Nijhoff [39]. Their system involves 3 fields u, v, w , satis-fying 5 equations on an elementary plaquette, 4 of which being the “same” equation on the4 sides of the plaquette. Using u, v, w for the fields, as in [39] (and not as in the rest of thispaper), the equations are w = uu − v ,w = uu − v ,w = u u − v ,w = u u − v ,w = uu − v + θ − θ u − u . We would argue that since the Tongas-Nijhoff system involves 5 relations between 12 quanti-ties on an elementary plaquette (3 fields at each of 4 vertices), whereas our system involves2 relations between 8 quantites, there is a fundamental difference. However, we suspectthere may be relations between solutions of the two systems. Likewise, we suspect thereare relations with the lattice modified Boussinesq system introduced in [32] and the latticeSchwarzian Boussinesq system that appears in [5], both of which are systems of 2 equationsfor 2 fields on an elementary plaquette.In checking the CAC property for (19)-(20) it emerges that it is possible to eliminatethe field g when considering the equations on a cube. So we introduce a third BT, with7 f f f f f f f Figure 1: 8 solutions around a cubeparameter θ , and denote by f the solution obtained from f via this BT, and consider theset of 8 solutions f, f , f , f , f , f , f , f associated with vertices of a cube, as indicatedin Figure 1. These solutions satisfy the equations f ( f − f ) + f ( f − f ) + f ( f − f ) = 0 (21)and f = f + ( θ − θ ) f + ( θ − θ ) f + ( θ − θ ) f ( f − f ) f f + ( f − f ) f f + ( f − f ) f f . (22)Once again, we have 2 relations between 8 quantities. The first of these equations has asuperficial similarity to the Hirota DAGTE equation [22], which was used as a starting pointto find discrete Boussinesq systems in [20]. We wonder in what sense the system (21)-(22)is integrable. As explained in section 2, we wish to look at solutions of the BEq with (7) with u → β atspatial infinity, with β >
0. Equivalently, we want solutions of the potential BEq (4) forwhich f ∼ βx + γ + as x → ∞ and f ∼ βx + γ − as x → −∞ , where γ ± are constants. Weobtain such solutions by applying the BT to the starting solution f = βx . Applying the BTonce gives new solutions f = βx − y x y where y = C e λ x + λ t + C e λ x + λ t + C e λ x + λ t . (23)Here λ , λ , λ are the three roots of the cubic equation λ = 3 βλ + θ , and C , C , C areconstants, not all zero, which can be jointly rescaled without changing the solution. Thereare two main situations to look at: the case θ < β when λ , λ , λ are all real and distinct,and the case θ > β when one is real and the other two are a complex conjugate pair.8Note that the first situation can only happen if β > C , C , C are zero is trivial. If one is zero, say C ,then the new solution is f = βx + c − p tanh ( p ( x − ct ) + α )or f = βx + c − p coth ( p ( x − ct ) + α )where c = − ( λ + λ ) = λ , p = ( λ − λ ) and α is an arbitrary constant. The correspondingsolutions of the B. equation are u = β − p sech ( p ( x − ct ) + α ) , u = β + p csch ( p ( x − ct ) + α ) . These are the standard soliton and singular soliton solutions. A direct calculation confirmsthat these are solutions provided c p β . (24)From this we again deduce the need for β to be positive, and obtain bounds on both thevelocity c and the amplitude parameter p for fixed β . Note that there are solutions withboth positive and negative velocity, but that the solutions do not depend on the sign of p .Note also that for solitons u < β and for singular solitons u > β .Proceeding to the case where all three constants C , C , C