A novel (2+1)-dimensional integrable KdV equation with peculiar solution structures
aa r X i v : . [ n li n . S I] J a n A novel (2+1)-dimensional integrable KdV equation with peculiarsolution structures
S. Y. Lou ∗ School of Physical Science and Technology, Ningbo University, Ningbo, 315211, China
January 24, 2020
Abstract
The celebrated (1+1)-dimensional Korteweg de-Vries (KdV) equation and its (2+1)-dimensional ex-tention, the Kadomtsev-Petviashvili (KP) equation, are two of the most important models in physicalscience. The KP hierarchy is explicitly written out by means of the linearized operator of the KP equa-tion. A novel (2+1)-dimensional KdV extension, the cKP3-4 equation, is obtained by combining thethird member (KP3, the usual KP equation) and the fourth member (KP4) of the KP hierarchy. Theintegrability of the cKP3-4 equation is guaranteed by the existence of the Lax pair and dual Lax pair.The cKP3-4 system can be bilinearized by using Hirota’s bilinear operators after introducing an addi-tional auxiliary variable. Exact solutions of the cKP3-4 equation possess some peculiar and interestingproperties which are not valid for the KP3 and KP4 equations. For instance, the soliton molecules andthe missing D’Alembert type solutions (the arbitrary travelling waves moving in one direction with afixed model dependent velocity) including periodic kink molecules, periodic kink-antikink molecules, fewcycle solitons and envelope solitons are existed for the cKP3-4 equation but not for the separated KP3equation and the KP4 equation.
Keywords: (2+1)-dimensional KdV equations, Lax and dual Lax pairs, soliton and soliton molecules,arbitrary travelling waves
PACS:05.45.Yv, 02.30.Ik, 47.20.Ky, 52.35.Mw, 52.35.Sb
In linear science, the wave motion equation u tt − c u xx = 0 (1)is used to describe almost all the waves in natural science because the general solution, the D’Alembertsolution, u = f ( x − | c | t ) + g ( x + | c | t ) , (2)includes arbitrary waves ( f and g are arbitrary functions of the indicated variables) moving in the x directionor in the − x direction with a fixed velocity | c | .The simplest extension of the wave motion equation (1) in nonlinear science is the so-called Kortewegde-Vries (KdV) equation [1] u t + u xxx + 6 uu x = 0 . (3)The KdV equation possesses various applications in physics and other scientific fields. It approximatelydescribes the evolution of long, one-dimensional waves in many physical settings, including shallow-waterwaves with weakly non-linear restoring forces, long internal waves in a density-stratified ocean, ion acousticwaves in a plasma, acoustic waves on a crystal lattice [1] and the 2-dimensional quantum gravity [2]. TheKdV equation can be solved by using the inverse scattering transform [3] and other powerful methods suchas the Hirota’s bilinear method [4] and the Darboux transformation [5]. ∗ E-mail:[email protected].
Can we find a nonlinear extension such that the missing D’Alembert type waves can be found?
