(2+1) -Dimensional Local and Nonlocal Reductions of the Negative AKNS System: Soliton Solutions
aa r X i v : . [ n li n . S I] A ug (2 + 1)-Dimensional Local and Nonlocal Reductions ofthe Negative AKNS System: Soliton Solutions 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
We first construct a (2 + 1)-dimensional negative AKNS hierarchy and then we giveall possible local and (discrete) nonlocal reductions of these equations. We find Hirotabilinear forms of the negative AKNS hierarchy and give one- and two-soliton solutions.By using the soliton solutions of the negative AKNS hierarchy we find one-soliton so-lutions of the local and nonlocal reduced equations.
Keywords.
Ablowitz-Musslimani reduction, (2 + 1)-dimensional negative AKNS hier-archy, Hirota bilinear method, Soliton solutions
Let R be the recursion operator of an integrable equation. Then the integrable hierarchy ofequations are defined as v t n = R n v x n = 0 , , , . . . . (1.1)In [1], we proposed a system of equations R [ v t n − a R n σ ] = bσ , n = 0 , , , . . . , (1.2)where σ , σ are some classical symmetries of the same integrable equation. This hierarchyrepresents the negative hierarchy of the integrable system defined in (1.1). For some specificchoices of the constants a, b, and σ , σ we have studied the existence of three-soliton solutionsand Painlev´e property of the KdV equation where the recursion operator is R = D + 8 v + ∗ [email protected] † Email:[email protected] v x D − . The equation (1.2) becomes the KdV(6) equation when a = − , b = 0 , n = 1 andby letting v = u x to get rid of nonlocal terms containing D − . We have also obtained (2 + 1)-extension of this equation, the (2 + 1)-KdV(6) equation by choosing a = − , b = − , n = 1,and σ = v x , σ = v y . The expanded form with v = u x of is the (2 + 1)-KdV(6) equation [2] u xxxt + u xxxxxx + 40 u xx u xxx + 20 u x u xxxx + 8 u x u xt + 120 u x u xx + 4 u t u xx + u xy = 0 . (1.3)We showed that (2 + 1)-KdV(6) equation possesses three-soliton solution having the samestructure with the KdV equation’s three-soliton solution and also Painlev´e property.By using our approach (1.2), we obtain negative hierarchy of integrable equations whichare nonlocal in general. Here nonlocality is due to the existence of the terms containing theoperator D − . In the KdV case the nonlocal terms disappear by redefinition of the dynamicalvariable. This may not be possible for other integrable systems.A new type of nonlocal reductions are obtained by relating one of the dynamical variableto the time and space reflections of the other one which was first introduced by Ablowitzand Musslimani [3]-[5]. Ablowitz-Muslimani type of nonlocal reductions attracted many re-searchers [6]-[32] to investigate new nonlocal integrable equations and find their solitonicsolutions. These nonlocal integrable equations have been obtained by the nonlocal reduc-tions of the AKNS and other systems of equations. First example was the nonlocal nonlinearSchr¨odinger (NLS) equation and then nonlocal modified KdV (mKdV) equation. Ablowitzand Musslimani proposed later some other nonlocal integrable equations such as reversespace-time and reverse time nonlocal NLS equation, sine-Gordon equation, (1 + 1)- and(2+1)- dimensional three-wave interaction, Davey-Stewartson equation, derivative NLS equa-tion, ST-symmetric nonlocal complex mKdV and mKdV equations arising from symmetryreductions of general AKNS scattering problem [3]-[5]. They discussed Lax pairs, an infinitenumber of conservation laws, inverse scattering transforms and found one-soliton solutionsof these equations. Ma, Shen, and Zhu showed that ST-symmetric nonlocal complex mKdVequation is gauge equivalent to a spin-like model in Ref. [22]. Ji and Zhu obtained soliton,kink, anti-kink, complexiton, breather, rogue-wave solutions, and nonlocalized solutions withsingularities of ST-symmetric nonlocal mKdV equation through Darboux transformation andinverse scattering transform [23], [24]. In [25], the authors showed that many nonlocal in-tegrable equations like Davey-Stewartson equation, T-symmetric NLS equation, nonlocalderivative NLS equation, and ST-symmetric complex mKdV equation can be converted tolocal integrable equations by simple variable transformations.Recently we studied all possible nonlocal reductions of the AKNS system. We have obtainedone-, two-, and three-soliton solutions of the nonlocal NLS [31] and mKdV equations [32].We also studied nonlocal reductions of Fordy-Kulish [29] and super integrable systems [30],[33].In this work, by the use of the formula (1.2) we obtain negative AKNS hierarchy denotedby AKNS( − n ) for n = 0 , , , . . . with one time t and two space variables x and y . This2ew system constitutes one of the few examples of (2 + 1)-dimensional integrable systemof equations [6], [7]. All these systems are nonlocal due to the term D − in the recursionoperator. We obtain the Hirota bilinear form of these systems and obtain one- and two-soliton solutions for n = 0 , ,
2. We then find all possible local and nonlocal reductions ofthe negative AKNS hierarchy for n = 0 , ,
2. There are in total 30 reduced equations for n = 0 , ,
2. All these equations constitute new examples of (2 + 1)-dimensional integrablesystem of equations [6], [7]. There exists only one type of local reductions where the seconddynamical variable is related to the complex conjugation of the other variable. By the useof constraint equations we obtain one-soliton solutions of the local and nonlocal reducedequations from the one-soliton solutions of the negative AKNS system of equations. Thereare solutions which develop singularities in a finite time and there are also solutions whichare finite and bounded depending on the parameters of the one-soliton solutions.
