Reductions of the (4 + 1)-dimensional Fokas equation and their solutions
NNonlinear Dynamics manuscript No. (will be inserted by the editor)
Reductions of the (4 + 1) -dimensional Fokas equationand their solutions
Yulei Cao · Jingsong He ∗ · Yi Cheng · Dumitru Mihalache
Received: date / Accepted: date
Abstract
An integrable extension of the Kadomtsev-Petviashvili (KP) andDavey-Stewartson (DS) equations is investigated in this paper. We will refer tothis integrable extension as the (4 + 1)-dimensional Fokas equation. The deter-minant expressions of soliton, breather, rational, and semi-rational solutions ofthe (4 + 1)-dimensional Fokas equation are constructed based on the Hirota’sbilinear method and the KP hierarchy reduction method. The complex dynam-ics of these new exact solutions are shown in both three-dimensional plots andtwo-dimensional contour plots. Interestingly, the patterns of obtained high-order lumps are similar to those of rogue waves in the (1 + 1)-dimensions bychoosing different values of the free parameters of the model. Furthermore,three kinds of new semi-rational solutions are presented and the classificationof lump fission and fusion processes is also discussed. Additionally, we give anew way to obtain rational and semi-rational solutions of (3 + 1)-dimensionalKP equation by reducing the solutions of the (4 + 1)-dimensional Fokas equa-tion. All these results show that the (4 + 1)-dimensional Fokas equation isa meaningful multidimensional extension of the KP and DS equations. Theobtained results might be useful in diverse fields such as hydrodynamics, non-linear optics and photonics, ion-acoustic waves in plasmas, matter waves inBose-Einstein condensates, and sound waves in ferromagnetic media. ∗ Corresponding author, Jingsong He: [email protected]; [email protected] Cao, Yi ChengSchool of Mathematical Sciences, USTC, Hefei, Anhui 230026, P. R. ChinaJingsong HeInstitute for Advanced Study, Shenzhen University, Shenzhen, Guangdong 518060, P. R.ChinaDumitru MihalacheHoria Hulubei National Institute of Physics and Nuclear Engineering, Magurele, RO 077125,Romania a r X i v : . [ n li n . S I] J u l Yulei Cao et al.
Keywords
4D Fokas equation · KP hierarchy reduction method · Rationalsolution · Semi-rational solution ·
3D KP equation
The research field of solitary waves is in fact an interdisciplinary research areathat has been deeply studied both theoretically and experimentally. Solitarywaves in hydrodynamics originated from the accidental discovery of Russell in1834 [1]. Nevertheless, he failed to give a rigorous proof of the existence of suchspecial type of waves. More than 60 years later, Korteweg and de Vries [2] madea comprehensive analysis of these solitary waves and established a mathemat-ical model of shallow water waves that adequately describes their complex dy-namics. About 70 years later, in 1965, Zabusky and Kruskal [3] found numeri-cally that such solitary waves have the property of elastic scattering, and calledthem ”solitons”. This pioneering work of Zabusky and Kruskal was a milestonein the area of solitons and all that. Since then, the study of solitons has begunto flourish in various fields such as nonlinear optics and optical fibers [4, 5],condensed matter, fluid mechanics, and plasma physics. The current researchmainly focuses on (1 + 1)-dimensional (1D) and (2 + 1)-dimensional (2D) sys-tems. However the physical space in reality is (3 + 1)-dimensional (3D) andone of the most important open problems in soliton theory is to constructintegrable nonlinear partial differential equations (NPDEs) in higher than twospatial dimensions. Therefore, the research value of high-dimensional nonlinearsystems is enormous. In order to seek new high-dimensional integrable NPDEs,many researchers have made great efforts during the past decades [6–26]. But,there are still many meaningful open problems to be addressed. With the in-creasing number of variables, solving high-dimensional NPDEs will be verydifficult. Therefore, it is a challenging work to obtain exact solutions of high-dimensional nonlinear systems. Furthermore, a natural problem is whetherthe exact solutions of high-dimensional NPDEs can be reduced to the exactsolutions of low-dimensional NPDEs?Inspired by the above problems, we consider the (4 + 1)-dimensional (4D)Fokas equation [27]: u x t − u x x x x + 14 u x x x x + 32 ( u ) x x − u y y = 0 . (1)This equation was introduced by Fokas in 2006 [27], being an integrable ex-tension of the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equa-tions. Because of the important physical applications of KP and DS equations,the 4D Fokas equation may be used to describe surface and internal waves inrivers with different physical situations. Solitons [28, 29], quasi-periodic so-lutions [30], lumps [31, 32] and lump-soliton solutions [33] for the 4D Fokasequation have been investigated. However, these studies are far from beingcomplete. To the best of authors’ knowledge, high-order rational and semi-rational solutions for the 4D Fokas equation have never been reported. In this eductions of the (4 + 1)-dimensional Fokas equation and their solutions 3 paper, we mainly focus on the new exact solutions of the 4D Fokas equation,and how to reduce the exact solutions of