To describe two dimensional nonlinear waves which are the extensions of the (1+1)-dimensional KdVwaves, there are some different versions of the (2+1)-dimensional integrable KdV equations including theKadomtsev-Petviashvili (KP, or KP3) equation [6] u t − a (6 uu x + u xxx − w y ) = 0 , u y = w x , (4)the Nizhnik-Novikov-Veselov (NNV) equation[7, 8, 9] u t + u xxx + u yyy + 3 ( uv ) x + 3 ( uw ) y = 0 , u x = v y , u y = w x , (5)the asymmetric NNV equation (ANNV) [10, 11], or named Boiti-Leon-Manna-Pempinelli equation[12] u t + u xxx + 3 ( uv ) x = 0 , u x = v y , (6)the Ito equation [13] au t + u y + u xxx + 3 uu x + 3 v x + bu x = 0 , u y = w x , v y = wu x , (7)and the breaking soliton equation [14, 15] u t + u xxy + 4 uw x + 2 wu x = 0 , u y = w x . (8)It is clear that when y = x all the (2+1)-dimensional integrable KdV equations (4)–(8) are reduced back tothe equivalent KdV equation (3).The stability and dynamics of soliton molecules (soliton bound states) have attracted considerable at-tention in several areas such as the optics [16, 17, 18, 19, 21] and Bose-Einstein condensates [20]. Sometheoretical proposals to form soliton molecules have been established [22, 23]. Especially, in Refs. [24, 25], anew mechanism, the velocity resonance, to form soliton molecules is proposed. It is found that the standard(1+1)-dimensional KdV equation (3) does not possess soliton molecules. However, in real physics higherorder effects such as the higher order dispersions and higher order nonlinearities have been neglected whenthe KdV equation (3) is derived [26]. Whence the higher order effects are considered to the usual KdVequation, one can really find the soliton molecules [24]. By using the velocity resonance mechanism, someauthors find new types of soliton molecules such as the dromion molecules and half periodic kink moleculesin some physical systems [27, 28, 29].After detailed calculations, one can find that there is no soliton molecules for all the (2+1)-dimensionalKdV extensions (4)–(8). To find soliton molecules for (2+1)-dimensional KdV systems, we have to considertheir higher order effects. For integrable systems, we can directly add the integrable higher order systemsto the lower order ones. In section 2 of this paper, we write down the positive KP hierarchy in terms ofthe linearized operator of the KP equation (4). Then we write down a novel (2+1)-dimensional integrableKdV system, the combination of the KP3 and the KP4 equations (cKP3-4), which will be reduced back tothe usual (1+1)-dimensional KdV equation (3) when y = x . The Lax pair and the dual Lax pair of thecKP3-4 equation are also given in section 2. In section 3, we study the multiple soliton solutions of the novel(2+1)-dimensional KdV equation. The soliton molecules are investigated in section 4. It is found that asoliton molecule may include arbitrary number of solitons. For this model, a single soliton molecule withoutother solitons may possess arbitrary shape. This factor can be simply proved by studying its travellingwaves. The last section is a summary and discussion. For a (1+1)-dimensional integrable system, there is a so-called recursion operator Φ such that a set ofinfinitely many commute symmetries can be directly obtained K n = Φ n u x . Thus, an integrable hierarchy, u t n = Φ n u x , n = 0 , , . . . , ∞ , (9)2an be simply obtained.However, in (2+1)-dimensional case, there is no such kind of recursion operators to find infinitely manycommute symmetries and then the integrable hierarchy. Fortunately, by means of the formal series symmetryapproach [30], for many kinds of (2+1)-dimensional integrable systems including the KP equation (4) [31],the NNV equation (5)[32] and the ANNV equation (6), one can find infinitely many commute symmetriesby using their linearized operators. For instance, for the KP equation (4) with a = 1, a symmetry, σ , isdefined as a solution of its linearized equation6 σu x + 6 uσ x + σ xxx − ∂ − x