The AKNS hierarchy [34] can be written as u t n = R n u x , ( n = 0 , , , . . . ) , 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) , where R is the recursion operator, R = (cid:18) − pD − q + D − pD − pqD − q qD − p − D (cid:19) . Here D is the total x -derivative and D − = R x (standard anti-derivative).Writing (1.2) in the following form R ( u t n ) − a R n ( u x ) = b u y for n = 0 , , . . . , (2.1)where u = (cid:18) pq (cid:19) , here a, b are any constants, we obtain (2 + 1)-dimensional negativeAKNS( − n ) systems for n = 0 , , (1) ( n = 0) (2 + 1) -AKNS(0) System: When n = 0, Eq. (2.1) reduces to R ( u t ) − au x = bu y . This yields the system bp y = 12 p tx − a p x − pD − ( pq ) t , (2.2) bq y = − q tx − a q x + qD − ( pq ) t . (2.3)3 ( n = 1) (2 + 1) -AKNS(-1) System: When n = 1, Eq. (2.1) reduces to R ( u t − au x ) = bu y . Letting u t − au x = ω , where ω = (cid:18) ω ω (cid:19) we have u t − au x = ω, R ω = bu y . This yields the system ω = p t − ap x ω = q t − aq x bp y = 12 ω ,x − pD − ( qω + pω ) bq y = − ω ,x + qD − ( qω + pω ) . (2.4)Inserting ω and ω we obtain the system bp y = 12 p tx − a p xx + ap q − pD − ( pq ) t , (2.5) bq y = − q tx + a q xx − ap q + qD − ( pq ) t . (2.6) (3) ( n = 2) (2 + 1) -AKNS(-2) System: When n = 2, Eq. (2.1) reduces to R ( u t − a R u x ) = bu y . Letting u t − a R u x = ω , where ω = (cid:18) ω ω (cid:19) we have u t − a R u x = ω, R ω = bu y . This yields the system ω = p t − a ( − p q + 12 p xx ) ω = q t − a ( pq − q xx ) bp y = 12 ω ,x − pD − ( qω + pω ) bq y = − ω ,x + qD − ( qω + pω ) . (2.7)Inserting ω and ω we obtain the system bp y = 12 p tx − a p xxx + 3 a p q p x − pD − ( pq ) t , (2.8) bq y = − q tx − a q xxx + 3 a p q q x + qD − ( pq ) t . (2.9)4 Hirota Method for Negative AKNS System
To obtain the Hirota bilinear form for the negative AKNS( − n ) system, with n = 0 , , and n = 2, we let p = gf , q = hf , (3.1)and ghf = − (cid:18) f x f (cid:19) x . (3.2) (1) ( n = 0) Hirota Bilinear Form for (2 + 1) -AKNS(0) System:
Using (3.1) and (3.2) in Eqs. (2.2) and (2.3) give b ( f g y − gf y ) = 12 ( f g tx − g t f x − g x f t + gf tx ) − a ( f g x − gf x ) , (3.3) b ( f h y − hf y ) = −
12 ( f h tx − h t f x − h x f t + hf tx ) − a ( f h x − hf x ) . (3.4)Hence we obtain the Hirota bilinear form as P ( D ) { g · f } ≡ ( bD y − D t D x + a D x ) { g · f } = 0 , (3.5) P ( D ) { h · f } ≡ ( bD y + 12 D t D x + a D x ) { h · f } = 0 , (3.6) P ( D ) { f · f } ≡ D x { f · f } = − gh. (3.7) (2) ( n = 1) Hirota Bilinear Form for (2 + 1) -AKNS(-1) System:
Using (3.1) and (3.2) in Eqs. (2.5) and (2.6) give b ( f g y − gf y ) = 12 ( f g tx − g t f x − g x f t + gf tx ) − a f g xx − f x g x + gf xx ) , (3.8) b ( f h y − hf y ) = −
12 ( f h tx − h t f x − h x f t + hf tx ) + a f h xx − f x h x + hf xx ) . (3.9)Hence we obtain the Hirota bilinear form as P ( D ) { g · f } ≡ ( bD y − D t D x + a D x ) { g · f } = 0 , (3.10) P ( D ) { h · f } ≡ ( bD y + 12 D t D x − a D x ) { h · f } = 0 , (3.11) P ( D ) { f · f } ≡ D x { f · f } = − gh. (3.12) (3) ( n = 2) Hirota Bilinear Form for (2 + 1) -AKNS(-2) System: b ( f g y − gf y ) = 2( f g tx − g t f x − g x f t + gf tx ) − a ( f g xxx + 3 f xx g x − g xx f x − gf xxx ) , (3.13)4 b ( f h y − hf y ) = − f h tx − h t f x − h x f t + hf tx ) − a ( f h xxx − f x h xx + 3 h x f xx − hf xxx ) . (3.14)Hence we obtain the Hirota bilinear form as P ( D ) { g · f } ≡ ( bD y − D t D x + a D x ) { g · f } = 0 , (3.15) P ( D ) { g · f } ≡ ( bD y + 12 D t D x + a D x ) { h · f } = 0 , (3.16) P ( D ) { f · f } ≡ D x { f · f } = − gh. (3.17)After having Hirota bilinear forms (3.5)-(3.7), (3.10)-(3.12), and (3.15)-(3.17), next step isto find the functions g , h , and f by using the Hirota method (see Sec. VI). It is straightforward to show that there exist no consistent local reductions in the form of q ( x, y, t ) = σ p ( x, y, t ) for all n = 0 , ,
