Coherent structure of Alice-Bob modified Korteweg de-Vries Equation
aa r X i v : . [ n li n . S I] J un Coherent structure ofAlice-Bob modified Korteweg de-Vries Equation
Congcong Li , S. Y. Lou , ∗ and Man Jia Center for Nonlinear Science and Department of Physics,Ningbo University, Ningbo, 315211, China Shanghai Key Laboratory of Trustworthy Computing,East China Normal University, Shanghai 200062, China
Abstract
To describe two-place events, Alice-Bob systems have been established by means of the shiftedparity and delayed time reversal in Ref. [1]. In this paper, we mainly study exact solutions of theintegrable Alice-Bob modified Korteweg de-Vries (AB-mKdV) system. The general Nth Darbouxtransformation for the AB-mKdV equation are constructed. By using the Darboux transformation,some types of shifted parity and time reversal symmetry breaking solutions including one-soliton,two-soliton and rogue wave solutions are explicitly obtained. In addition to the similar solutionsof the mKdV equation (group invariant solutions), there are abundant new localized structures forthe AB-mKdV systems.
PACS numbers: 02.30.Ik ∗ Corresponding author: [email protected] . INTRODUCTION In 2013, Ablowitz and Musslimani [2] proposed a new integrable nonlocal nonlinearSchr¨odinger (NLS) equation iA t + A xx ± A B = 0 , B = ˆ f A = ˆ P ˆ CA = A ∗ ( − x, t ) , (1)where the operators ˆ P and ˆ C are the usual parity and charge conjugation. In literature,the nonlocal nonlinear Schr¨odinger equation (1) is also called parity-time reversal (PT)symmetric. PT symmetry plays an important role in the quantum physics [3] and manyother areas of physics, such as quantum chromodynamics [4], electric circuits [5], optics[6, 7] and Bose-Einstein condensates [8], etc.It is well known that there are various correlated and/or entangled events that may behappened in different times and places. To describe two-place physical problems, Alice-Bob(AB) systems [1] are proposed by using the AB-BA equivalence principle and the shiftedparity ( ˆ P s ), delayed time reversal ( ˆ T d ) and charge conjugate ( ˆ C ) symmetries. If one event(A, Alice event) is correlated/entangled to another (B, Bob event), we denote the correlatedrelation as B = ˆ f A for suitable ˆ f operators. Usually, the event A = A ( x, t ) happened at { x, t } and event B = B ( x ′ , t ′ ) happened at { x ′ , t ′ } = ˆ f { x, t } . In fact, { x ′ , t ′ } is usually faraway from { x, t } . Hence, the intrinsic two-place models or Alice-Bob systems are nonlocal.In addition to the nonlocal nonlinear Schr¨odinger equation (1), there are many other typesof two-place nonlocal models, such as the nonlocal KdV systems [9], the nonlocal modifiedKdV systems [10, 11], the discrete nonlocal NLS systems [12], the coupled nonlocal NLSsystems [13] and the nonlocal Davey-Stewartson systems [14–16], etc.In [1], one of us (Lou) proposed a series of integrable AB systems including the AB-KdVsystems, AB-mKdV systems, AB-KP systems, AB-sine Gordon systems, AB-NLS systemsand AB-Toda systems. Furthermore, by using the ˆ P s , ˆ T d and C symmetries, their ˆ P s , ˆ T d andˆ C invariant muti-soliton solutions are obtained in elegant forms. In addition, Lou establisheda most general