(2+1) -dimensional AKNS( −N ) Systems: N=3,4
aa r X i v : . [ n li n . S I] O c t (2 + 1)-dimensional AKNS( − N ) Systems: N = 3 , Metin G¨urses ∗ Department of Mathematics, Faculty of ScienceBilkent University, 06800 Ankara - Turkey
Aslı Pekcan † Department of Mathematics, Faculty of ScienceHacettepe University, 06800 Ankara - Turkey
Abstract
In this work we continue to study negative AKNS( N ) that is AKNS( − N ) system for N = 3 , The AKNS hierarchy [1] is cu t N = R N u x where u = (cid:18) pq (cid:19) i . e . (cid:18) p t N q t N (cid:19) = R N (cid:18) p x q x (cid:19) , (1)for N = 0 , , , · · · , where R is the recursion operator, R = (cid:18) − pD − q + D − pD − pqD − q qD − p − D (cid:19) (2)and c is an arbitrary constant. Here D is the total x -derivative and D − = R x is the standard anti-derivative. In the last decade there have been considerable amount of works on nonlocal reductionsof this system. Nonlinear Schr¨odinger (NLS) system ( N = 1) and its reductions [2]-[17], multidimen-sional versions of NLS and their reductions [18]-[22], modified KdV (mKdV) system ( N = 2) and itsreductions [5], [13], [15]-[17], [23]-[27] have attracted many researchers to obtain interesting solitonsolutions by using inverse scattering method, Darboux transformations, and Hirota bilinear method.There are also some interests on higher members of the system (1) for N ≥ N ≥
3. Similardifficulties arise also for negative AKNS hierarchy in (2 + 1)-dimensions [41].Letting p = gf , q = hf , and t N → − t N N then the coupled partial differential equations for N = 0 , , cD t N + D N +1 x ) { g · f } = 0 , (3)( cD t N + ( − N D N +1 x ) { h · f } = 0 , (4) D x { f · f } = − gh. (5) ∗ [email protected] † Email:[email protected]
1n our previous work [41] (see also [42]) we started to investigate (2 + 1)-dimensional negative AKNSsystem, namely AKNS( − N ) system defined as R ( u t N ) − a R N ( u x ) = b u y for N = 0 , , , · · · , (6)Here a and b are any constants. In [41] we have considered only the cases for N ≤
2, and constructedthe Hirota bilinear forms only for these negative AKNS systems. We obtained soliton solutions, andfound all local and nonlocal reductions of these systems. Similar to AKNS( N ) systems by letting p = gf , q = hf , ghf = − (cid:18) f x f (cid:19) x , (7)we obtain the Hirota bilinear form of (2 + 1)-dimensional AKNS( − N ) system as [41]( bD y − D x D t N + a N D N +1 x ) { g · f } = 0 , (8)( bD y + 12 D x D t N + ( − N a N D N +1 x ) { h · f } = 0 , (9) D x { f · f } = − gh. (10)for N = 0 , , N ) system are simpler and can be written in anice form for N = 0 , ,
2. Due to this property of the Hirota bilinear forms for AKNS( N ) systemfor N = 0 , , N = 0 , ,
2. This simplicity is valid only for N = 0 , ,