2. Here we will give the local reductions in the form of q ( x, y, t ) = σ ¯ p ( x, y, t ) for all n = 0 , , σ is any real constant. (1) Local Reductions for the System n = 0 : Let q ( x, y, t ) = σ ¯ p ( x, y, t ) then two coupled equations (2.2) and (2.3) reduce consistentlyto the following single equation bp y = 12 p tx − a p x − σ pD − ( p ¯ p ) t , (4.1)where σ is any real constant and a bar over a letter denotes complex conjugation. Here a and b are pure imaginary numbers. (2) Local Reductions for the System n = 1 : Let q ( x, y, t ) = σ ¯ p ( x, y, t ) then two coupled equations (2.5) and (2.6) reduce consistentlyto the following single equation bp y = 12 p tx − a p xx + aσ p ¯ p − σ pD − ( p ¯ p ) t , (4.2)where σ is any real constant and a bar over a letter denotes complex conjugation. Here a isa real and b is a pure imaginary number. 6
3) Local Reductions for the System n = 2 : Let q ( x, y, t ) = σ ¯ p ( x, y, t ) then two coupled equations (2.8) and (2.9) reduce consistentlyto the following single equation bp y = 12 p tx − a p xxx + 3 a σ p ¯ p p x − σ pD − ( p ¯ p ) t , (4.3)where σ is any real constant and a bar over a letter denotes complex conjugation. Here a and b are pure imaginary numbers. In order to have consistent nonlocal reductions we use the following representation for D − D − F = 12 (cid:18)Z x −∞ − Z ∞ x (cid:19) F ( x ′ , y, t ) dx ′ . (5.1)We define the quantity ρ ( x, y, t ) which is invariant under the discrete transformations x → ǫ x , y → ǫ y, and t → ǫ t as ρ ( x, y, t ) = D − p p ǫ ≡ (cid:18)Z x −∞ − Z ∞ x (cid:19) p ( x ′ , y, t ) p ( ǫ x ′ , ǫ y, ǫ t ) dx ′ , (5.2)where ǫ = ǫ = ǫ = 1. It is easy to show that ρ ( ǫ x, ǫ y, ǫ t ) = ǫ ρ ( x, y, t ) . (5.3) (1) Nonlocal Reductions for the System n = 0 :(a) Let q ( x, y, t ) = σ p ( ǫ x, ǫ y, ǫ t ) then two coupled equations (2.2) and (2.3) reduce con-sistently to the following single equation bp y = 12 p tx − a p x − σ pD − ( p p ǫ ) t , (5.4)where σ is any real constant and p ǫ = p ( ǫ x, ǫ y, ǫ t ). The above reduced equation is validonly when ǫ = − ǫ ǫ = 1. We have only two possible cases: p ǫ = p ( x, y, − t ) and p ǫ = p ( − x, − y, − t ) for time reversal and time and space reversals respectively. (b) Let q ( x, y, t ) = σ ¯ p ( ǫ x, ǫ y, ǫ t ) then two coupled equations (2.2) and (2.3) reduce con-sistently to the following single equation bp y = 12 p tx − a p x − σ pD − ( p ¯ p ǫ ) t , (5.5)7here σ is any real constant. This reduction is valid only when ǫ ǫ ǫ ¯ b = − b, ǫ ¯ a = − a. (5.6)In this case we have seven different time and space reversals:(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 above gives a nonlocal equation in the form of (5.5) in 2+1 dimensions. (2) Nonlocal Reductions for the System n = 1 :(a) Let q ( x, y, t ) = σ p ( ǫ x, ǫ y, ǫ t ) then two coupled equations (2.5) and (2.6) reduce con-sistently to the following single equation bp y = 12 p tx − a p xx + aσp p ǫ − σpD − ( pp ǫ ) t , (5.7)where σ is any real constant and p ǫ = p ( ǫ x, ǫ y, ǫ t ). The above reduced equation is validonly when ǫ = − ǫ ǫ = 1. We have only two