AB-KdV equation and presented its ˆ P s , ˆ T d and ˆ C invariant Painlev´e IIreduction and soliton-cnoidal periodic wave interaction solutions for the AB-KdV system[17]. However, to find ˆ P s , ˆ T d and ˆ C symmetry breaking solutions is much more difficult.2n this paper, influenced by the idea of Lou in [17], we try to investigate ˆ P s , ˆ T d and ˆ C symmetry breaking solutions for a special AB-mKdV equation which has been also proposedin [1]. On the one hand, we will show that it can be derived from the third order AKNSsystem. On the other hand, we will construct its Nth Darboux transformation and giveits one-soliton solutions and two-soliton solutions through Darboux transformation. Theseexplicit solutions possess some new properties that are different from the ones for the mKdVequation. II. AB-MKDV SYSTEMS AND THEIR COMMON SHIFTED PARITY ANDTIME REVERSAL INVARIANT SOLUTIONS
The most general AB-mKdV system may have the form K ( A, B ) = 0 ,B = ˆ f A = ± ˆ P s ˆ T d ˆ C a A = ± ˆ C a A ( − x + x , − t + t ) , a = 0 , or 1 , (2)where x and t are arbitrary constants and K ( A, B ) is an arbitrary functional of A and B with the condition K ( u, u ) = u t + u xxx + 6 u u x = 0 . (3)In fact here ˆ f = ± ˆ P s ˆ T d ˆ C a is a discrete symmetry of the mKdV equation (3) with ˆ f = 1.A concrete differential polynomial form of (2) reads A t + b A xxx + b B xxx + ( a A + a AB + a B ) A x + ( a A + a AB + a B ) B x = 0 ,B = ˆ f A = ± ˆ C a A ( − x + x , − t + t ) , a = 0 , or 1 , (4)where a , a , a , a , a and b are arbitrary constants while b = 1 − b and a = 6 − a − a − a − a − a . The AB-mKdV system (4) can be considered as a special reduction ofthe coupled mKdV equation which can be derived from the two layer fluid dynamic systems[18]. 3 more special form of the AB-mKdV system, A t + A xxx + 6 ABA x = 0 ,B = ˆ f A = ± ˆ P s ˆ T d ˆ C a A = ± ˆ C a A ( − x + x , − t + t ) , a = 0 , or 1 , (5)can also be considered as a special reduction of the third order AKNS system [1], AKN S ≡ A t + A xxx + 6 ABA x = 0 ,B t + B xxx + 6 ABB x = 0 . (6)Because of the property (3), all the shifted parity and delayed time reversal invariant(for simplicity, ˆ f -invariant) solutions of the general AB-mKdV system (2) possess the sameform of the usual mKdV equation (3). Thus, to find ˆ f -invariant solutions of the AB-mKdVsystems is equivalent to select out the ˆ f -invariant solutions from known solutions (which areusually ˆ f symmetry breaking) of the usual mKdV equation.It is fortunate that the multiple soliton solutions of the mKdV equation can be recon-structed as [1], u soliton = ± ∂∂x tan − P ν e K ν sinh (cid:16)P Nj =1 ν j η j (cid:17)P ν o K ν cosh (cid:16)P Nj =1 ν j η j (cid:17) , (7)where the summation of ν o should be done for all non-dual odd permutations of ν i =1 , − , i = 1 , , . . . , N with odd number of ν i = 1, the summation of ν e should bedone for all non-dual even permutations of ν i = 1 , − , i = 1 , , . . . , N with even numberof ν i = 1, K ν ≡ Y i>j ( k i − ν i ν j k j ) , (8)and η j being defined as η j = k j (cid:18) x − x (cid:19) − k j (cid:18) t − t (cid:19) + η j , (9) k j , j = 1 , , . . . , N and η j , j = 1 , , . . . , N are arbitrary constants.It is clear that