are nonzero, we need todistinguish between the case that all have the same sign, in which case the solution will benonsingular, and the case that there are differing signs, in which case there is singularity.We start with the former. Looking at the expression for y in (23), at a given time t andposition x we will “see” a soliton if two of the terms balance and are much bigger thanthe third term. So for example, we will see a soliton determined by the first two terms atposition x ≈ λ − λ log (cid:18) C C (cid:19) − ( λ + λ ) t (this is obtained from balance between the first two terms), provided t ( λ − λ )( λ − λ ) (cid:28) K where K is some constant (this being the condition that at the given x , the first two termsare much bigger than the third one). Clearly a critical role is played by the sign of theproduct ( λ − λ )( λ − λ ). If this is positive we “see” a soliton determined by the first two9igure 2: The merging soliton. Parameter values β = 5 and θ = −
10, so λ , λ , λ ≈− . , . , .
48. The constants C , C , C are all taken to be 1. Plots of u = f x (with f given by (23)), displayed for times t = − , − . , − . , − . , . , . , .
8. Note smallersolitons are faster (see Equation (24)).terms for large negative times, if it is negative we will see the soliton for large positive times.It is straightforward to check that this product is positive for two of the three possible pairsof terms in y and negative for the other pair. Thus the solutions describe the merger oftwo solitary waves into a single one. Furthermore, if we choose, without loss of generality, λ < λ < λ , then the incoming solitary waves are those of velocity λ and λ , one of whichis negative and one positive, and the outgoing one has velocity λ . Since λ + λ + λ = 0there is a “law of conservation of speed”. See Figure 2 (compare with Figures 3.1 and 5.1 in[29] and Figure 11 in [7]). In this plot, as in all subsequent plots in this section, u is plottedas a function of x . We call this solution a “merging soliton”.Similar considerations can be applied in the case that all three constants C , C , C in(23) are nonzero, but their signs differ. There will be one pair with the same sign and twopairs with opposite signs. The solutions can describe the absorption of a standard soliton bya singular soliton, see Figure 3, or the merger of two singular solitons to a standard soliton,see Figures 4 and 5 (compare Figure 12 in [7]).In summary, we have obtained 5 types of solution by a single application of the BTto the starting solution f = βx : standard solitons, singular solitons, a merging soliton, asingular soliton absorbing a soliton, and the merger of a pair of singular solitons to a singlesoliton. We refer to these solutions collectively as “1 BT solutions”.10igure 3: Absorption of a soliton by singular soliton. Parameters and times identical toFigure 2 except C = − C = −
1. 11igure 5: Singular soliton merger, more detail. Times t = − . , − . , − . , − . f = βx − y y xx − y y xx y y x − y y x . (25)We have not succeeded to give a complete (analytic) classification of these solutions, but wereport cases in which we have found superpositions without singularities: • For certain parameter values, a pair of standard soliton solutions can be superposedto give a colliding 2-soliton solution. The 2 solitons should be taken with velocities ofdiffering signs; this is a necessary, but not sufficient, condition for such a superpositionto be possible. • For certain parameter values, a standard soliton and a singular soliton can be super-posed to give a 2-soliton