possible cases: p ǫ = p ( x, − y, t ) and p ǫ = p ( − x, − y, − t ) for space reversal and time and space reversals respectively. (b) Let q ( x, y, t ) = σ ¯ p ( ǫ x, ǫ y, ǫ t ) then two coupled equations (2.5) and (2.6) reduce con-sistently to the following single equation bp y = 12 p tx − a p xx + aσp ¯ p ǫ − σpD − ( p ¯ p ǫ ) t , (5.8)where σ is any real constant. This reduction is valid only when ǫ ǫ ǫ ¯ b = − b, ǫ ǫ ¯ a = a. (5.9)In this case we have seven different time and space reversals:(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.8v) 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 above gives a nonlocal equation in the form of (5.8) in 2+1 dimensions. (3) Nonlocal Reductions for the System n = 2 :(a) Let q ( x, y, t ) = σ p ( ǫ x, ǫ y, ǫ t ) then two coupled equations (2.8) and (2.9) reduce con-sistently to the following single equation bp y = 12 p tx − a p xxx + 3 a σpp ǫ p x − σpD − ( pp ǫ ) t , (5.10)where σ is any real constant and p ǫ = p ( ǫ x, ǫ y, ǫ t ). The above reduced equation is validonly when ǫ = − ǫ ǫ = 1. We have only two possible cases: p ǫ = p ( x, y, − t ) and p ǫ = p ( − x, − y, − t ) for time reversal and time and space reversals respectively. (b) Let q ( x, y, t ) = σ ¯ p ( ǫ x, ǫ y, ǫ t ) then two coupled equations (2.5) and (2.6) reduce con-sistently to the following single equation bp y = 12 p tx − a p xxx + 3 a σp ¯ p ǫ p x − σpD − ( p ¯ p ǫ ) t , (5.11)where σ is any real constant. This reduction is valid only when ǫ ǫ ǫ ¯ b = − b, ǫ ¯ a = − a. (5.12)In this case we have seven different time and space reversals:(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 above gives a nonlocal equation in the form of (5.11) in 2+1 dimensions. At theend we obtain 27 nonlocal equations from negative AKNS hierarchy in 2+1 dimensions. Remark.
In all the above nonlocal equations we can use D − = R x when there exist only y and t reversals, p ǫ = p ( x, ǫ y, ǫ t ). 9 Soliton Solutions for Negative AKNS Hierarchy (2 + 1) -AKNS( − n ) System ( n = 0 , , Here we will present how to find one-soliton solution of (2 + 1)-AKNS(0) system. For n = 1and n = 2 the steps for finding one-soliton solution are same with n = 0 case except thedispersion relations.Consider the system (3.5)-(3.7). To find one-soliton solution we use the following expansionsfor the functions g , h , and f , g = εg , h = εh , f = 1 + ε f , (6.1)where g = e θ , h = e θ , θ i = k i x + τ i y + ω i t + δ i , i = 1 , . (6.2)When we substitute (6.1) into the equations (3.5)-(3.7), we obtain the coefficients of ε as P ( D ) { g · } = bg ,y − g ,xt + ag ,x = 0 , (6.3) P ( D ) { h · } = bh ,y + 12 h ,xt + ah ,x = 0 , (6.4)yielding the dispersion relations τ = 1 b ( 12 k ω − ak ) , τ = 1 b ( − k ω − ak ) . (6.5)From the coefficient of ε f ,xx = − g h , (6.6)we obtain the function f as f = − e ( k + k ) x +( τ + τ ) y +( ω + ω ) t + δ + δ ( k + k ) . (6.7)The coefficients of ε vanish with the dispersion relations and (6.7). From the coefficient of ε we get f f ,xx − f ,x = 0 , (6.8)and this equation also vanishes directly due to the dispersion relations and (6.7). Withoutloss of generality let us also take ε = 1. Hence a pair of solutions of (2 + 1)-AKNS(0) system(2.2)-(2.3) is given by ( p ( x, y, t ) , q ( x, y, t )) where p ( x, y, t ) = e θ Ae θ + θ , q ( x, y, t ) = e θ Ae θ + θ , (6.9)with θ i = k i x + τ i y + ω i t + δ i , i = 1 , τ = b ( k ω − ak ), τ = b ( − k ω − ak ), and A = − k + k ) . Here k i , ω i , and δ i , i = 1 , n = 1 that is