if some of η j are nonzero, then the multiple solutions (7) is ˆ f symmetrybreaking. The arbitrariness of η j is introduced by the space-time translation invariants.4owever, all the AB-mKdV systems are space-time translation symmetry breaking. Thus, η j should be fixed. From the expressions (7) and (9), it is straightforward to find that A = u soliton | η j =0 , j =1 , ..., N (10)is just the ˆ f -invariant N -soliton solution of all the real AB-mKdV systems (2).Because all the ˆ f -invariant solutions of the AB-mKdV systems only constitute a subset ofthe solutions of the mKdV equation, it is more interest to find ˆ f -symmetry breaking solutionsof the AB-mKdV systems. In order to find some nontrivial ˆ f -symmetry breaking solutions,we restrict to study the Darboux transformations of the special AB-mKdV equation (5) with a = 0 and the lower negative sign, i.e., ˆ B = − A ( − x + x , − t + t ). III. DARBOUX TRANSFORMATION OF THE AB-MKDV SYSTEM
The Darboux transformation method can be traced back to the way of thinking in thestudy of the linear problem of Darboux. It is an effective method to obtain exact solutions forintegrable nonlinear systems [20]. In this section, we will give the Darboux transformationfor the AB-mKdV system (5). First, we start with the following Lax pair of the AB-mKdVsystem (5): ϕ x = U ϕ = − iλ AB iλ ϕ, ϕ t = V ϕ = α βγ − α ϕ, (11)where ϕ = ( ϕ ( x, t ) , ϕ ( x, t )) T , λ is the spectral parameter and α , β and γ are given by( i = √− α = − iλ − iABλ + AB x − BA x ,β = 4 Aλ + 2 iA x λ + 2 A B − A xx ,γ = 4 Bλ − iB x λ + 2 AB − B xx . (12)The compatibility condition of equation (11), U t − V x + [ U, V ] = 0results in (5). 5econdly, imitating the procedure of Darboux transformation for general integrable mKdVequation [21, 22], we will construct the Darboux transformation of the AB-mKdV equation(5). Taking the gauge transformation, ϕ [1] = T [1] ϕ, (13)the spectral problem (11) turns into ϕ [1] x = ( T [1] x + T [1] U )( T [1] ) − ϕ [1] = U [1] ϕ [1] ,ϕ [1] t = ( T [1] t + T [1] V )( T [1] ) − ϕ [1] = V [1] ϕ [1] . (14)Letting T [1] = λI + S [1] , (15)with S [1] = ( s [1] ij ) × , s [1] ij ( i, j = 1 ,
2) are functions of x and t , I being the identity matrix.After that, we get the relationship between the new potentials { A [1] , B [1] } and the old ones { A, B } , A [1] = A + 2 is [1]12 ,B [1] = B − is [1]21 . (16)From the correlation relation B = ˆ f A = − A ( − x + x , − t + t ), we obtain the followingconstraint: s [1]12 ( − x + x , − t + t ) = s [1]21 ( x, t ) . (17)The eigenfunctions corresponding to the seed solution are f ( λ j ) = ( f ( λ j ) , f ( λ j )) T , g ( λ j ) = ( g ( λ j ) , g ( λ j )) T and the eigenvalues are λ = λ j ( j = 1 ,
2) in (11). Then we get λ j + s [1]11 + α j s [1]12 = 0 ,s [1]21 + α j ( λ j + s [1]22 ) = 0 ,α j = f ( λ j ) + γ j g ( λ j ) f ( λ j ) + γ j g ( λ j ) , (18)with γ j ( j = 1 ,