solution. The resulting solutions include both colliding pairsand pairs moving in the same direction. • For certain parameter values, a standard soliton solution can be superposed with oneof the two types of singular solution describing a merger, to give a solution with threesolitary waves merging to two. See Figure 7, in which the soliton with parameters θ = 8 , C = 1 , C = 1 , C = 0 is superposed with the 1 BT solution with θ = − , C = − , C = 1 , C = 1 (for β = 5). In the cases of this that we have found,the initial configuration always has two solitary waves moving in one direction, andthe other in the opposite direction, but the final configuration can have two movingin one direction, or one in each direction. We have not found cases of mergers of 3moving in the same direction to 2, but we cannot currently exclude this possibility.So far we have not found any cases of the merger of 4 solitary waves to 2, though we cannotcurrently exclude this possibility. The superpositions of a pair of solutions describing a12igure 6: 3 solitary waves merge to 2. Superposition of two solutions of type (23) with θ = 8 , C = 1 , C = 1 , C = 0 and θ = − , C = − , C = 1 , C = 1, for β = 5. Plots of u against x for times t = − . , − . , − . , , . , . , . , . f = βx − θ y ( y y x − y y x ) + θ y ( y y x − y y x ) + θ y ( y y x − y y x ) y ( y x y xx − y x y xx ) + y ( y x y xx − y x y xx ) + y ( y x y xx − y x y xx ) . (26)Once again, we do not have a full analytic classification of solutions, but numerical experi-ments indicate that this is simpler than for superpositions of 2 solutions. 3-soliton solutionsare obtained from (certain) superpositions of 2 standard solitons and 1 singular soliton.Solutions describing the merger of 4 waves to 3 are obtained from (certain) superpositionsof a standard soliton, a singular soliton, and a merging soliton. See Figure 7 for an example.We have not succeeded in obtaining nonsingular solutions from a superposition usingmore than one merger-type solution. However there are some remarkable singular solutions.In Figure 8 we present plots of the superposition of three merger-type solutions, describingthe evolution of 6 solitary waves into 3, via a brief, finite time duration singularity. Thesingularity forms after t = − . t = 0 . , n applications of13igure 7: 4 waves merge to 3. Superposition of three solutions of type (23) with θ = − , C = 1 , C = 1 , C = 0 (a soliton) θ = − , C = 1 , C = − , C = 0 (a singular soliton)and θ = − , C = 1 , C = 1 , C = 1 (a merging soliton) for β = 5. Plots of u against x fortimes t = − . , − . , − . , , . , . , . , f = βx should be f = βx − W x W (27)where W is the Wronskian W = det y y . . . y n y (cid:48) y (cid:48) . . . y (cid:48) n ... ... ... y ( n − y ( n − . . . y ( n − n (28)and each of the functions y i are of the form in (23), i.e. y i is a general solution of the differ-ential equation y (cid:48)(cid:48)(cid:48) i = 3 βy (cid:48) i + θ i y i . (In this paragraph we use primes to denote differentiationwith respect to x .) We prove this as follows. Assuming f = βx − W (cid:48) W , f = βx − W (cid:48) W , f = βx − W (cid:48) W , f = βx − W (cid:48) W , substituting in (14), simplifying and integrating once gives therequirement W W = K ( W W (cid:48) − W W (cid:48) )where K is an arbitrary constant (note that each of the W ’s is only defined up to anoverall constant). Now if W, W , W , W all have the form of Wronskians, of dimensions n − , n − , n − , n respectively, with W being exactly the determinant of the matrix in(28), then 14igure 8: 6 solitary waves merge