for the system (2.5)-(2.6) one-soliton solution is given by (6.9) where θ i = k i x + τ i y + ω i t + δ i , i = 1 , τ = 1 b ( 12 k ω − a k ) , τ = 1 b ( − k ω + a k ) . (6.10)For n = 2 that is for the system (2.8)-(2.9) one-soliton solution is again given by (6.9) where θ i = k i x + τ i y + ω i t + δ i , i = 1 , τ = 1 b ( 12 k ω − a k ) , τ = 1 b ( − k ω − a k ) . (6.11) (2 + 1) -AKNS( − n ) System ( n = 0 , , Here as in the previous section, we will only deal with (2 + 1)-AKNS(0) system and findtwo-soliton solution of this system. For n = 1 and n = 2 we have the same form of two-solitonsolution only with difference of the dispersion relations.Consider the system (3.5)-(3.7). For two-soliton solution, we take g = εg + ε g , h = εh + ε h , f = 1 + ε f + ε f , (6.12)where g = e θ + e θ , h = e η + e η , (6.13)with θ i = k i x + τ i y + ω i t + δ i , η i = ℓ i x + s i y + m i t + α i for i = 1 ,
2. When we insert aboveexpansions into (3.5)-(3.7), we get the coefficients of ε n , 1 ≤ n ≤ ε : bg ,y − g ,xt + ag ,x = 0 , (6.14) bh ,y + 12 h ,xt + ah ,x = 0 , (6.15) ε : f ,xx + g h = 0 , (6.16) ε : b ( g ,y f − g f ,y ) −
12 ( g ,xt f − g ,t f ,x − g ,x f ,t + g f ,xt ) + a ( g ,x f − g f ,x )+ bg ,y − g ,xt + ag ,x = 0 , (6.17) b ( h ,y f − h f ,y ) + 12 ( h ,xt f − h ,t f ,x − h ,x f ,t + h f ,xt ) + a ( h ,x f − h f ,x )+ bh ,y + 12 h ,xt + ah ,x = 0 , (6.18) ε : f f ,xx − f ,x + f ,xx + g h + g h = 0 , (6.19)11 : b ( g ,y f − g f ,y ) −
12 ( g ,xt f − g ,t f ,x − g ,x f ,t + g f ,xt ) + a ( g ,x f − g f ,x )+ b ( g ,y f − g f ,y ) −
12 ( g ,xt f − g ,t f ,x − g ,x f ,t + g f ,xt ) + a ( g ,x f − g f ,x ) = 0 , (6.20) b ( h ,y f − h f ,y ) + 12 ( h ,xt f − h ,t f ,x − h ,x f ,t + h f ,xt ) + a ( h ,x f − h f ,x )+ b ( h ,y f − h f ,y ) + 12 ( h ,xt f − h ,t f ,x − h ,x f ,t + h f ,xt ) + a ( h ,x f − h f ,x ) = 0 , (6.21) ε : f ,xx f − f ,x f ,x + f f ,xx + g h = 0 , (6.22) ε : b ( g ,y f − g f ,y ) −
12 ( g ,xt f − g ,t f ,x − g ,x f ,t + g f ,xt ) + a ( g ,x f − g f ,x ) = 0 , (6.23) b ( h ,y f − h f ,y ) + 12 ( h ,xt f − h ,t f ,x − h ,x f ,t + h f ,xt ) + a ( h ,x f − h f ,x ) = 0 . (6.24) ε : f f ,xx − f ,x = 0 . (6.25)The equations (6.14) and (6.15) give the dispersion relations τ i = 1 b ( 12 k i ω i − ak i ) , s i = 1 b ( − ℓ i m i − aℓ i ) , i = 1 , . (6.26)From the coefficient of ε we obtain the function f , f = e θ + η + α + e θ + η + α + e θ + η + α + e θ + η + α = X ≤ i,j ≤ e θ i + η j + α ij , (6.27)where e α ij = − k i + ℓ j ) , ≤ i, j ≤ . (6.28)The equations (6.17) and (6.18) give the functions g and h , g = A e θ + θ + η + A e θ + θ + η , h = B e θ + η + η + B e θ + η + η , (6.29)where A i = − ( k − k ) ( k + ℓ i ) ( k + ℓ i ) , B i = − ( ℓ − ℓ ) ( ℓ + k i ) ( ℓ + k i ) , i = 1 , . (6.30)The equation (6.19) yields the function f as f = M e θ + θ + η + η , (6.31)where M = ( k − k ) ( l − l ) ( k + l ) ( k + l ) ( k + l ) ( k + l ) . (6.32)Other equations (6.20)-(6.25) vanish immediately by the dispersion relations (6.26) and thefunctions f , f , F , and G . 12et us also take ε = 1. Then two-soliton solution of the system (2.2)-(2.3) is given with thepair ( p ( x, y, t ) , q ( x, y, t )), p ( x, y, t ) = e θ + e θ + A e θ + θ + η + A e θ + θ + η e θ + η + α + e θ + η + α + e θ + η + α + e θ + η + α + M e θ + θ + η + η , (6.33) q ( x, y, t ) = e η + e η + B e θ + η + η + B e θ + η + η e θ + η + α + e θ + η + α + e θ + η + α + e θ + η + α + M e θ + θ + η + η , (6.34)with θ i = k i x + τ i y + ω i t + δ i , η i = ℓ i x + s i y + m i t + α i for i = 1 , τ i = b ( k i ω i − ak i ), s i = b ( − ℓ i m i − aℓ i ), i = 1 ,