2) being arbitrary constants. 6hus, the matrix T [1] can be written as T [1] = λ λ + 1 α − α λ α − λ α λ − λ α α ( λ − λ ) λ α − λ α . (19)Finally, we construct the n-fold Darboux transformation for the AB-mKdV system (2) tofind the P s T d symmetry breaking soliton solutions. ϕ [ n ] = T n ( λ ) ϕ, T n ( λ ) = T [ n ] ( λ ) T [ n − ( λ ) · · · T [ k ] ( λ ) · · · T [1] ( λ ) , (20)with T [ k ] ( λ ) = λI + S [ k ] = λI + 1 α k − α k − λ k α k − − λ k − α k λ k − − λ k α k − α k ( λ k − λ k − ) λ k − α k − − λ k α k , (21)where α j = f [ k − ( λ j ) + γ j g [ k − ( λ j ) f [ k − ( λ j ) + γ j g [ k − ( λ j ) , ( j = 2 k − , k, k = 1 , , · · · , n ) f [ k ] ( λ ) = f [ k ]1 ( λ ) f [ k ]2 ( λ ) = T [ k ] ( λ ) f [ k − ( λ , λ , · · · , λ k − , λ k ) ,g [ k ] ( λ ) = g [ k ]1 ( λ ) g [ k ]2 ( λ ) = T [ k ] ( λ ) g [ k − ( λ , λ , · · · , λ k − , λ k ) , (22)and the matrix S [ k ] meets the following constraint condition s [ k ]12 ( − x + x , − t + t ) = s [ k ]21 ( x, t ) ( k = 1 , , · · · , n ) . (23)The new solution A [ n ] ( x, t ) and old one A ( x, t ) should satisfy A [ n ] = A + 2 i n X k =1 s [ k ]12 . (24) IV. SOLITON SOLUTIONS OF THE AB-MKDV SYSTEM
In this section, we will describe how to obtain the exact solutions of the AB-mKdV system(5) in detail, including one-soliton solutions and two-soliton solutions with the help of theDarboux transformation. 7 . One-soliton solutions from zero seed
The well known solution with exponential form of the AB-mKdV equation (5) can bewritten in the following form, A = ρe κ (( x − x ) − ( κ +6 ρ )( t − t )) , (25)where κ and ρ are two complex parameters.First of all, we choose zero seed solution A = 0. By solving the spectral equation corre-sponding to zero seed, we get f ( x, t ; λ ) = e − iλ ( x +4 λ t ) , g ( x, t ; λ ) = e iλ ( x +4 λ t ) . (26)Hence, we obtain α j = γ j e iλ j ( x +4 λ j t ) = γ j e ξ j , j = 1 , s ( x, t ) = λ − λ γ e ξ − γ e ξ , s ( x, t ) = ( λ − λ ) γ γ e ξ + ξ γ e ξ − γ e ξ . (27)The constraint condition (17) results in γ = e − ξ ( x ,t ) , γ = e − ξ ( x ,t ) , (28)where ξ j ( x , t ) = 2 iλ j ( x + 4 λ j t ) , ( j = 1 , γ = − e − ξ ( x ,t ) , γ = e − ξ ( x ,t ) , thus, A [1] = 2 i ( λ − λ ) e ξ − ξ ( x ,t ) + e ξ − ξ ( x ,t ) (29)Note that | e ξ + e ξ | = 2 e ξ R + ξ R (cosh( ξ R − ξ R ) + cos( ξ I − ξ I )) , (30)where λ j = µ j + iν j , µ j , ν j ∈ R ( j = 1 , ξ jR and ξ jI are real and imaginary parts of ξ j respectively, ξ jR = ℜ ( ξ j ) = − ν j x + 8 ν j ( ν j − µ j ) t,ξ jI = ℑ ( ξ j ) = 2 µ j x + 8 µ j ( µ j − ν j ) t. (31)8 FIG. 1: Plot of the single soliton (33) with µ = 0 . ν = 0 . x = 2 and t = 2 for the quantity AB . In order to guarantee the solution does not include any singular points, λ j (j=1,2) shouldsatisfy 2 µ ν + µ ν + µ ν + 2 µ ν = 0 , ( ν − ν ) + (cid:2) ν ( ν − µ ) − ν ( ν − µ ) (cid:3) = 0 . (32)Letting µ = µ and ν = − ν , then ξ R + ξ R = 0 holds for all ( x, t ) ∈ R , and meets theabove conditions. Therefore, we gain a typical soliton, A [1] = − iν e − iµ ζ sech( ν ζ ) ,ζ = 2 (cid:16) x − x (cid:17) + 8 (cid:0) µ − ν (cid:1) (cid:18) t − t (cid:19) ,ζ = 2 (cid:16) x − x (cid:17) − (cid:0) ν − µ (cid:1) (cid:18) t − t (cid:19) . (33)Thus, the soliton propagates to the right when ν − µ > ν − µ <