2. For N ≥ N ) system for N = 3 , , , N ) system for N ≥ N = 3 and N = 4. Hence to constructthe Hirota bilinear forms of the systems AKNS(-3) and AKNS(-4) are now practically possible.In this work we construct first the Hirota bilinear forms of AKNS(-3) and AKNS(-4) systems.In Section 3 we obtain one-soliton solutions of these two systems. We present local and nonlocalreductions of these negative AKNS systems in Sections 4 and 5 respectively. In Section 6 we giveone-soliton solutions of these reduced nonlocal equations. (2 + 1) -dimensional AKNS(-N) systemsfor N = 3 , In this section we study the (2 + 1)-dimensional AKNS(-3) and AKNS(-4) systems and obtain theHirota bilinear forms of them. For N = 3 the system (6) gives the (2 + 1)-dimensional AKNS(-3)system as bp y = 12 p xt − a p xxxx + 3 a qp x + a pp x q x + apqp xx + a p q xx − a p q − pD − ( pq ) t , (11) bq y = − q xt + a q xxxx − a pq x − a qq x p x − aqpq xx − a q p xx + 3 a p q + qD − ( pq ) t . (12)2f we use (7) in the above system, we obtain the Hirota bilinear form of the (2 + 1)-dimensionalAKNS(-3) system as ( bD y − D x D t + a D x ) { g · f } = − ahs, (13)( bD y + 12 D x D t − a D x ) { h · f } = 38 agτ, (14) D x { f · f } = − gh, (15) D x { g · g } = f s, (16) D x { h · h } = f τ, (17)where s and τ are auxiliary functions.For N = 4 the system (6) gives the (2 + 1)-dimensional AKNS(-4) system as bp y = 12 p xt − a p x + 5 a pqp xxx + 5 a pp x q xx − a p q p x + 5 a pp xx q x + 5 a qp x p xx + 5 a p x q x − pD − ( pq ) t (18) bq y = − q xt − a q x + 5 a qpq xxx + 5 a qq x p xx − a q p q x + 5 a qq xx p x + 5 a pq x q xx + 5 a q x p x + qD − ( pq ) t . (19)By using (7) in the system (18) and (19) we get the Hirota bilinear form of the (2 + 1)-dimensionalAKNS(-4) system as ( bD y − D x D t + a D x ) { g · f } = 516 aD x { h · s } , (20)( bD y + 12 D x D t + a D x ) { h · f } = 516 aD x { g · τ } , (21) D x { f · f } = − gh, (22) D x { g · g } = f s, (23) D x { h · h } = f τ, (24)where s and τ are auxiliary functions. (2 + 1) -dimensional AKNS(-N) systems for N = 3 , To obtain one-soliton solutions of AKNS(-3) and AKNS(-4) we take g = εg , h = εh , f = 1 + ε f , s = ε s , and τ = ε τ , where g = e θ , h = e θ (25)for θ i = k i x + ρ i y + ω i t + δ i , i = 1 ,
2. For simplicity, we use t for both t and t . We insert these expansionsin the Hirota bilinear forms of AKNS(-3) and AKNS(-4) systems and consider the coefficients of ε n , n = 1 , , ,
4. The coefficients of ε give the dispersion relations as ρ = 1 b ( 12 k ω − a k ) , ρ = 1 b ( − k ω + a k ) (26)3or AKNS(-3) system, and ρ = 1 b ( 12 k ω − a k ) , ρ = 1 b ( − k ω − a k ) (27)for AKNS(-4) system. From the coefficients of ε we have f = − e θ + θ ( k + k ) , (28)and s = τ = 0. The coefficients of ε and ε vanish directly.Let us take ε = 1. Hence one-soliton solutions of the AKNS(-3) system given by (11) and (12) and theAKNS(-4) system given by (18) and (19) are ( p ( x, y, t ) , q ( x, y, t )) where p ( x, y, t ) = e θ − e θ θ ( k + k ) , q ( x, y, t ) = e θ − e θ θ ( k + k ) (29)for θ i = k i x + ρ i y + ω i t + δ i , i = 1 ,