2. Here k i , ℓ i , ω i , m i , δ i , and α i , i = 1 , n = 1 i.e. for the system (2.5)-(2.6) two-soliton solution is given by (6.33)-(6.34) where θ i = k i x + τ i y + ω i t + δ i , η i = ℓ i x + s i y + m i t + α i for i = 1 , τ i = 1 b ( 12 k i ω i − a k i ) , s i = 1 b ( − ℓ i m i + a ℓ i ) , i = 1 , . (6.35)For n = 2 that is for the system (2.8)-(2.9) two-soliton solution is also given by (6.33)-(6.34)where θ i = k i x + τ i y + ω i t + δ i , η i = ℓ i x + s i y + m i t + α i for i = 1 , τ i = 1 b ( 12 k i ω i − a k i ) , s i = 1 b ( − ℓ i m i − a ℓ i ) , i = 1 , . (6.36) In our studies of nonlocal NLS and nonlocal mKdV equations we introduced a general method[31]-[33] to obtain soliton solutions of nonlocal integrable equation. This method consists ofthree main steps: • Find a consistent reduction formula which reduces the integrable system of equationsto integrable nonlocal equations. • Find soliton solutions of the system of equations by use of the Hirota bilinear methodor by inverse scattering transform technique, or by use of Darboux Transformation. • Use the reduction formulas on the soliton solutions of the system of equations to obtainthe soliton solutions of the reduced nonlocal equations. By this way one obtains manydifferent relations among the soliton parameters of the system of equations.In the following sections we mainly follow the above method in obtaining the soliton solu-tions of AKNS( − n ) systems for n = 0 , , and n = 2 by using Type 1 and Type 2 approachesgiven in [32]. 13 .1 One-Soliton Solutions of Local Reduced Equations The constraints that one-soliton solutions of the local equations (4.1), (4.2), and (4.3) whichare reduced from AKNS( − n ) for n = 0 , , and n = 2 systems respectively can be found bythe local reduction formula q ( x, y, t ) = σ ¯ p ( x, y, t ) that is e k x + τ y + ω t + δ Ae ( k + k ) x +( τ + τ ) y +( ω + ω ) t + δ + δ = σe ¯ k x +¯ τ y +¯ ω t +¯ δ Ae (¯ k +¯ k ) x +(¯ τ +¯ τ ) y +(¯ ω +¯ ω ) t +¯ δ +¯ δ . (7.1)If we use the Type 1 approach, we obtain the following constraints:1) k = ¯ k , ω = ¯ ω , e δ = σe ¯ δ , (7.2)so that the equality (7.1) is satisfied for each n = 0 , , and n = 2. Note that under the aboveconstraints, the dispersion relations give τ = ¯ τ . Hence one-soliton solutions of (4.1), (4.2),and (4.3) are given by p ( x, y, t ) = e k x + τ y + ω t + δ − σ ( k +¯ k ) e ( k +¯ k ) x +( τ +¯ τ ) y +( ω +¯ ω ) t + δ +¯ δ , (7.3)wherei. for n = 0, a and b are pure imaginary numbers and τ = b ( k ω − ak ),ii. for n = 1, a is a real and b is a pure imaginary number and τ = b ( k ω − a k ),iii. for n = 2, a and b are pure imaginary numbers and τ = b ( k ω − a k ).If sign( σ ) < σ = − ( k + ¯ k ) e µ , (7.4)where µ is another real constant. Then the above one-soliton solution becomes p ( x, y, t ) = e φ θ (7.5)where θ = 12 [( k + ¯ k ) x + ( τ + ¯ τ ) y + ( w + ¯ w ) t + δ + ¯ δ + µ ] , (7.6) φ = 12 [( k − ¯ k ) x + ( τ − ¯ τ ) y + ( w − ¯ w ) t + δ − ¯ δ − µ ] , (7.7)Hence one-soliton solutions of the locally reduced equations for n = 0 , , σ ) <