the 4D Fokas equation to the exactsolutions of low-dimensional NPDEs.The structure of this paper is as follows. In section 2, the determinantexpressions of soliton and breather solutions are constructed by using the KPhierarchy reduction method. In sections 3 and 4, high-order rational and semi-rational solutions are generated for the 4D Fokas equation and the complexdynamic behavior of the corresponding solutions are shown by both three-dimensional plots and two-dimensional contour plots. Then in sections 5, anew way for obtaining rational and semi-rational solutions of 3D KP equationis presented. Finally, in section 6 we discuss and summarize our results. In this section, we introduce the determinant expression of soliton and breathersolutions for the 4D Fokas equation. Through the following transformation: x = k x + k x , the 4D Fokas equation (1) becomes the following 3D equation u xt + 14 k ( k − k ) u xxxx + 3 k u ) xx − k u y y = 0 . (2)Additionally, if we further make the transformation y = k y + k y , then the 4D Fokas equation becomes u xt + 14 k ( k − k ) u xxxx + 3 k u ) xx − k k k u yy = 0 . (3)Now we make the variable transformation u = ( k − k )(ln f ) xx . (4)Then the 4D Fokas equation (1) is transformed into the following bilinear form:[ D x + 4 k ( k − k ) D x D t − k k k k ( k − k ) D y ] f · f = 0 , (5)where D is Hirota’s bilinear differential operator [34]. Applying the change ofindependent variables z = x, z = (cid:115) k k ( k − k )2 k k iy, z = − k ( k − k ) t, (6) Yulei Cao et al. the bilinear form (5) can be transformed into the following bilinear equationof the KP hierarchy [8]:[ D z − D z D z + 3 D z ] f · f = 0 . (7)According to Sato theory [35,36], we construct the Gram determinant solutionsof the 4D Fokas equation. Theorem 1.
The 4D Fokas equation (1) admits the following soliton andbreather solutions: u = ( k − k )(ln f ) xx , x = k x + k x , (8)with f = det ≤ i,j ≤ N ( m (0) i,j ) , where m ( n ) i,j = δ ij + 1 p i + p ∗ j ( − p i p ∗ j ) n e ξ i + ξ ∗ j , (9) ξ j = k p i x + k p i x + (cid:115) k k ( k − k ) − k k p i y + (cid:115) k k ( k − k ) − k k p i y − k ( k − k ) p i t + ξ i . Here δ ij = 0 , p i and ξ i are arbitrary complex constants, i, j , and N arearbitrary positive integers, and the asterisk denotes the complex conjugation.We must emphasize that k k ( k − k ) − k k > Lemma 1.
The bilinear equation of KP hierarchy (7) has solutions τ n = det ≤ i,j ≤ N ( m ( n ) i,j ) , (10)with the matrix element m ( n ) i,j satisfying the following differential and differencerelations ∂ z m ( n ) i,j = ϕ ( n ) i ψ ( n ) j ,m ( n +1) i,j = m ( n ) i,j + ϕ ( n ) i ψ ( n +1) j ,∂ z m ( n ) i,j = ϕ ( n +1) i ψ ( n ) j + ϕ ( n ) i ψ ( n − j ,∂ z m ( n ) i,j = ϕ ( n +2) i ψ ( n ) j + ϕ ( n +1) i ψ ( n − j + ϕ ( n ) i ψ ( n − j ,∂ z k ϕ = ϕ ( n + k ) i , ∂ z k ψ i = − ψ ( n − k ) i , ( k = 1 , , . (11)Here m ( n ) i,j , ϕ ( n ) i , and ψ ( n ) j are functions of the variables z , z , and z . eductions of the (4 + 1)-dimensional Fokas equation and their solutions 5 Proof of Lemma 1.
Reusing the differential of determinant and the ex-pansion formula of bordered determinant [35,36], the derivatives of the τ func-tions can be expressed by the following bordered determinants: ∂ z τ n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ( n ) i,j ϕ ( n ) i − ψ ( n ) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,∂ z τ n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ( n ) i,j ϕ ( n +1) i − ψ ( n ) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ( n ) i,j ϕ ( n ) i − ψ ( n − j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,∂ z τ n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ( n ) i,j ϕ ( n +2) i − ψ ( n ) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ( n ) i,j ϕ ( n +1) i − ψ ( n − j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ( n ) i,j ϕ ( n ) i − ψ ( n − j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,∂ z ∂ z τ n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ( n ) i,j ϕ ( n +3) i − ψ ( n ) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ( n ) i,j ϕ ( n ) i ψ ( n − j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,∂ z τ n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ( n ) i,j ϕ ( n +3) i − ψ ( n ) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ( n ) i,j ϕ ( n +1) i ψ ( n − j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ( n ) i,j ϕ ( n +2) i − ψ ( n − j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ( n ) i,j ϕ ( n ) i ψ ( n − j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,∂ z τ n = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ( n ) i,j ϕ ( n +3) i − ψ ( n ) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 3 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ( n ) i,j ϕ ( n +2) i ψ ( n − j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + 3 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ( n ) i,j ϕ ( n +1) i − ψ ( n − j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m ( n ) i,j ϕ ( n ) i ψ ( n − j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . As a result: ( ∂ z − ∂ z ∂ z + 3 ∂ z ) τ n × τ n = 0 , ∂ z τ n × ( ∂ z τ n − ∂ z τ n ) + 6( ∂ z τ n ) − ∂ z τ n ) = 0 . This completes the proof of Lemma 1. Then, we will prove Theorem 1 withLemma 1.