0. As ν − µ = 0, the soliton is stationary. Fig.1 shows this situation.Setting ν = 0 and 2 µ + µ = 0. This results in ξ R = 0 for all ( x, t ) ∈ R . Now thesolution gives A [1] = ( − ν + 6 iµ ) e iµ ζ e (3 iµ − ν ) ζ ,ζ = 2( x − x µ ( t − t ,ζ = 2( x − x − ν − µ )( t − t . (34)9 a)–20 –10 0 10 20x –0.3–0.2–0.10 0.10.20.3t–20Re(A) (b)–20 –10 0 10 20x –0.3–0.2–0.10 0.10.20.3t–2–101Re(A)(c)–30 –20 –10 0 10 20 30x –0.3–0.2–0.10 0.10.20.3t012aba(A) FIG. 2: Display of the structure of the kink shape complexiton solution (34) with µ = 0 . ν = 0 . x = t = 0 for (a) the real part Re ( A ) = ℜ ( A ), (b) the imaginary part Im ( A ) = ℑ ( A ) and(c) the amplitude M ( A ). From this solution, we know that if ν >
0, the complexiton spreads like kink, if ν < ν − µ >
0, the wave travels to the right, as ν − µ <
0, it propagates to the left, while ν − µ = 0, it is stationary. Fig.2 describesthe situation. B. One-soliton solutions from nonzero seed.
To find the one soliton solution from nonzero seed of the AB-mKdV equation (5)with B = − A ( − x + x , t = − t + t ), we suppose that the nonzero seed A ( x, t ) = ρe κ (( x − x ) − ( κ +6 ρ )( t − t )) , ρ = 0. As the eigenvalue λ = i ( κ + 2 ρ ), we obtain eigenfunc-10ions ϕ ( x, t ) = (cid:26) c + ( c + c ) (cid:20) ρ ( x − x − δ + ( t − t (cid:21)(cid:27) e κ [ ( x − x ) − ( κ +6 ρ )( t − t ) ] ,ϕ ( x, t ) = (cid:26) c − ( c + c ) (cid:20) ρ ( x − x − δ + ( t − t (cid:21)(cid:27) e − κ [ ( x − x ) − ( κ +6 ρ )( t − t ) ] . (35)where δ + = 3 κ ρ + 6 κρ + 6 ρ , c , c are arbitrary constants.By taking c = 1, c = 0, which leads to α = e − κ [ ( x − x ) − ( κ +6 ρ )( t − t ) ] ( − γ + 11 + ( γ + 1) (cid:2) ρ ( x − x ) − δ + ( t − t ) (cid:3) ) , (36)In the case of λ = i ( κ − ρ ), the eigenfunctions have the following expression, ϕ ( x, t ) = (cid:26) c + ( c − c ) (cid:20) ρ ( x − x − δ − ( t − t (cid:21)(cid:27) e κ [ ( x − x ) − ( κ +6 ρ )( t − t ) ] ,ϕ ( x, t ) = (cid:26) c + ( c − c ) (cid:20) ρ ( x − x − δ − ( t − t (cid:21)(cid:27) e − κ [ ( x − x ) − ( κ +6 ρ )( t − t ) ] . (37)where δ − = 3 κ ρ − κρ + 6 ρ , c , c are arbitrary constants.By setting c = 0, c = 1, which yields to α = e − κ [ ( x − x ) − ( κ +6 ρ )( t − t ) ] ( γ −
11 + ( γ − (cid:2) ρ ( x − x ) − δ − ( t − t ) (cid:3) ) . (38)By solving the constraint condition (17), we get γ = 1, γ = 1. Then we set γ = 1, γ = −
1. Thus the general expression of one-soliton solutions is A [1] = ρe κ [ ( x − x ) − ( κ +6 ρ )( t − t ) ] (cid:18) − − ξ ρ ξ − ξ (cid:19) ,ξ = κρ (cid:18) t − t (cid:19) ,ξ = 3( κ + 2 ρ ) (cid:18) t − t (cid:19) − (cid:16) x − x (cid:17) (39)Usually, the expression (39) displays some kinds of rogue wave structure for the quantity AB .The solution (39) is ˆ P s ˆ T d ˆ C invariant for imaginary κ and real ρ . However, (39) is not − ˆ P s ˆ T d invariant for any selections of parameters.11 a)–1–0.500.51t –4 –2 0 2 4x (b) –4–2024 x–1 –0.5 0 0.5 1t02468AB FIG. 3: Rogue-wave solution (40) with ρ = 1, κ = i , x = 0 and t = 0. Its density plot and shapeare described in (a) and (b) respectively for the quantity AB . For pure imaginary κ and real ρ , setting κ = iκ I and ρ = ρ R , equation (39) is changed to A [1] = ρ R e iκ I [ ( x − x )+( κ I − ρ R )( t − t ) ] (cid:18) − − iξ I ρ R ξ R + 144 ξ I (cid:19) ,ξ I = κ I ρ R (cid:18) t − t (cid:19) ,ξ R = 3 (cid:0) κ I + 2 ρ R (cid:1) (cid:18) t − t (cid:19) + (cid:16) x − x (cid:17) (40)In Fig.3, we describe such a rogue-wave with ρ = 1 and κ = i . In this case, we see that aneye-shaped form which has a hump and two valleys.For general selections