0. The norm of p becomes | p ( x, y, t ) | = e − µ θ . (7.8)14 .2 One-Soliton Solutions of Nonlocal Reduced Equations Firstly let us consider the nonlocal reduction q ( x, y, t ) = σp ( ǫ x, ǫ y, ǫ t ). Here the con-straints that one-soliton solutions of the nonlocal equations (5.4), (5.7), and (5.10) whichare reduced from AKNS( − n ) for n = 0 , , and n = 2 systems respectively can be found by e k x + τ y + ω t + δ Ae ( k + k ) x +( τ + τ ) y +( ω + ω ) t + δ + δ = σe ǫ k x + ǫ τ y + ǫ ω t + δ Ae ǫ ( k + k ) x + ǫ ( τ + τ ) y + ǫ ( ω + ω ) t + δ + δ , (7.9)where A = − k + k ) and τ i , i = 1 , k i and ω i due to the dispersionrelations of each case n = 0 , , k = ǫ k , ω = ǫ ω , e δ = σe δ . (7.10)When we use these constraints with the possibilities for ( ǫ , ǫ , ǫ ) given in Sects. 7.4, 7.5,and 7.6 on the dispersion relations of the cases n = 0 , ,
2, we get τ = ǫ τ .For n = 0 we have ( ǫ , ǫ , ǫ ) = (1 , , −
1) and one-soliton solution of the reduced equation(5.4) is p ( x, y, t ) = e k x + τ y + ω t + δ − σ k , e k x +2 τ y +2 δ , (7.11)where τ = b ( k ω − ak ). Assume that all the parameters; k , ω , δ , a , and b so τ are real.Let σ = − k e µ then p ( x, y, t ) = e φ e θ +2 µ , (7.12)where µ is a real constant and φ = k x + τ y + ω t + δ , (7.13) θ = k x + τ y + δ . (7.14)Eq. (7.12) can further be simplified as p ( x, y, t ) = e ω t − µ θ + µ ) . (7.15)Hence for the defocusing case, sign( σ ) <
0, one-soliton solution is bounded for all t ≥ ω ≤ x, y, t ).For n = 1 we have ( ǫ , ǫ , ǫ ) = (1 , − ,
1) and one-soliton solution of the reduced equation(5.7) is p ( x, y, t ) = e k x + τ y + ω t + δ − σ k e k x +2 ω t +2 δ , (7.16)15here τ = b ( k ω − a k ). Assume that all the parameters; k , ω , δ , a , and b so τ are real.Let σ = − k e µ then p ( x, y, t ) = e φ e θ +2 µ (7.17)where µ is a real constant and φ = k x + τ y + ω t + δ , (7.18) θ = k x + ω t + δ , (7.19)which can be simplified as p ( x, y, t ) = e τ y − µ θ + µ ) . (7.20)Hence for sign( σ ) <
0, one-soliton solution is finite for all ( x, y, t ) but not bounded.For n = 2 we have ( ǫ , ǫ , ǫ ) = (1 , , −
1) and one-soliton solution of the reduced equation(5.10) is p ( x, y, t ) = e k x + τ y + ω t + δ − σ k e k x +2 τ y +2 δ , (7.21)where τ = b ( k ω − a k ). Hence, similar to n = 0 case, the solution (7.21) can be simplifiedto the form (7.15) with only difference in τ . And that solution is bounded for all t ≥ ω ≤ x, y, t ) when sign( σ ) < ǫ , ǫ , ǫ ) is ( − , − , − A , if we use Type 1 approach we obtain trivialsolution. Hence we use Type 2 on e θ Ae θ + θ = σ e θ − Ae θ − + θ − . (7.22)From the application of the cross multiplication we get e θ + Ae δ e θ − = ke θ − + Ake δ e θ , (7.23)where θ j = k j x + τ j y + ω j t + δ j , θ − = − k j x − τ j y − ω j t + δ j , j = 1 , . Hence we obtain the conditions1)
Aσe δ = 1 , Ae δ = σ, (7.24)yielding e δ = ξ i ( k + k ) √ σ and e δ = ξ i √ σ ( k + k ) for ξ j = ± j = 1 ,
2. Therefore one-solitonsolution of the equations (5.4), (5.7), and (5.10) is given by p ( x, y, t ) = iξ e k x + τ y + ω t ( k + k ) √ σ (1 + ξ ξ e ( k + k ) x +( τ + τ ) y +( ω + ω ) t ) , ξ j = ± , j = 1 , , (7.25)16ith corresponding dispersion relations; (6.5) for n = 0, (6.10) for n = 1, and (6.11) for n = 2. We can further simplify the solution (7.25) as p ( x, y, t ) = e φ + δ θ , (7.26)where φ = 12 [( k − k ) x + ( τ − τ ) y + ( ω − ω ) t ] , (7.27) θ = 12 [( k + k ) x + ( τ + τ ) y + ( ω + ω ) t ] . (7.28)The solution (7.26) is finite if k + k , τ + τ , and ω + ω are real. In addition to that it isbounded if k − k = 0, τ − τ = 0, and ω − ω ≤ t ≥