σ yy − σ t ≡ Lσ = 0 , (10)where L ≡ ∂ x + 6 ∂ x u − ∂ − x ∂ y − ∂ t is the linearized operator of the KP equation (4) and ∂ − x is defined as ∂ − x ∂ x = ∂ x ∂ − x = 1. In Ref. [31], it is proven that the following expressions σ n = 12( n − n L n y n − , n = 1 , , . . . ∞ , (11)are all commute symmetries of the KP equation (4) for arbitrary integers n ≥ u t n = 12( n − n L n y n − , n = 1 , , . . . ∞ . (12)The first five of (12) read ( u y = w x , u yy = z xx , u yyy = m xxx ),KP1: u t = u x , (13)KP2: u t = − u y , (14)KP3: u t = − uu x − u xxx + 3 w y , (15)KP4: u t = 12 (2 wu x − z y + u xxy + 4 uu y ) , (16)KP5: u t = u xxxxx + 10[( u + uu xx − zu − w − z xx ) x + u x u xx − uz x − ∂ − x ( uz xx + w x )] + 5 m y . (17)It is not difficult to find that the KP3 equation (15) is just the KP equation (4) by taking t = − at . Itis interesting that in addition to the KP3 equation, the KP4 equation is also an extension of the KdVequation. In other words, when y = x the KP4 equation (16) is also reduced back to an equivalent form ofthe usual (1+1)-dimensional KdV equation (3). Because the commute KP3 and KP4 equations are all the(2+1)-dimensional KdV extensions, we can obtain a novel integrable (2+1)-dimensional KdV equation, thecKP3-4 equation u t = a ( uu x + u xxx − w y ) + b (2 wu x − z y + u xxy + 4 uu y ) , (18) u y = w x , u yy = z xx . It is trivial that when y → x , w → u, z → u , the (2+1)-dimensional KdV equation (18) is reduced back tothe usual KdV equation (3) up to some suitable scaling and Galileo transformations.To see the integrability of the model (18), we directly write down its Lax pair ψ y = i( ψ xx + uψ ) , i ≡ √− , (19) ψ t = 2i bψ xxxx + 4 aψ xxx + 4i buψ xx + 2(3 au + 2i bu x + bw ) ψ x − i(3 aw + bz − bu + 3 a i u x − bu xx + i bw x ) ψ, (20)and the dual Lax pair φ y = − i( φ xx + uφ ) , (21) φ t = − bφ xxxx + 4 aφ xxx − buφ xx + 2(3 au − bu x + bw ) φ x +i(3 aw + bz − bu − a i u x − bu xx − i bw x ) φ. (22)One can directly verified that the compatibility condition ψ yt = ψ ty (and/or φ yt = φ ty ) is nothing but thefield u is a solution of the cKP3-4 equation (18). 3imilar to the KP equation (4), one can also directly verify that u t − = ( ψφ ) x , (23)where spectrum functions ψ and φ defined by (19)–(22), is a negative flow of the cKP3-4 equation. By usingthe method proposed in Refs. [33, 34], the whole negative KP hierarchy can be obtained from (23) in termsof the Lax operators defined in (19) and (21).The existence of the Lax pairs allows one to study the cKP3-4 equation by means of the standard methodssuch as the inverse scattering transformation, Darboux transformation, ¯ ∂ approach and so on. To study themultiple soliton solutions, the Hirota’s bilinear method is a simplest way. (18) By making the transformation u = 2(ln f ) xx , w = 2(ln f ) xy , z = 2(ln f ) yy , (24)the cKP3-4 equation (18) becomes(2 f xxx f f xx + f xxxxx f − f f xyy − f xx f x − f xxxx f f x + 3 f f x f yy + 6 f f y f xy + 8 f xxx f x − f x f y ) a +(2 f xx f xxy f − f yyy f − f xx f y − f xx f xy f x + f xxxxy f − f xxxy f f x − f xxxx f y f + 3 f yy f y f +4 f xxy f x + 4 f xxx f y f x − f y ) b − f xxt f + 2 f xt f x f + f t f xx f − f t f x = 0 . (25)The trilinear equation (25) can not be directly bilinearized. However, if we introduce an auxiliary variable τ such that [ D x D τ + a (3 D y − D x )] f · f = 0 , (26)then the trilinear equation (25) can be rewritten as,( f ∂ x − f x )[ a (2 bD x D y − D x D t + 3 D x D τ ) + bD y D τ ] f · f = 0 , which can be solved by the bilinearized equation[ a (2 bD x D y − D x D t + 3 D x D τ ) + bD y D τ ] f · f = 0 . (27)In (26)–(27), the Hirota’s bilinear operator D x is defined by D nx f · g = ( ∂ x − ∂ x ′ ) n f ( x ) g ( x ′ ) | x ′ = x , (28)and the other operators { D y , D t , D τ } are defined in the same way.Finally, the multiple soliton solutions of the cKP3-4 equation (18) are given by (24) with f being asolution of the trilinear equation (25) which can be solved by the bilinear equation system (26) and (27).The procedure solving the bilinear equation system (26) and (27) is standard and well known. We just writedown the final result in a fully space-time reversal symmetric form [35, 36], f = X { ν } K { ν } cosh N X i =1 ν i ξ i ! , (29) ξ i = k i x + l i y + ak i ( k i − l i )( t + τ ) + bl i k i ( k i − l i ) t + ξ i , (30) K { ν } = Y i 1] tanh( η + y ) tanh( η − x ) . (50)The expression (50) displays the dissipative soliton (or kink-antikink molecule) structure for c = 0 or c = 0)and the PK-PAK molecule (or periodic dissipative soliton) structure for cc = 0. Fig. 4f is a plot of a specialPK-PAK molecule expressed by (50) with the parameter selections x = − , y = − , c = 0 . , c = 5at time t = 0.Without the condition (45), the solution of (44) can be written as U ( ξ ) = 3 akl + bl + k ω k ( ak + bl ) − c k (2 m − 1) + 2 c k m cn ( cξ − ξ , m ) (51)with arbitrary constants c, ξ , m, k, l and ω . In additional to the arbitrary travelling wave solution (46),the general travelling wave of the cKP3-4 (51) is a periodic wave for m = 1 and/or a soliton solution for m = 1. 7 a) –3–2–10123 x–3 –2 –1 0 1 2 3y–0.400.40.8 (b) –3–2–10123 x–1–0.5 0 0.5 1y–0.8–0.400.40.8 (c) –4–2024 x–4 –2 0 2 4y–1–0.500.51 (d) –15–10–5051015 x–20 –10 0 10y00.511.52(e) –15–10–5051015 x–30 –20 –10 0 10y00.511.52 (f) –10–50510 x–10 –5 0 5 10y00.51 Figure 3: (a) Few cycle soliton structure expressed by (47) with c = 4 at time t = 0. (b) Few cycle solitonstructure expressed by (47) with c = 50 at time t = 0. (c) Periodic kink given by (48) with c = 0 . t = 0. (d) Kink-kink molecule given by (49) with c = 0 , k = − , k = − x = 8. (e) Kink-kinkmolecule given by (49) with c = 0 . , k = − , k = − , k = 5 and x = 10 at t = 0. (f) PK-APK moleculegiven by (50) with x = y = − c = − c = 0 . t = 0.8 Conclusions and discussions In summary, there are some different types of (2+1)-dimensional KdV extensions. In this paper, it isfound that in addition to the KP3 equation, the KP4 equation is also an extension of the KdV equationin (2+1) dimensions. Thus, the combination of the KP3 equation and the KP4 equation (i.e. the cKP3-4equation (18)) is also a (2+1)-dimensional KdV extension. The Lax pair and the dual Lax pair of thecKP3-4 equation are explicitly given. The cKP3-4 equation can be bilinearized by introducing an auxiliaryparameter. The multiple soliton solutions can be directly written down with help of the bilinearized equationsystem and the auxiliary parameter can be absorbed by the soliton position parameters.The cKP3-4 equation is quite different from any member of the KP hierarchy, say, the KP3 and KP4equations of the hierarchy (12). Some interesting properties are valid only for the combined system (18) butnot for the separated KP3 and KP4 systems. For instance, the soliton molecules exist only for the cKP3-4 butnot for the KP3 and KP4 equations. Any numbers of solitons can be involved in one soliton molecule. TheD’Alembert type solutions (arbitrary travelling wave moving in one direction with fixed model dependentvelocity) including various new types of solitons and soliton molecules can be found for the cKP3-4 equation.This kind of arbitrariness exists for linear waves but it has not yet found for any other known nonlinearsystems. The more about the cKP3-4 equation (18), say the interactions among the special local excitationslike shown in Fig. 3 and the usual solitons (24) with (29), should be further studied. Acknowledgements The work was sponsored by the National Natural Science Foundations of China (Nos.11975131,11435005)and K. C. Wong Magna Fund in Ningbo University. References [1] D. G. Crighton, Appl. Math. 39 (1995).[2] H. Y. Guo, Z. H. Wang and K. Wu, Phys. Lett. B 277 (1991).[3] C. S. Gardner, J. M. Greene, M. D. Kruskal and R. M. Miura, Phys. Rev. Lett. 539 (1970).[7] L.P. Nizhnik, Sov. Phys. 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