of parameters, the structure of the rogue wave (40) may be com-plicate and even singular. In Fig. 4, a different analytic rogue wave is displayed for its realpart, imaginary part and amplitude with the parameter selections, κ = i + 0 . , ρ = 1 − . i, x = t = 0 . (41)For the general complex parameter selections of κ and ρ , the solution (39) may still bean analytic rogue wave or a singular rogue wave dependent on the selections of parameters.Fig. 4 displays the structure for an analytical rogue wave (39) with the complex parameterselections. 12 a) –4–2024 x–1 –0.5 0 0.5 1t02468Re(AB) (b) –4–2024 x–1 –0.5 0 0.5 1t00.51Im(AB)(c) –4–2024 x–1 –0.5 0 0.5 1t00.511.5M(AB) (d)–1–0.500.51t –4 –2 0 2 4x FIG. 4: Rogue-wave solution (39) with the parameter selections (41). (a) Real part, (b) imaginarypart, (c) amplitude and (d) the density plot of (c).
C. Two-soliton solutions from zero seed.
In this section, we study some kinds of two-soliton solutions. To find two-soliton solutionsof the AB-mKdV equation (5), by means of the 2-fold Darboux transformation, after solvingthe constraint condition (23), we have γ = − e − ξ ( x ,t ) , γ = e − ξ ( x ,t ) , γ = − e − ξ ( x ,t ) and γ = e − ξ ( x ,t ) . Finally, we can obtain the following soliton-periodic solution A [2] = − λ λ λ e ξ + λ λ λ e ξ + λ λ λ e ξ + λ λ λ e ξ )( e ξ + ξ + e ξ + ξ ) λ λ + ( e ξ + ξ + e ξ + ξ ) λ λ + ( e ξ + ξ + e ξ + ξ ) λ λ , (42)where ξ j = 2 iλ j (cid:20)(cid:16) x − x (cid:17) + 4 λ j (cid:18) t − t (cid:19)(cid:21) , ( j = 1 , , , λ ij = λ i − λ j ( i, j = 1 , , , and i < j ) (43)13 FIG. 5: The density plot of the interaction between a soliton (analytic) and a singular periodic wavedescribed by (42) with the parameter selections λ = 2 , λ = 2 . , λ = 2 i, λ = − i, x = t = 0. It is interesting that the solution (42) possesses many interesting properties. Usually it is asingular periodic-solitary wave. For instance, if we select the spectral parameters as λ = 2 , λ = 2 . , λ = − λ = 2 i, x = t = 0 , (44)then the solution (42) becomes an interaction solution between an soliton and a singularperiodic wave as shown in density plot Fig.5 for the quantity M ( A ) ≡ q ℜ ( A [2] ) + ℑ ( A [2] ) . If two pairs of spectral parameters are complex conjugate each other, say, λ = λ ∗ , λ = λ ∗ ,then the solution becomes an analytic interaction two soliton solution. Fig. 6 displays suchkind of interaction with the parameter selections λ = 1 + i, λ = 1 − i, λ = 2 i, λ = − i, x = t = 0 . (45)Fig. 6a, Fig. 6b and Fig. 6c display the structures of (42) with (45) for the real part ℜ ( A [2] ), imaginary part ℑ ( A [2] ) and the amplitude (cid:12)(cid:12) A [2] (cid:12)(cid:12) ≡ M ( A ) respectively. From Fig.6c, we know that this kind of interaction display the usual elastic interaction property. Twosolitons have not changed their velocities and directions of propagation while a phase shiftfor every soliton will be companied. 14 a)–4 –2 0 2 4x –0.6–0.4–0.20 0.2 0.4 0.6t–202Re(A) (b)–4 –2 0 2 4x –0.6–0.4–0.20 0.2 0.4 0.6t–4–202Im(A)(c)–4 –2 0 2 4x –0.6–0.4–0.20 0.2 0.4 0.6t024M(A) FIG. 6: The plot of a standard two soliton interaction solution described by (42) with the parameterselections (45). (a) Real part, (b) imaginary part and (c) amplitude.