2, where the dispersion relations are given in (26) and (27),respectively. Here k i , δ i , i = 1 , − N ) systems for N = 3 , t ≥ x ∈ R . Before presenting the soliton solutions of thenonlocal (and local) reduced (2 + 1)-dimensional AKNS( − N ) equations for N = 3 ,
4, we will first givelocal and nonlocal consistent reductions. (2 + 1) -dimensional AKNS( − N ) systems for N = 3 , In this section we present local reductions of (2 + 1)-dimensional AKNS(-3) and AKNS(-4) systems.We first obtain the constraints to have these local reductions consistently. Then we give reduced localequations under these constraints. Note that both AKNS(-3) and AKNS(-4) systems do not possessthe local reduction q ( x, y, t ) = kp ( x, y, t ), k is a real constant. q ( x, y, t ) = k ¯ p ( x, y, t ) , k is a real constant Under this reduction the AKNS(-3) system (11) and (12) reduces to the local complex AKNS(-3)equation bp y = 12 p xt − a p xxxx + 3 a k ¯ pp x + a kp | p x | + ak | p | p xx + a kp ¯ p xx − a k p | p | − kpD − ( | p | ) t , (30)where ¯ b = − b , ¯ a = a , consistently.When we apply this reduction to the AKNS(-4) system (18) and (19) the system consistently reducesto the local complex AKNS(-4) equation bp y = 12 p xt − a p x + 5 a kpp x ¯ p xx − a k | p | p x + 5 a kp ¯ p x p xx + 5 a k ¯ pp x p xx + 5 a k | p x | p x − kpD − ( | p | ) t , (31)where ¯ b = − b , ¯ a = − a . 4 Nonlocal reductions of (2 + 1) -dimensional AKNS( − N ) systems for N = 3 , In order to have consistent nonlocal reductions we use the following representation for D − [41] D − F = 12 (cid:18)Z x −∞ − Z ∞ x (cid:19) F ( x ′ , y, t ) dx ′ . (32)We define ξ ( x, y, t ) which is invariant under the discrete transformations x → ε x , y → ε y, and t → ε t as ξ ( x, y, t ) = D − p p ε ≡ (cid:18)Z x −∞ − Z ∞ x (cid:19) p ( x ′ , y, t ) p ( ε x ′ , ε y, ε t ) dx ′ , (33)where ε j = 1 , j = 1 , ,
3. It is clear that ξ ( ε x, ε y, ε t ) = ε ξ ( x, y, t ) . (34)In this following part we present nonlocal reductions of (2+ 1)-dimensional AKNS(-3) and AKNS(-4)systems. We obtain the conditions on the parameters to have these nonlocal reductions consistently.Then we give reduced nonlocal equations under these conditions. q ( x, y, t ) = kp ( ε x, ε y, ε t ) , ε = ε = ε = 1 , k is a real constant Under this reduction the AKNS(-3) system given by (11) and (12) reduces consistently to the followingnonlocal AKNS(-3) equation: bp y = 12 p tx − a p xxxx + 3 a kp ε p x + a kpp x p εx + akpp ε p xx + a kp p εxx − a k p ( p ε ) − kpD − ( pp ǫ ) t , (35)where k is any real constant. Here p ǫ = p ( ǫ x, ǫ y, ǫ t ). The above equation is valid only when ǫ = − ǫ ǫ = 1. Therefore we have only p ǫ = p ( x, − y, t ) and p ǫ = p ( − x, − y, − t ) cases as for the AKNS(-1) system [41].When we apply this reduction to the AKNS(-4) system (18) and (19), it reduces