to 3 via a singularity. Superposition of three solu-tions of type (23) with θ = 2 , C = C = C = 1, θ = 9 , C = 1 , C = − , C = 1and θ = 15 , C = C = C = 1 for β = 5. Plots of u against x for times t = − , − , − . , − . , − . , , . , . , . , . , . , W is the determinant of the same matrix with the ( n − n ’th rows andcolumns deleted , • W is the determinant of the same matrix with the n ’th row and n ’th column deleted, • W is the determinant of the same matrix with the n ’th row and ( n − • W (cid:48) is the determinant of the same matrix with the ( n − n ’th columndeleted , • W (cid:48) is the determinant of the same matrix with the ( n − n − A is an arbitrary n × n matrix, C is the same matrix with the ( n − n ’th rows and columns deleted, B is the same matrix with the n ’th row and n ’th columndeleted, B is the same matrix with the ( n − n ’th column deleted, B is thesame matrix with the n ’th row and ( n − B is the same matrixwith the ( n − n − C det A = det B det B − det B det B . The general result on the form of the solution follows by induction.A similar result appeared for the bad BEq in [34], however taking each y i to be thesum of only two exponentials. In [21], Hirota gave the general solution of the bad BEqusing the “Hirota method” and it is interesting to see how this works for the good BEq.For this paragraph we work directlty with the BEq in the form (1), with β >
0. Writing U = − (log τ ) xx , the equation becomes τ τ tt − τ t − β (cid:0) τ τ xx − τ x (cid:1) + 13 (cid:0) τ τ xxxx − τ x τ xxx + 3 τ xx (cid:1) = 0which is in “Hirota bilinear form”. This has “multisoliton” solutions in the usual form τ = 1 + (cid:88) i c i e η i + (cid:88) i 16o guarantee that all these solutions are nonsingular requires φ ij > a i , a j , b i , b j (in addition to choosing the constants c i > 0) and that is not the casehere. Furthermore, it is possible to choose a i , a j , b i , b j such that φ ij = 0. A straightforwardcalculation shows that if this happens then b i + b i b j + b j = 3 β , implying that there is someconstant θ for which b i , b j are distinct solutions of the cubic equation b = 3 βb + θ . This isthe origin of the merging soliton solutions in the the Hirota framework. The remarkably simple method for finding conservation laws from a BT is very old, see forexample [40]. For the BEq, we simply need to observe that (11)-(12) implies s t + (2 u − s x − s ) x = 0 . Thus s , which depends on θ , provides a generating function for (densities of) conservationlaws. To obtain the standard conservation laws, observe that the solution s to (11) can bewritten as an asymptotic series in θ in the form s ∼ ∞ (cid:88) i = − θ − i/ s i . Each of the coefficients s i is the density for a conservation law. The first few coefficients aregiven as follows: s − = 1 , s = 0 , s = us − , s = − v + u x s − . Further terms can be computed using the recurrence relation s k +2 = 13 s − us k − k +1 (cid:88) j = − min( k +1 ,k − j +1) (cid:88) i =max( − , − j − s k − i − j s i s j − k +1 (cid:88) i = − s i s ( k − i ) x − s kxx , k ≥ . So for example s = 23 u xx + v x ,s = 13 s − (3 uv − u xxx − v xx ) ,s = 19 s − ( u xxxx + 3 v xxx − u + 3 uu xx − uv x − v − u x v ) . For each i = 1 , , . . . , s i is the density F of a conservation law F t + G x = 0. For i = 3 theconservation law is evidently trivial ( F = H x , G = − H t for some H ). Indeed we will shortly17how that all the conservation laws for i = 3 , , , . . . are trivial. The associated flux G isthe coefficient of θ − i/ in 2 u − s x − s . Thus for i = 1 , , , G = 1 s − ( u