Proof of Theorem 1.
In order to construct soliton and breather solutionsfor the bilinear equation (7), we choose functions m ( n ) i,j , ϕ ( n ) i , and ψ ( n ) j as follows m ( n ) i,j = δ ij + 1 p i + q j ϕ ( n ) i ψ ( n ) j ,ϕ ( n ) i = p ni e ξ i ,ψ ( n ) j = ( − q j ) − n e η j , (12)where ξ i = p i z + p i z + p i z + ξ i ,η j = q j z − q j z + q j z + η j , and p i , q j , ξ i , and η j are arbitrary complex constants. Through the followingrestrictions: z = k x + k x , z = (cid:115) k k ( k − k )2 k k i ( k y + k y ) , Yulei Cao et al. z = − k ( k − k ) t, q ∗ j = p j , η ∗ j = ξ j , and then setting f = τ , δ ij = 0 ,
1, the solutions of bilinear equation (7) can betransformed into the solutions of the 4D Fokas equation. This completes theproof of Theorem 1. Without losing generality, we take k = 1 , k = 2 , k = 1,and k = 1 in this section.2.1 N -soliton solutionsEquation (1) admits N -soliton solutions, assuming δ ij = 1 when i = j , and δ ij = 0 when i (cid:54) = j in (8). The one-soliton solution u is generated by taking N = 1 and p = p R − ip I : u = 24 p R e p R [ x + 2 x − √ p I ( y + y ) − p R − p I ) t ] (cid:0) p R + e p R [ x + 2 x − √ p I ( y + y ) − p R − p I ) t ] (cid:1) . (13)From the above expressions, it is not difficult to calculate that the maximumamplitude of the one-soliton solution is 3 p R ; when x , y −→ ±∞ solution u approaches to the constant background plane 0 in the ( x , y )-plane [seeFig. 1(a)]. The velocity and center of the soliton are 6 p R − p I and x +2 x − √ p I ( y + y ) − p R − p I ) t , respectively. By taking the parameters N = 2 , p = 1 − i and p = 1 + i in equation (8) we obtain the expression ofthe two-soliton solution u [see Fig. 1(b)]: u = 120 e ι + 120 e ι + 960 e ι + 2400 e ι + 2400 e ι ( e ζ + 10 e ζ + 10 e ζ + 20 e ζ ) , (14) ζ = 4 x + 8 x + 2 √ y + 2 √ y , ζ = 2 √ y + 2 √ y + 6 t,ζ = 2 x + 4 x + 4 √ y + 4 √ y + 3 t, ζ = 2 x + 4 x + 3 t,ι = ζ + ζ , ι = ζ + ζ , ι = ζ + ζ ,ι = ζ + ζ , ι = ζ + ζ . Additionally, taking the parameters N = 3 , p = 1 − i , p = 1 + i , and p = 1 in equation (8), the three-soliton solution is obtained. We also give theexpression of the three-soliton solution in the ( x , y )-plane [see Fig. 1(c)] inwhich f is expressed as f =1 + 12 e x +4 x +2 √ y +2 √ y − t + 168 e x +8 x +2 √ y +2 √ y − t + 168 e x +8 x − √ y − √ y − t + 12 e x +4 x − √ y − √ y − t + 111560 e x +12 x − t + 12 e x +4 x − t + 120 e x +8 x − t . (15) eductions of the (4 + 1)-dimensional Fokas equation and their solutions 7(a) t=0 (b) t=0 (c) t=0 Fig. 1
Dynamic behavior of solutions of the 4D Fokas equation defined in equation (8). (a):one-soliton solution with parameters N = 1 , δ = 1 , p = 1 − i , x = 0 , y = 0, and t = 0;(b): two-soliton solution with parameters N = 2 , δ = 1 , δ = 0 , δ = 0 , δ = 1 , p =1 − i , p = 1 + i , x = 0 , y = 0, and t = 0; (c): three-soliton solution with parameters N = 3 , δ jj = 1 , δ ij = 0( i, j = 1 , , , p = 1 − i , p = 1 + i , p = 1 , x = 0 , y = 0, and t = 0. (a) t=0 (b) t=0 Fig. 2