consistently to thefollowing nonlocal AKNS(-4) equation: bp y = 12 p tx − a p x + 5 a kpp ε p xxx + 5 a kpp x p εxx − a k p ( p ε ) p x + 5 a kpp xx p εx + 5 a kp ε p x p xx + 5 a kp x p εx − kpD − ( pp ǫ ) t , (36)where k is any real constant and p ǫ = p ( ǫ x, ǫ y, ǫ t ). This equation is valid only when ǫ = − ǫ ǫ = 1. Here we have also two possible cases; p ǫ = p ( x, y, − t ) and p ǫ = p ( − x, − y, − t ) as for theAKNS(0) and AKNS(-2) systems [41]. q ( x, y, t ) = k ¯ p ( ε x, ε y, ε t ) , ε = ε = ε = 1 , k is a real constant When we apply this reduction to the AKNS(-3) system (11) and (12), it reduces consistently to thenonlocal complex AKNS(-3) equation, bp y = 12 p tx − a p xxxx + 3 a k ¯ p ε p x + a kpp x ¯ p εx + akp ¯ p ε p xx + a kp ¯ p εxx − a k p (¯ p ε ) − kpD − ( p ¯ p ǫ ) t , (37)5here k is any real constant, if the following constraints hold: ǫ ǫ ǫ ¯ b = − b, ǫ ǫ ¯ a = a. (38)Therefore we have seven different time and space reversals as in the AKNS(-1) case [41]:(i) p ǫ ( x, y, t ) = p ( − x, y, t ), where a is pure imaginary and b is real.(ii) p ǫ ( x, y, t ) = p ( x, − y, t ), where a and b are real.(iii) p ǫ ( x, y, t ) = p ( x, y, − t ), where a is pure imaginary and b are real.(iv) p ǫ ( x, y, t ) = p ( − x, − y, t ), where a and b are pure imaginary.(v) p ǫ ( x, y, t ) = p ( − x, y, − t ), where a is real and b is pure imaginary.(vi) p ǫ ( x, y, t ) = p ( x, − y, − t ), where a and b are pure imaginary.(vii) p ǫ ( x, y, t ) = p ( − x, − y, − t ), where a and b are real.Each case (i)-(vii) gives a (2 + 1)-dimensional nonlocal equation in the form of (37).Under this reduction the AKNS(-4) system (11) and (12) reduces consistently to the nonlocal complexAKNS(-4) equation, bp y = 12 p tx − a p x + 5 a kp ¯ p ε p xxx + 5 a kpp x ¯ p εxx − a k p (¯ p ε ) p x + 5 a kpp xx ¯ p εx + 5 a k ¯ p ε p x p xx + 5 a kp x ¯ p εx − kpD − ( p ¯ p ǫ ) t , (39)where k is any real constant, if the following constraints hold: ǫ ǫ ǫ ¯ b = − b, ǫ ¯ a = − a. (40)Therefore we have seven different time and space reversals as in the AKNS(0) and AKNS(-2) cases[41]:(i) p ǫ ( x, y, t ) = p ( − x, y, t ), where a is pure imaginary and b is real.(ii) p ǫ ( x, y, t ) = p ( x, − y, t ), where a is pure imaginary and b is real.(iii) p ǫ ( x, y, t ) = p ( x, y, − t ), where a and b are real.(iv) p ǫ ( x, y, t ) = p ( − x, − y, t ), where a and b are pure imaginary.(v) p ǫ ( x, y, t ) = p ( − x, y, − t ), where a is real and b is pure imaginary.(vi) p ǫ ( x, y, t ) = p ( x, − y, − t ), where a is real and b is pure imaginary.(vii) p ǫ ( x, y, t ) = p ( − x, − y, − t ), where a and b are real.Each case (i)-(vii) gives a (2 + 1)-dimensional nonlocal equation in the form of (39). In this part we first give the constraints on the parameters of one-soliton solutions of the reduced localand nonlocal equations. Then we present one-soliton solutions obeying these constraints.
Let us use the local reduction formula q ( x, y, t ) = k ¯ p ( x, y, t ) with Type 1 approach to obtain theconstraints on the parameters of one-soliton solutions of the local equations (30) and (31). We have e k x + ρ y + ω t + δ Ae ( k + k ) x +( ρ + ρ ) y +( ω + ω ) t + δ + δ = ke ¯ k x +¯ ρ y +¯ ω t +¯ δ Ae (¯ k +¯ k ) x +(¯ ρ +¯ ρ ) y +(¯ ω +¯ ω ) t +¯ δ +¯ δ . (41)6ere the constraints are obtained as1) k = ¯ k , ω = ¯ ω , e δ = ke ¯ δ , (42)for both (2 + 1)-dimensional local reduced complex AKNS(-3) and AKNS(-4) equations. Under theseconstraints, we get the relation ρ = ¯ ρ directly. Therefore one-soliton solutions of (30) and (31) aregiven by p ( x, y, t ) = e k x + ρ y + ω t + δ − k ( k +¯ k ) e ( k +¯ k ) x +( ρ +¯ ρ ) y +( ω +¯ ω ) t + δ +¯ δ . (43)Here for N = 3, a is a real and b is a pure imaginary number, and ρ = b ( k ω − a k ). For N = 4, a and b are pure imaginary numbers, and ρ = b ( k ω − a k ).Let k = − ( k + ¯ k ) e α < α is a real constant. Then the solution (43) can be written as p ( x, y, t ) = e ψ θ ) , (44)where θ = 12 [( k + ¯ k ) x + ( ρ + ¯ ρ ) y + ( ω + ¯ ω ) t + δ + ¯ δ + α ] , (45) ψ = 12 [( k − ¯ k ) x + ( ρ − ¯ ρ ) y + ( ω − ¯ ω ) t + δ − ¯ δ − α ] . (46)The real-valued solution | p ( x, y, t ) | is | p ( x, y, t ) | = e − α ( θ ) . (47)Hence for k <
0, one-soliton solutions of the local reduced complex equations (30) and (31) arenonsingular and bounded.
We have two types of nonlocal reductions for (2 + 1)-dimensional AKNS(-3) and AKNS(-4) systems.We consider each case separately. a) q ( x, y, t ) = kp ( ǫ x, ǫ y, ǫ t ), k is a real constant. From that relation, when we use the Type 1approach we obtain the constraints on the parameters of the one-soliton solutions as1) k = ε k , ω = ε ω , e δ = ke δ . (48)Under these conditions, the relation ρ = ǫ ρ holds directly.For N = 3, we have the case when ( ε , ε , ε ) = (1 , − , p ( x, y, t ) = e k x + ρ y + ω t + δ − k k e k x +2 ω t +2 δ , (49)where ρ = b ( k ω − a k ). Let us take the parameters k , ω , δ , a , and b real. Hence ρ is also real.Now let k = − k e α < α is a real constant then the solution (49) can be written as p ( x, y, t ) = e ψ e θ +2 α , (50)7here ψ = k x + ρ y + ω t + δ , (51) θ = k x + ρ t + δ . (52)The solution (50) can be rewritten as p ( x, y, t ) = e ρ y − α θ + α ) . (53)Hence for k <
0, one-soliton solution of the (2 + 1)-dimensional reduced nonlocal AKNS(-3) equation(35) is nonsingular for all ( x, y, t ) but unbounded.For N = 4, we have the case when ( ε , ε , ε ) = (1 , , − p ( x, y, t ) = e k x + ρ y + ω t + δ − k k , e k x +2 ρ y +2 δ , (54)where ρ = b ( k ω − a k ). Let us assume that all the parameters that k , ω , δ , a , and b are real.In that case ρ is also real. Let now k = − k e α < α is a real constant then the solution (55)becomes p ( x, y, t ) = e ψ e θ +2 α , (55)where ψ = k x + ρ y + ω t + δ , (56) θ = k x + ρ y + δ . (57)The solution (55) can be rewritten as p ( x, y, t ) = e ω t − α θ + α ) . (58)Therefore for k <
0, one-soliton solution is bounded for all t ≥ ω ≤