If we select one spectral parameter as zero and the left two spectral parameter are complexconjugate, then the solution (42) becomes an interaction solution between a soliton and akink. Fig. 7 and Fig. 8 exhibits two different types of interaction modes.Fig.7 is a plot of (42) with the parameter selection λ = x = t = 0 , λ = − λ = i, λ = 1 . i. (46)From Fig.7, we find that before the interaction we have a dark (gray) soliton and a kink.However, after the interaction, the soliton becomes a bright soliton while the kink remainsits shape. This kind of transition comes from the nonlocal interaction of the model.Fig.8 is a plot of (42) with the parameter selection λ = x = t = 0 , λ = − λ = 2 i, λ = i. (47)15 FIG. 7: The plot of an interaction solution between a soliton and kink described by (42) with theparameter selections (46) for the amplitude. –4–2024 x–1 –0.5 0 0.5 1t0123M(A)
FIG. 8: The plot of the second type of interaction solution between a soliton and kink describedby (42) with the parameter selections (47).
From Fig. 8 we can find that the kink is unchanged by the interaction except for thephase shift and the soliton is a bright soliton before and after interaction. However, thesoliton is a bright soliton without background before interaction while after the interactionthe background of the soliton is not zero because of the existence of the kink.
V. SUMMARY AND DISCUSSION
It is shown that the AB systems are important in not only mathematics but also physics.In this paper, we investigate only a special AB-mKdV system which is directly obtained16rom the third order AKNS system to describe two-place physical events. The Nth Darbouxtransformation of the AB-mKdV system is constructed. Some special kind of exact solutionsrelated to the first and second Darboux transformation are explicitly discussed in detail. Itis found that the complex AB-mKdV equation possesses abundant solution structures whichhave not yet be noticed before. For instance, for single soliton solution of the complex AB-mKdV equation which may be a bell-ring shape soliton (bright soliton) and a kink soliton(dark soliton) dependent on the spectral parameter (but not model parameters) selections.For the single bell shape soliton, there may be some quite different ones. In this paper twosingle bell shape bright solitons are obtained. The first one is comes from the first Darbouxtransformation (Fig. 1) and the second one comes from the second Darboux transformation.For the single kink shape soliton, there may be some interesting structures. Especially, thekink solutions of the complex AB-mKdV system may possesses an oscillated tail as shownin Fig. 2c.For the rogue wave solutions of the complex AB-mKdV system, its structure is also quitecomplicated. Two types of structures of the single rogue wave are plotted in the figures 3and 4 respectively.For the interactions between two solitons, there are also some interesting phenomena. Forinstance, the dark soliton may be transition to bright soliton after interaction.From the results of this paper, we can see that there are various problems should berevealed in the future studies.
Acknowledgements
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