x + 2 v ) ,G = − s − (cid:18) u + v x + 13 u xx (cid:19) ,G = 1 s − (cid:18) u − uv x − uu xx − u x + u x v + v + 19 u xxxx (cid:19) ,G = 19 s − (cid:0) u v − u u x − uu xxx − uv xx − u xx v − vv x − u x u xx − u x v x +3 v xxxx + u xxxxx ) . We note there are 3 possible series for s , corresponding to the 3 possible choices of s − . The dependence of s , s , . . . on the choice of s − is clear, and can be verified to beconsistent with the recursion relation. We denote the three solutions of (11) with these threeasymptotic series by s (1) , s (2) , s (3) . If we define σ = s (1) + s (2) + s (3) then σ has asymptoticseries (cid:80) ∞ i =1 s i θ − i . However, if we define A = ( s (2) − s (3) ) s (1) x +( s (3) − s (1) ) s (2) x +( s (1) − s (2) ) s (3) x − ( s (1) − s (2) )( s (2) − s (3) )( s (3) − s (1) ) , (29)then it can be verified (using (11) for each of the functions s (1) , s (2) , s (3) ) that σ = s (1) + s (2) + s (3) = − (log A ) x . (30)It follows that s i is a total x derivative for all i , and the associated conservation laws aretrivial.The use of a BT to generate symmetries is rather newer [37]. The critical observationmade in [37] for the KdV, Sine Gordon and Camassa Holm equations, was that whileindividual BTs are not “small” transformation (and thus not directly related to symmetries,which are transformations of solutions that are infinitesimally close to the identity), thecomposition of two BTs can be small in this sense. For the BEq this is not the case, andit is necessary to consider the composition of 3 BTs. This has its origins in the fact thatthe Lax pair for the BEq is a 3 × f = f + ( θ − θ ) f + ( θ − θ ) f + ( θ − θ ) f ( f − f − f x + f x ) f + ( f − f − f x + f x ) f + ( f − f − f x + f x ) f = f − ( θ − θ ) s + ( θ − θ ) s + ( θ − θ ) s ( s − s ) s x + ( s − s ) s x + ( s − s ) s x − ( s − s )( s − s )( s − s ) . (31)18he critical observation is that as θ , θ tend to θ , the numerator of the second term becomessmall, but the denominator can remain large by taking s , s , s to be distinct solutions of(11)-(12). Thus, writing θ = θ, θ = θ + b(cid:15), θ = θ + a(cid:15) and taking the limit (cid:15) → 0, weobtain the following generator for infinitesimal symmetries acting on f (via f → f + (cid:15)Q f ( θ ): Q f ( θ ) = a ( s (1) − s (2) ) + b ( s (3) − s (1) )( s (2) − s (3) ) s (1) x + ( s (3) − s (1) ) s (2) x + ( s (1) − s (2) ) s (3) x − ( s (1) − s (2) )( s (2) − s (3) )( s (3) − s (1) ) . (32)Here s (1) , s (2) , s (3) are distinct solutions of (11)-(12). The generator for the field h can bewritten down but is long and complicated. The generator for w (see Section 2) takes asimpler form: Q w ( θ ) = as (3) ( s (2) − s (1) ) + bs (2) ( s (1) − s (3) )( s (2) − s (3) ) s (1) x + ( s (3) − s (1) ) s (2) x + ( s (1) − s (2) ) s (3) x − ( s (1) − s (2) )( s (2) − s (3) )( s (3) − s (1) ) . (33)The generators for the fields u and v are x -derivatives of the generators for f and w re-spectively. In computing these derivatives, it is useful to notice that the quantity in thedenominator Q f ( θ ) and Q w ( θ ) is the quantity A introduced above in the discussion of con-servation laws, see (29), which satisfies A x = − A ( s (1) + s (2) + s (3) ). Using the asymptoticexpansions for s (1) , s (2) , s (3) obtained in the discussion of conservation laws we obtain thefirst few local symmetries: X = ∂∂w ,X = ∂∂f ,X = u ∂∂f + v ∂∂w ,X = ( − v − u x ) ∂∂f + (cid:18) v x + 23 u xx − u (cid:19) ∂∂w (cid:18) = f t ∂∂f + w t ∂∂w (cid:19) ,X = 3(6 uu x + 12 uv − u xxx − v xx ) ∂∂f +(12 u − u xx u − uv x − u x + 18 v + 2 u xxxx + 3 v xxx ) ∂∂w ,X = (15 u − uu xx + 45 u x v + 45 v + u xxxx ) ∂∂f +(18 u u x + 45 vu − u xxx u − v xx u − u xx u x − u xx v − u x v x − vv x ) ∂∂w . Here we have taken, without loss of generality, a = 1 , b = 0. The index i on the vectorfield X i indicates that it is obtained from the coefficient of θ − i/ in the expansions of thegenerators. Note that the vector fields X , X , . . . vanish, in analog of the situation forconservation laws. 19he local symmetries listed above are generated from the (32)-(33) by expansion inpowers of θ . Using the identity (30) the full symmetry can be written (for the case a =1 , b = 0) in the form X = ( s (1) − s (2) ) e (cid:82) s (1) + s (2) + s (3) dx (cid:18) ∂∂f − s (3) ∂∂w (cid:19) . This is a symmetry provided s (1) , s (2) , s (3) are solutions of (11)-(12). Taking, for example, s (3) = s (1) we obtain the nonlocal symmetry X = ( s (1) − s (2) ) e (cid:82) s (1) + s (2) dx (cid:18) ∂∂f − s (1) ∂∂w (cid:19) . There are 6 distinct versions of this symmetry arising from permutations of s (1) , s (2) , s (3) .Nonlocal symmetries are useful as it is possible to construct invariant solutions with respectto nonlocal, as well as local, symmetries [27].Returning to local symmetries, it is straightforward to verify, using just (11), that thesymmetry generators Q f ( θ ) = ( s (1) − s (2) ) e (cid:82) s (1) + s (2) + s (3) dx , Q w ( θ ) = − s (3) Q f ( θ )satisfy the linear differential equations (cid:32) − D + Du + uD − D + D u + 2 Dv + vD D − uD + 2 vD + Dv D − ( uD + D u ) + u D + Du − ( v x D + Dv x ) (cid:33) (cid:32) Q w Q f (cid:33) = θ (cid:32) − DD (cid:33) (cid:32) Q w Q f (cid:33) . (Here D denotes differentiation with respect to x .) Denoting the matrix differential operatoron the LHS of this equation as P , and the one on the RHS as P , we have P − P (cid:32) Q w Q f (cid:33) = θ (cid:32) Q w Q f (cid:33) implying that the operator P − P can be identified as the recursion operator [35] for thepotential BEq. Since Q u ( θ ) = Q f ( θ ) x and Q v ( θ ) = Q w ( θ ) x , the operator P P − can beidentified as the recursion operator of the BEq. (Note that the recursion operator differsfrom the standard one for the BEq, as given, for example, in [36], as our form of the BEq(5)-(6) is slightly different.) 20 Conclusion The theme of this paper has been how the B¨acklund transformation, and particularly itssuperposition principles, can give so much insight into the properties of the Boussinesqequation. Specifically, we have obtained two systems of lattice equations associated withthe superposition principles, we have used the superposition principle to study the solitonsolutions of the equation, which have a rich structure that has not yet been full explored,and we have given a concise and complete account of the theory of conservation laws andsymmetries of the equation, using a generating function for symmetries derived immediatelyfrom the superposition principle of 3 BTs.The novelty in this work in the context of the theory of B¨acklund transformations, incomparison, say, to our recent work on the BT for the Camassa-Holm equation [38], is inthe need to look at the superposition principle for 3 BTs. For the BEq, the superpositionprinciple of 2 BTs is not purely algebraic, whereas for 3 BTs it is. We expect this structureto be shared by the many interesting equations associated with the Lie algebra SL (3) (i.e.with 3 × References [1] Ablowitz, M. J., and Satsuma, J. Solitons and rational solutions of nonlinearevolution equations. J. Math. Phys. 19 , 10 (1978), 2180–2186.212] Adler, V. E. B¨acklund transformation for the Krichever-Novikov equation. Internat.Math. Res. Notices , 1 (1998), 1–4.