A hybrid of a V-type soliton and one breather of the 4D Fokas equation withparameters N = 2 , p = + i, p = − i, δ ij = 1 ( i, j = 1 , , x = 0 , y = 0, and t = 0 in(8). Panel (b) is the contour plot of panel (a). N ≥ δ ij = 1 and some param-eters p i are complex in equation (8). We first consider the case of N = 2 and δ ij = 1. The following parameters are further taken in equation (8): p = 12 + i, p = 12 − i, δ = 1 , δ = 1 , and the mixed solution consisting of a V-type soliton and one breather solutionis derived, see Fig. 2. For this mixed solution the expression of f is as follows: f = cosh[ γ − √ y + y )] + sinh[ γ − √ y + y )] + cosh[ γ + 2 √ y + y )]+ sinh[ γ − √ y + y )] + 45 [cosh(2 γ ) + sinh(2 γ )] −
45 [cosh( γ )+ sinh( γ )] sin( γ ) −
25 [cosh( γ ) + sinh( γ )] cos( γ ) , (16) Yulei Cao et al. where γ = x + 2 x + 332 t, γ = 2 x + 4 x + 3 t. (17)Furthermore, for larger N , we can derive the mixed solution consisting of aV-type soliton and more breathers. For example, when we take the parameters N = 3 , p = + i, p = − i, p = and δ ij = 1 ( i, j = 1 , ,
3) in equation(8) the mixed solution consisting of a V-type soliton and two breathers ispresented in Fig. 3. (a) t=0 (b) t=0
Fig. 3
A hybrid of a V-type soliton and two breathers of the 4D Fokas equation withparameters N = 3 , p = + i, p = − i, p = , δ ij = 1 ( i, j = 1 , , , x = 0 , y = 0, and t = 0 in (8). Panel (b) is the contour plot of panel (a). The rational solutions of low-dimensional integrable systems have been ex-tensively investigated. However, there are few studies of rational solutions inhigh-dimensional systems. Inspired by the works of Ohta and Yang [36–38],the rational solutions of the 4D Fokas equation are constructed by introducingthe following Lemma.
Lemma 2.
The bilinear equation of KP hierarchy (7) has solutions τ (cid:48) n = det ≤ i,j ≤ N ( M ( n )2 i − , j − ) , (18)with the matrix element M ( n ) i,j satisfying the following differential and differ-ence relations ∂ z M ( n ) i,j = Φ ( n ) i Ψ ( n ) j ,m ( n +1) i,j = M ( n ) i,j + Φ ( n ) i Ψ ( n +1) j ,∂ z M ( n ) i,j = Φ ( n +1) i Ψ ( n ) j + Φ ( n ) i Ψ ( n − j ,∂ z M ( n ) i,j = Φ ( n +2) i Ψ ( n ) j + Φ ( n +1) i Ψ ( n − j + Φ ( n ) i Ψ ( n − j ,∂ z k Φ = Φ ( n + k ) i , ∂ z k Ψ i = − Ψ ( n − k ) i , ( k = 1 , , . (19) eductions of the (4 + 1)-dimensional Fokas equation and their solutions 9 Here M ( n ) i,j , Φ ( n ) i , and Ψ ( n ) j are functions of the variables z , z , and z . Theabove relations were proven in [36], hence we omit here the proof. The functions Φ ( n ) i , Ψ ( n ) j , and M ( n ) ij are defined by Φ ( n ) i = A i p n e ξ , Ψ ( n ) j = B j ( − q ) e η ,M ( n ) i,j = (cid:90) z Φ ( n ) i Ψ ( n ) j dz = A i B j p + q ( − pq ) n e ξ + η , (20)where A i = i (cid:88) k =0 c k ( i − k )! ( p∂ p ) i − k , B j = j (cid:88) l =0 d l ( j − l )! ( q∂ q ) j − l ,ξ = pz + p z + p z + ξ , η = qz − q z + q z + η . (21)For simplicity, we can rewrite the functions M ( n ) i,j as M ( n ) i,j = i (cid:88) k =0 c k ( i − k )! ( p∂ p + ξ (cid:48) + n ) i − k × j (cid:88) l =0 d l ( j − l )! ( q∂ q + η (cid:48) − n ) j − l p + q , (22)where ξ (cid:48) = pz + 2 p z + 3 p z , η (cid:48) = qz − q z + 3 q z , (23)and p , q , c k , and d l are arbitrary complex constants. Further, taking the pa-rameter constraints p = q = 1 , c k = d ∗ k , setting τ (cid:48) = f , z = k x + k x , z = (cid:113) k k ( k − k )2 k k i ( k y + k y ), and z = − k ( k − k ) t , the rational solutions of the 4D Fokas equation can begenerated from equation (7). Based on the above results, the rational solutionsof the 4D Fokas equation are presented in the following Theorem. Theorem 2.