0, and nonsingular for all( x, y, t ).For both nonlocal reduced AKNS(-3) and AKNS(-4) equations we also have the case when ( ǫ , ǫ , ǫ ) =( − , − , − p ( x, y, t ) = 0.Therefore here we use Type 2 approach [23], [15], which is based on the cross multiplication. Fromthe reduction formula we have e θ Ae θ + θ = k e θ − Ae θ − + θ − , (59)where θ j = k j x + ρ j y + ω j t + δ j , θ − j = − k j x − ρ j y − ω j t + δ j , j = 1 , . By the cross multiplication we get e θ + Ae δ e θ − = ke θ − + Ake δ e θ , (60)yielding the conditions Ake δ = 1 , Ae δ = k. (61)8ence we get e δ = σ i ( k + k ) √ k , e δ = σ i √ k ( k + k ) , σ j = ± , j = 1 , . (62)Therefore one-soliton solutions of the nonlocal equations (35) and (36) for p ε = p ( − x, − y, − t ) aregiven by p ( x, y, t ) = iσ e k x + ρ y + ω t ( k + k ) √ k (1 + σ σ e ( k + k ) x +( ρ + ρ ) y +( ω + ω ) t ) , σ j = ± , j = 1 , , (63)where ρ and ρ satisfy the dispersion relation (26) for N = 3 and the relation (27) for N = 4. Wecan rewrite the solution (63) as p ( x, y, t ) = e ψ + δ θ ) , (64)where ψ = 12 [( k − k ) x + ( ρ − ρ ) y + ( ω − ω ) t ] , (65) θ = 12 [( k + k ) x + ( ρ + ρ ) y + ( ω + ω ) t ] . (66)Let all the parameters be real. Then the solution (64) for nonlocal reduced AKNS(-3) equation (35)is nonsingular and bounded if k = k , ρ = ρ yielding ω = a k − ω and ω ≤ ω for t ≥
0. Thesolution (64) for nonlocal reduced AKNS(-4) equation (36) is nonsingular and bounded if k = k , ρ = ρ yielding ω = − ω and ω ≤ t ≥ b) q ( x, y, t ) = k ¯ p ( ǫ x, ǫ y, ǫ t ), k is a real constant. When we use Type 1, we obtain the constraintsfor the parameters of one-soliton solutions of the nonlocal reduced complex AKNS(-3) and AKNS(-4)equations as 1) k = ε ¯ k , ω = ε ¯ ω , e δ = ke ¯ δ , (67)which are same as in the cases of AKNS(0), AKNS(-1), and AKNS(-2) systems [41]. The above relationsyield ρ = ε ¯ ρ directly. Hence one-soliton solutions of the nonlocal reduced complex AKNS(-3) andAKNS(-4) equations given by (37) and (39) respectively, are p ( x, y, t ) = e k x + ρ y + ω t + δ − k ( k + ε ¯ k ) e ( k + ε ¯ k ) x +( ρ + ε ¯ ρ ) y +( ω + ε ¯ ω ) t + δ +¯ δ , (68)with the corresponding dispersion relations.One can find nonsingular and bounded solutions for different choices of ( ε , ε , ε ). There are 14nonlocal complex reduced equations obtained by the reduction q ( x, y, t ) = k ¯ p ( ǫ x, ǫ y, ǫ t ). Let usconsider x -reversal case that is when ( ε , ε , ε ) = ( − , , k = α + iβ , τ = α + iβ , ω = α + iβ , and e δ = α + iβ , where α j , β j ∈ R for j = 1 , , ,
4. Then the solution (68)for both nonlocal reduced complex AKNS(-3) and AKNS(-4) equations given by (37) and (39) for( ε , ε , ε ) = ( − , ,
1) becomes p ( x, y, t ) = e ( α + iβ ) x +( α + iβ ) y +( α + iβ ) t ( α + iβ )1 + k ( α + β )4 β e iβ x +2 α y +2 α t (69)so | p ( x, y, t ) | = e α x +2 α y +2 α t ( k ( α + β )4 β e α y +2 α t + cos(2 β x )) + sin (2 β x ) . (70)For x = nπ β and k ( α + β )4 β e α y +2 α t + ( − n = 0, n integer, the above solution is singular.9 Conclusion
In this work we constructed the Hirota bilinear forms of the (2 + 1)-dimensional systems AKNS(-3)and AKNS(-4) and found one-soliton solutions of these systems. We then found all possible local andnonlocal reductions of these systems of equations. By using these solutions with the reduction formulaswe presented one-soliton solutions of all reduced nonlocal equations.
This work is partially supported by the Scientific and Technological Research Council of Turkey(T ¨UB˙ITAK).