[3] Adler, V. E., Bobenko, A. I., and Suris, Y. B. Classification of integrableequations on quad-graphs. The consistency approach. Comm. Math. Phys. 233 , 3(2003), 513–543.[4] Ankiewicz, A., Bassom, A. P., Clarkson, P. A., and Dowie, E. Conservationlaws and integral relations for the Boussinesq equation. ArXiv e-prints (Nov. 2016).[5] Atkinson, J. B¨acklund transformations for integrable lattice equations. J. Phys. A41 , 13 (2008), 135202, 8.[6] Atkinson, J., Lobb, S. B., and Nijhoff, F. W. An integrable multicomponentquad-equation and its Lagrangian formulation. Theoret. and Math. Phys. 173 , 3 (2012),1644–1653. Russian version appears in Teoret. Mat. Fiz. 73 (2012), no. 3, 363–374.[7] Bogdanov, L. V., and Zakharov, V. E. The Boussinesq equation revisited. Phys.D 165 , 3-4 (2002), 137–162.[8] Boiti, M., and Pempinelli, F. Similarity solutions and B¨acklund transformationsof the Boussinesq equation. Nuovo Cimento B (11) 56 , 1 (1980), 148–156.[9] Boussinesq, J. Th´eorie de lintumescence liquide appel´ee onde solitaire ou de trans-lation se propageant dans un canal rectangulaire. Comptes Rendus Acad. Sci (Paris)72 (1871), 755–759.[10] Boussinesq, J. Th´eorie des ondes et des remous qui se propagent le long d’un canalrectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitessessensiblement pareilles de la surface au fond. J. Math. Pures Appl. (2) 17 (1872), 55–108.[11] Chen, H. H. Relation between B¨acklund transformations and inverse scattering prob-lems. In B¨acklund transformations, the inverse scattering method, solitons, and theirapplications (Workshop Contact Transformations, Vanderbilt Univ., Nashville, Tenn.,1974) . Springer, Berlin, 1976, pp. 241–252. Lecture Notes in Math., Vol. 515.[12] Clarkson, P. A. Rational solutions of the Boussinesq equation. Anal. Appl. (Singap.)6 , 4 (2008), 349–369.[13] Clarkson, P. A. Rational solutions of the classical Boussinesq system. NonlinearAnal. Real World Appl. 10 , 6 (2009), 3360–3371.2214] Clarkson, P. A., and Dowie, E. Rational solutions of the Boussinesq equationand applications to rogue waves. arXiv preprint arXiv:1609.00503 (2016).[15] Clarkson, P. A., and Kruskal, M. D. New similarity reductions of the Boussinesqequation. J. Math. Phys. 30 , 10 (1989), 2201–2213.[16] Gantmacher, F. R. The theory of matrices. Vols. 1, 2 . Translated by K. A. Hirsch.Chelsea Publishing Co., New York, 1959.[17] Hietarinta, J. Boussinesq-like multi-component lattice equations and multi-dimensional consistency. J. Phys. A 44 , 16 (2011), 165204, 22.[18] Hietarinta, J., and Zhang, D.-j. Multisoliton solutions to the lattice Boussinesqequation. J. Math. Phys. 51 , 3 (2010), 033505, 12.[19] Hietarinta, J., and Zhang, D.-j. Soliton taxonomy for a modification of the latticeBoussinesq equation. SIGMA Symmetry Integrability Geom. Methods Appl. 7 (2011),Paper 061, 14.[20] Hietarinta, J., and Zhang, D.-j. Hirota’s method and the search for integrablepartial difference equations. 1. Equations on a 3 × J. Difference Equ. Appl.19 , 8 (2013), 1292–1316.[21] Hirota, R. Exact N -soliton solutions of the wave equation of long waves in shallow-water and in nonlinear lattices. J. Mathematical Phys. 14 (1973), 810–814.[22] Hirota, R. Discrete analogue of a generalized Toda equation. J. Phys. Soc. Japan50 , 11 (1981), 3785–3791.[23] Hirota, R., and Satsuma, J. Nonlinear evolution equations generated from theB¨acklund transformation for the Boussinesq equation. Progr. Theoret. Phys. 57 , 3(1977), 797–807.