The (4 + 1)-dimensional Fokas equation (1) has rational so-lutions u = ( k − k )(ln f ) xx , x = k x + k x , (24)where f = det ≤ i,j ≤ N ( M ( n )2 i − , j − ) | n =0 . (25)The matrix elements in f are defined by M ( n ) i,j = i (cid:88) k =0 c k ( i − k )! ( p∂ p + ξ (cid:48) + n ) i − k × j (cid:88) l =0 c ∗ l ( j − l )! ( p ∗ ∂ p ∗ + ξ (cid:48) ∗ − n ) j − l p + p ∗ | p =1 , (26) ξ (cid:48) = k x + k x + (cid:115) k k k ( k − k ) − k y + (cid:115) k k k ( k − k ) − k y − k ( k − k ) t, the asterisk denotes the complex conjugation, i, j, k , and l are arbitrary pos-itive integers, and c k and c l are arbitrary complex constants. We take k =1 , k = , k = 1, and k = 1 in this section.3.1 Fundamental rational solutionAccording to Theorem 2, taking the parameters N = 1, c = 1, and c = 0in equation (24), we first derive the fundamental rational solution of the 4DFokas equation: u = 2( k − k ) k k k k ( k − k )[ k y + k y ] + k k l lump − k k (cid:16) k k ( k − k )[ k y + k y ] + 9 k k l lump + k k (cid:17) , (27)where l lump = k x + k x + 3( k k − k ) t + 12 . (28)As can be seen from the above expressions, in order to ensure that the funda-mental rational solution is non-singular, k k k k ( k − k ) > x , y )-plane, see Fig. 4: Λ = ( x , y ) = (cid:18) − k k x + 3( k k − k k ) t + 12 k , y − k k y (cid:19) ,Λ = ( x , y ) = (cid:32) − k k x + 3( k k − k k ) t + 1 + √ k , y − k k y (cid:33) ,Λ = ( x , y ) = (cid:32) − k k x + 3( k k − k k ) t + 1 − √ k , y − k k y (cid:33) . After simple calculations, we get a maximum value H Max = H ( x , y ) =8( k − k ) and two minimum values H Min = H ( x , y ) = H ( x , y ) = k − k of the lump solution. The lump trajectory is k x + k x + 3( k k − k ) t + = 0.3.2 High-order rational solutionsIn this section, we consider the high-order rational solutions of the 4D Fokasequation. N -order lump solutions are derived in the ( x , y )-plane from The-orem 2 for any given N . For example, taking N = 2 , k = 1 , k = , k = 1, eductions of the (4 + 1)-dimensional Fokas equation and their solutions 11(a) t=0 (b) t=0 Fig. 4
First-order rational solution for the 4D Fokas equation in the ( x , y )-plane withparameters N = 1 , c = 1 , c = 0 , k = 1 , k = , k = 1 , k = 1 , x = 0 , y = 0, and t = 0in equation (24). Panel (b) is the contour plot of panel (a).(a) t=0 (b) t=0(c) t=0 (d) t=0 Fig. 5
Second-order rational solutions of the 4D Fokas equation with parameters N =2 , c = 0 , c = 0 , k = 1 , k = , k = 1 , k = 1 , x = 0 , y = 0, and t = 0 in equation (24).(a): a fundamental pattern with c = 1 and c = − ; (b): a triangular pattern with c = 1and c = −
30. Panels (c) and (d) are the contour plots of panels (a) and (b), respectively. and k = 1, the second-order lump solutions u are obtained u = ( k − k ) (cid:32) ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M (0)11 M (0)13 M (0)31 M (0)33 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:33) xx , (29)where M ( n ) i,j are defined in Theorem 2. As shown in Fig. 5, the second-orderlumps have two types of patterns, which are controlled by four free parameters.Similarly, the third-order lump solutions are derived by taking N = 3 , k =1 , k = , k = 1, and k = 1. The third-order lumps have three types ofpatterns, which are controlled by six parameters, see Fig. 6. For larger values of N , as more free parameters will be generated, the patterns of the lumps willbe more abundant and their dynamic behavior will be more complicated. Wenote that the pattern dynamics of high-order lumps is similar to rogue wavesdynamics in (1 + 1)-dimensional systems. (a) t=0 (b) t=0 (c) t=0(d) t=0 (e) t=0 (f) t=0 Fig. 6
Third-order rational solutions of the 4D Fokas equation with parameters N =3 , c = 0 , c = 0 , c = 0 , k = 1 , k = , k = 1 , k = 1 , x = 0 , y = 0, and t = 0 inequation (24). (a): a fundamental pattern with c = 1 , c = − and c = − ; (b): atriangular pattern with c = 1 , c = − and c = 0; (c): a ring pattern with c = 1 , c = 0and c = 20. Panels (d), (e), and (f) are the contour plots of panels (a), (b), and (c),respectively. In this section, we present a Theorem for constructing the semi-rational solu-tions of the 4D Fokas equation. In order to obtain the semi-rational solutionsof the 4D Fokas equation, we will first introduce the following differential op-erators Ξ i = n i (cid:88) k =0 a ik ( p i ∂ p i ) n i − k , (cid:102) j = n j (cid:88) l =0 a ∗ jl ( p ∗ j ∂ p ∗ j ) n j − l . (30) eductions of the (4 + 1)-dimensional Fokas equation and their solutions 13 We choose the following functions ϕ ( n ) i = Ξ i p ni e ξ i ,ψ ( n ) j = (cid:102) j ( − q j ) − n e η j ,K ( n ) i,j = Ξ i (cid:102) j p i + p ∗ j [ δ ij + ( − p i p ∗ j ) n e ξ i + ξ ∗ j ] . (31)The functions ϕ ( n ) i and ψ ( n ) j also satisfy the equation (11). For simplicity, werewrite the matrix element K ( n ) i,j as K ( n ) i,j = ( − p i p ∗ j ) e ξ i + ξ ∗ j n i (cid:88) k =0 a ik ( p i ∂ p i + ξ (cid:48) i + n ) n i − k × n j (cid:88) l =0 a ∗ jl ( p ∗ j ∂ p ∗ j + ξ (cid:48) ∗ j − n ) n j − l p i + p ∗ j + δ ij a in i a ∗ jn j , (32)where ξ i = p i z + p i z + p i z + ξ i ,ξ (cid:48) i = p i z + 2 p i z + 3 p i z . (33)Here p i and a ik are arbitrary complex constants, δ ij = 0 ,
1, and n i are ar-bitrary positive integers. Furthermore, taking τ = f , z = k x + k x , z = (cid:113) k k ( k − k )2 k k i ( k y + k y ), and z = − k ( k − k ) t , then, the semi-rational solutions of the 4D Fokas equation would be derived. Thus, semi-rational solutions of 4D Fokas equation can be determined by the followingTheorem. Theorem 3.