0. For n = 0 and n = 2 cases, theseconditions are satisfied if k , τ , ω are real, k = k , τ = τ , ω = − ω , and ω ≤ t ≥ n = 1 case, they are satisfied if k , τ , ω , a are real, k = k , τ = τ , ω = 2 ak − ω , and ω − ak ≤ t ≥ q ( x, y, t ) = σ ¯ p ( ǫ x, ǫ y, ǫ t ). The constraints thatone-soliton solutions of the nonlocal equations (5.5), (5.8), and (5.11) which are reducedfrom AKNS( − n ) for n = 0 , , and n = 2 systems respectively can be found by e k x + τ y + ω t + δ Ae ( k + k ) x +( τ + τ ) y +( ω + ω ) t + δ + δ = σe ǫ ¯ k x + ǫ ¯ τ y + ǫ ¯ ω t +¯ δ Ae ǫ (¯ k +¯ k ) x + ǫ (¯ τ +¯ τ ) y + ǫ (¯ ω + ¯ ω ) t +¯ δ +¯ δ , (7.29)where A = − k + k ) and τ i , i = 1 , n = 0 , , k = ǫ ¯ k , ω = ǫ ¯ ω , e δ = σe ¯ δ . (7.30)Using these constraints besides the conditions (5.6), (5.9), and (5.12) in the dispersion rela-tions of the cases n = 0 , , τ = ǫ ¯ τ .Thus one-soliton solution of the reduced equations (5.5), (5.8), and (5.11) is given by p ( x, y, t ) = e k x + τ y + ω t + δ − σ ( k + ǫ ¯ k ) e ( k + ǫ ¯ k ) x +( τ + ǫ ¯ τ ) y +( ω + ǫ ¯ ω ) t + δ +¯ δ , (7.31)with the corresponding dispersion relations τ = b ( k ω − ak ), τ = b ( k ω − a k ), and τ = b ( k ω − a k ) given respectively. It is clear that there are finite and singular solutions(7.31) depending on the parameters of the solutions.Note that since there are 21 nonlocal reduced equations by the reduction formula q ( x, y, t ) = σ ¯ p ( ǫ x, ǫ y, ǫ t ) for n = 0 , , y -reflection that is when ( ǫ , ǫ , ǫ ) =171 , − ,
1) as an example. Let σ = − ( k + ¯ k ) e µ , µ is a real constant. Then one-solitonsolutions of the nonlocal equations:( n = 0) , bp y ( x, y, t ) = 12 p tx ( x, y, t ) − ap x ( x, y, t ) − σp ( x, y, t ) D − ( p ( x, y, t )¯ p ( x, − y, t )) t , (7.32)where a is a pure imaginary, b is a real number,( n = 1) , bp y ( x, y, t ) = 12 p tx ( x, y, t ) − a p xx ( x, y, t ) + aσp ( x, y, t )¯ p ( x, − y, t ) − σp ( x, y, t ) D − ( p ( x, y, t )¯ p ( x, − y, t )) t , (7.33)where a and b are real numbers,( n = 2) , bp y ( x, y, t ) = 12 p tx ( x, y, t ) − a p xxx ( x, y, t ) + 3 a σp ( x, y, t )¯ p ( x, − y, t ) p x ( x, y, t ) − σp ( x, y, t ) D − ( p ( x, y, t )¯ p ( x, − y, t )) t , (7.34)where a is a pure imaginary, b is a real number, become p ( x, y, t ) = e φ θ ) , (7.35)where φ = 12 [( k − ¯ k ) x + ( τ + ¯ τ ) y + ( ω − ¯ ω ) t + ( δ − ¯ δ − µ ))] , (7.36) θ = 12 [( k + ¯ k ) x + ( τ − ¯ τ ) y + ( ω + ¯ ω ) t + ( δ + ¯ δ − µ ))] . (7.37)The solution (7.35) is finite if τ − ¯ τ ∈ R which happens when τ ∈ R . In addition to thatit is bounded if k − ¯ k = 0, τ + ¯ τ = 2 τ = 0, and ω − ¯ ω ≤ t ≥
0. This occurs onlywhen k ∈ R and τ = 0. But taking τ = 0 reduces the dimension of the solution from 2 + 1to 1 + 1. In this work we obtained a new negative AKNS hierarchy denoted by AKNS( − n ) for n = 0 , , , . . . in 2 + 1 dimensions. We obtained the Hirota bilinear forms of these sys-tems and found one- and two-soliton solutions for n = 0 , ,
2. We then found all possiblelocal and nonlocal reductions of these systems. Using the constraint equations among the dy-namical variables for n = 0 , , D − (integro-differential equations) and terms p ( ǫ x, ǫ y, ǫ t )(mirror symmetric terms) where ǫ = ǫ = ǫ = 1. From the one-soliton solutions of the neg-ative AKNS system of equations we obtained one-soliton solutions of the local and nonlocal18educed equations. Among all these one-soliton solutions there are solutions which developsingularities in a finite time and there are also solutions which are finite and bounded de-pending on the parameters of the solutions. This work is partially supported by the Scientific and Technological Research Council ofTurkey (T ¨UB˙ITAK).
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