[24] Huang, X. C. A two-parameter B¨acklund transformation for the Boussinesq equation. J. Phys. A 15 , 10 (1982), 3367–3372.[25] Kalantarov, V. K., and Ladyˇzenskaja, O. A. Formation of collapses in quasi-linear equations of parabolic and hyperbolic types. Zap. Nauˇcn. Sem. Leningrad. Otdel.Mat. Inst. Steklov. (LOMI) 69 (1977), 77–102, 274. Boundary value problems of math-ematical physics and related questions in the theory of functions, 10.2326] Levi, D., and Winternitz, P. Nonclassical symmetry reduction: example of theBoussinesq equation. J. Phys. A 22 , 15 (1989), 2915–2924.[27] Lou, S., Hu, X., and Chen, Y. Nonlocal symmetries related to B¨acklund transfor-mation and their applications. Journal of Physics A: Mathematical and Theoretical 45 ,15 (2012), 155209.[28] Lou, S. Y. A note on the new similarity reductions of the Boussinesq equation. Phys.Lett. A 151 , 3-4 (1990), 133–135.[29] Manoranjan, V. S., Mitchell, A. R., and Morris, J. L. Numerical solutionsof the good Boussinesq equation. SIAM J. Sci. Statist. Comput. 5 , 4 (1984), 946–957.[30] Manoranjan, V. S., Ortega, T., and Sanz-Serna, J. M. Soliton and antisolitoninteractions in the “good” Boussinesq equation. J. Math. Phys. 29 , 9 (1988), 1964–1968.[31] Maruno, K.-I., and Kajiwara, K. The discrete potential Boussinesq equation andits multisoliton solutions. Appl. Anal. 89 , 4 (2010), 593–609.[32] Nijhoff, F. W. Discrete Painlev´e equations and symmetry reduction on the lattice.In Discrete integrable geometry and physics (Vienna, 1996) , vol. 16 of Oxford LectureSer. Math. Appl. Oxford Univ. Press, New York, 1999, pp. 209–234.[33] Nijhoff, F. W., Papageorgiou, V. G., Capel, H. W., and Quispel, G. R. W. The lattice Gel (cid:48) fand-Diki˘ı hierarchy. Inverse Problems 8 , 4 (1992), 597–621.[34] Nimmo, J. J. C., and Freeman, N. C. A method of obtaining the N -soliton solutionof the Boussinesq equation in terms of a Wronskian. Phys. Lett. A 95 , 1 (1983), 4–6.[35] Olver, P. J. Evolution equations possessing infinitely many symmetries. J. Mathe-matical Phys. 18 , 6 (1977), 1212–1215.[36] Olver, P. J. Applications of Lie groups to differential equations , vol. 107 of GraduateTexts in Mathematics . Springer-Verlag, New York, 1986.[37] Rasin, A. G., and Schiff, J. The Gardner method for symmetries. J. Phys. A 46 ,15 (2013), 155202, 15.[38] Rasin, A. G., and Schiff, J. B¨acklund transformations for the Camassa–Holmequation. Journal of Nonlinear Science (2016), 1–25.2439] Tongas, A., and Nijhoff, F. The Boussinesq integrable system: compatible latticeand continuum structures. Glasg. Math. J. 47 , A (2005), 205–219.[40] Wadati, M., Sanuki, H., and Konno, K. Relationships among inverse method,B¨acklund transformation and an infinite number of conservation laws. Progr. Theoret.Phys. 53 (1975), 419–436.[41] Weiss, J. The Painlev´e property and B¨acklund transformations for the sequence ofBoussinesq equations. J. Math. Phys. 26 , 2 (1985), 258–269.[42] Xenitidis, P., and Nijhoff, F. Lattice Schwarzian Boussinesq equation and two-component systems. ArXiv e-prints (Feb. 2012).[43] Xenitidis, P., and Nijhoff, F. Symmetries and conservation laws of lattice Boussi-nesq equations. Phys. Lett. A 376 , 35 (2012), 2394–2401.[44] Zhang, Y., Chang, X., Hu, J., Hu, X., and Tam, H.-W. Integrable discretizationof soliton equations via bilinear method and B¨acklund transformation. Science ChinaMathematics 58 , 2 (2015), 279–296.[45] Zhang, Y., and Chen, D.-y. A modified B¨acklund transformation and multi-solitonsolution for the Boussinesq equation.