The (4 + 1)-dimensional Fokas equation (1) has semi-rationalsolutions u = ( k − k )(ln f ) xx , x = k x + k x , (34)where f = det ≤ i,j ≤ N ( K ( n ) i,j ) | n =0 , (35)and the matrix elements in f are defined by K ( n ) i,j = ( − p i p ∗ j ) e ξ i + ξ ∗ j n i (cid:88) k =0 c ik ( p i ∂ p i + ξ (cid:48) i + n ) n i − k × n j (cid:88) l =0 c ∗ jl ( p ∗ j ∂ p ∗ j + ξ (cid:48) ∗ j − n ) n j − l p i + p ∗ j + δ ij a in i a ∗ jn j , (36) ξ i = k p i x + k p i x + (cid:115) k k k ( k − k ) − k p i y + (cid:115) k k k ( k − k ) − k p i y − k ( k − k ) p i t + ξ i ,ξ (cid:48) i = k p i x + k p i x + (cid:115) k k k ( k − k ) − k p i y + (cid:115) k k k ( k − k ) − k p i y − k ( k − k ) p i t. The asterisk denotes the complex conjugation, i, j, k , and l are arbitrary pos-itive integers, and k , k , k , and k are arbitrary real constants. It is notdifficult to find that the semi-rational solutions will become rational solutionswhen δ ij = 0. We note that the patterns of rational solutions are similar tothose corresponding to rational solutions of the Davey-Stewartson equation,reported in Refs. [39–41].4.1 Lumps on one-soliton backgroundThe semi-rational solution u ls consisting of a lump and a soliton is derived bytaking N = 1 , n i = 1 , a = 0 , a = 1 , p = 1 , δ ii = 1, and δ ij = 0 ( i (cid:54) = j ) inequation (34). The expression of u ls is as follows u ls = 2( k − k ) k k (cid:26) [2 k k ( k − k ) l + k k l lump − k k ] e ξ − k k ( l lump + 1) + 8 k k ( k − k ) l + k k (cid:27) e ξ (cid:16) [18 k k ( k − k ) l + 9 k k l lump + k k ] e ξ + 18 k k (cid:17) , (37)where l = k y + k y , ξ = 2 k x + 2 k x + 2( k k − k ) t,l lump = k x + k x + 3( k k − k ) t. By choosing different parameters k , k , k . and k , we derive a lump fusinginto or fissioning from a dark soliton or from a bright soliton, see Fig. 7. Theclassification of four different types of interaction between lumps and one-soliton solutions is given in Table 1. From the above results, we can easilycalculate that the velocities of lump and soliton are V lump = 3 k k − k k and V soliton = k k − k k , respectively. The velocity of lump is always greater thanthat of soliton, see Fig. 7.The semi-rational solutions consisting of more lumps and a soliton aregenerated for N = 1 and n i ≥ N =1 , n i = 2 , a = 0 , a = 4 , a = 1 , p = 1 , δ = 1, and δ = 0 in equation(34), we can derive the semi-rational solutions composed of two lumps and asoliton. There are also four distinct types of such semi-rational solutions. To eductions of the (4 + 1)-dimensional Fokas equation and their solutions 15(a) t=-5 (b) t=0 (c) t=5(d) t=-5 (e) t=0 (f) t=5(g) t=-5 (h) t=0 (i) t=5(j) t=-5 (k) t=0 (l) t=5 Fig. 7
Time evolution of semi-rational solution u ls of the 4D Fokas equation, (a,b,c): Abright lump is annihilated by a bright soliton with parameters k = − , k = − . , k =1 , k = 1 , x = 0, and y = 0; (d,e,f): A bright lump is created from a bright soliton withparameters k = 1 , k = − . , k = 1 , k = 1 , x = 0, and y = 0; (g,h,i): A dark lump isannihilated by a dark soliton with parameters k = − . , k = 1 , k = 1 , k = − , x = 0,and y = 0; (j,k,l): A dark lump is created from a dark soliton with parameters k = − . , k = − , k = 1 , k = − , x = 0, and y = 0. have an idea of the dynamics of such semi-rational solutions, we show hereonly the process of fission of two lumps from a bright soliton, see Fig. 8.4.2 Lumps on multi-solitons backgroundFor N ≥ , n i = 1 , δ = 1 , δ = 0 , δ = 0, and δ = 1 in equation (34), thesemi-rational solutions consisting of more lumps and more solitons are derived.For example, taking N = 1 , n i = 2 , k = 1 , k = 2 , k = 1 , k = 1 , a =1 , a = 1 , a = 1 , a = 1 , p = , p = , δ = 1 , δ = 0 , δ = 0, and δ =1 in equation (34), we obtain the interaction of local wave structures describedby two lumps and two solitons. The exact expression of the corresponding k k > k < k < k k < | k | > k > > k by a bright soliton k k > k > k > k k < k > | k | > > k from a bright soliton k k > | k | > k > > k a lump is annihilated k k < k > k > k k > k > | k | > > k a lump is created k k < k < k < Table 1
Classification of four different types of interactions between lumps and one-solitonsolution. (a) t=-7 (b) t=-1(c) t=1 (d) t=7
Fig. 8
Time evolution of the process of fission of two lumps from a bright soliton of the4D Fokas equation with parameters N = 1 , n i = 2 , a = 0 , a = 4 , a = 1 , p = 1 , δ ii =1 , δ = 0 , k = 1 , k = 1 . , k = 1 , k = 1 , x = 0, and y = 0 in equation (34). solution u ls is as follows u = ( k − k ) (cid:32) ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K (0)11 K (0)12 K (0)21 K (0)22 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:33) xx , (38)where K (0)11 = 1 + (cid:18)
34 ( y + y ) + 14 ( x + 2 x − t + 1) + 14 (cid:19) e x +2 x − t ,K (0)12 = (cid:18) y + y ) x + 2 x − t + 1310 ) −
516 ( t + 25 ) + 36125 + A (cid:19) e (cid:37) ,K (0)21 = (cid:18) y + y ) x + 2 x − t + 1310 ) −
516 ( t + 25 ) + 36125 − A (cid:19) e (cid:37) , eductions of the (4 + 1)-dimensional Fokas equation and their solutions 17 K (0)22 = 1 + (cid:18)
29 ( y + y ) + 6( x + 2 x − t + 32 ) + 38 (cid:19) e x + x − t ,A = √− y + y )( 13 x + 23 x + t + 915 ) ,(cid:37) = 56 x + 53 x − t + 5 √−
336 ( y + y ) ,(cid:37) = 56 x + 53 x − t − √−
336 ( y + y ) . The five panels in Fig. 9 describe the process of creation of two lumps fromthe background of two solitons. As shown in Fig. 9, with time evolution morepeaks are created during the interaction between lumps and solitons around t = 2 .
5, then two lumps and two solitons are completely separated around t = 10. (a) t=-10 (b) t=-2 (c) t=0(d) t=2 (e) t=10 Fig. 9
The time evolution of fission of two lumps from two solitons for the 4D Fokasequation with parameters N = 1 , n i = 2 , a = 1 , a = 1 , a = 1 , a = 1 , p = , p = , δ = 1 , δ = 0 , δ = 0 , δ = 1 , k = 1 , k = 2 , k = 1 , k = 1 , x = 0, and y = 0 inequation (34). N = 2 , n i = 2 and δ ij = 1 in equation(34). The corresponding semi-rational solution is shown in Fig. 10, Obviously, the process of their interaction is elastic, the amplitudes and shapes of soli-ton, breather, and lumps did not change after the interaction. This type ofsemi-rational solution has never been reported elsewhere, to the best of ourknowledge. (a) t=-4 (b) t=4 Fig. 10
Time evolution of semi-rational solution consisting of two lumps, a breather, anda V-type soliton of the 4D Fokas equation with parameters N = 2 , n i = 2 , k = 1 , k =2 , k = 1 , k = 1 , a = 1 , a = 1 , a = 1 , a = 1 , p = 1 + i , p = 1 − i , δ = 1 , δ =1 , δ = 1 , δ = 1 , x = 0, and y = 0 in equation (34). The 3D KP equation can be read as follows [42]( W ς + 6 W W υ + W υυυ ) υ − α ( W ϕϕ + W ZZ ) = 0 . (39)It describes the dynamic behavior of nonlinear waves and solitons in plasmaand fluids [43, 44]. The rational and semi-rational solutions of the 3D KPequation can be expressed in Theorems 4 and 5 as follows. Theorem 4.
The 3D KP equation (39) has rational solutions W = 2(ln f ) υυ , (40)where f = det ≤ i,j ≤ N ( H ( n )2 i − , j − ) | n =0 , (41)and the matrix elements in f are defined by H ( n ) i,j = i (cid:88) k =0 d k ( i − k )! ( p∂ p + E (cid:48) + n ) i − k × j (cid:88) l =0 d ∗ l ( j − l )! ( p ∗ ∂ p ∗ + E (cid:48) ∗ − n ) j − l p + p ∗ | p =1 , (42) eductions of the (4 + 1)-dimensional Fokas equation and their solutions 19 E (cid:48) i = υ + (cid:114) α iϕ + (cid:114) α iz − ς. Theorem 5.
The 3D KP equation (39) has semi-rational solutions W = 2(ln f ) υυ , (43)where f = det ≤ i,j ≤ N ( H ( n ) i,j ) | n =0 , (44)and the matrix elements in f are defined by H ( n ) i,j = ( − p i p ∗ j ) e E i + E ∗ j n i (cid:88) k =0 d ik ( p i ∂ p i + E (cid:48) i + n ) n i − k × n j (cid:88) l =0 d ∗ jl ( p ∗ j ∂ p ∗ j + E (cid:48) ∗ j − n ) n j − l p i + p ∗ j + δ ij b in i b ∗ jn j , (45) E i = p i υ + ip i (cid:114) α ϕ + ip i (cid:114) α z − p i ς + E i ,E (cid:48) i = υ + (cid:114) α iϕ + (cid:114) α iz − ς. Here the asterisk denotes the complex conjugation and d k , d ik , d jl , and d l arearbitrary complex constants.The proofs of Theorems 4 and 5 are similar to those of Theorems 2 and 3.It is not difficult to see that the rational and semi-rational solutions of the 4DFokas equation can degenerate to the rational and semi-rational solutions ofthe 3D KP equation. The corresponding transformation is as follows W DKP I ( υ, ϕ, Z, ς ) = 2 k − k u DF okas ( x , x , y , y , t ) , (46)where k x + k x = υ, k y + k y = (cid:115) k k k k ( k − k ) ( ϕ + Z ) , t = 4 k ( k − k ) ς. In this paper, the determinant expression of N -solitons is constructed for the4D Fokas equation by using the KP hierarchy reduction method. New typesof mixed solutions composed of breathers and V-type solitons are obtainedby choosing the appropriate parameters in Theorem 1 (see Fig. 2 and Fig.3). High-order rational solutions of the 4D Fokas equation are also derived bymeans of Theorem 2, as well as we give the condition k k k k ( k − k ) > x , y )-plane, which is a traveling wave localized in space and time, see Fig. 4. High-order rational solutions displaythe interaction between several lumps in the ( x , y )-plane, and exhibit similardynamical patterns to those of rogue waves in the (1+1)-dimensions by alteringthe free parameters c k in Theorem 2 (see Fig. 5 and Fig. 6).Furthermore, three kinds of new semi-rational solutions of the 4D Fokasequation are generated by introducing differential operators Ξ i and (cid:102) j . For N = 1 , n i ≥ , δ ii = 1, and δ ij = 0 ( i (cid:54) = j ) in Theorem 3, the semi-rationalsolutions composed of lumps and one-soliton solutions are derived. There arefour distinct dynamical patterns of these semi-rational solutions, which areobtained by changing the values of parameters k , k , k , and k (see Fig. 7and Fig. 8). The specific classification of these patterns is shown in Table 1.For N ≥ , n i = 1 , δ ii = 1 and δ ij = 0 ( i (cid:54) = j ) in Theorem 3, the semi-rationalsolutions consisting of more lumps and more solitons are also generated (seeFig. 9). Also a new kind of semi-rational solution composed of two lumps, abreather, and a V-type soliton is derived, for N = 2 , n i = 2, and δ ij = 1. Wepoint out that the interaction between the mentioned entities of such semi-rational solution is elastic. This kind of semi-rational solution that is illustratedin Fig. 10, has never been reported elsewhere, to the best of our knowledge.Additionally, using our rational and semi-rational solutions of the 4D Fokasequation, we derived the rational and semi-rational solutions of the 3D KPequation. These results indicate that the 4D Fokas equation is a valuable multi-dimensional extension of the KP and DS equations. In addition, this paperprovides an idea for seeking the exact solutions of high-dimensional solitonequations, and also provide a reference for how to reduce the exact solutionsof high-dimensional systems to the exact solutions of low-dimensional ones.These results are useful to the study of the dynamics of nonlinear waves indiverse physical settings in hydrodynamics, nonlinear optics and photonics,plasmas, quantum gases (Bose-Einstein condensates), and solid state physics. Funding
This work is supported by the NSF of China under Grant No.11671219 and No. 11871446.
Compliance with ethical standardsConflict of interest
The authors declare that they have no conflict of in-terest.
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