# CTE Solvability, Nonlocal Symmetry, and Interaction Solutions of Coupled Integrable Dispersionless System

Authors: Jun Yu; Bo Ren; Ping Liu; Jia-Li Zhou
Journal: Complexity
 (2022)

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## Abstract

The consistent tanh expansion (CTE) method is successfully applied to the coupled integrable dispersionless (CID) system. A nonauto-Bäcklund transformation (BT) theorem includes two fieldsf and v1 is obtained by using the CTE method. One obtains the consistent condition in the nonauto-BT theorem by means of the relation between the fields f and v1. The CID system possesses the CTE solvability property by some detailed analysis. Many interactions between one soliton and multiple resonant solitons, and between one soliton and cnoidal waves are generated by using the nonauto-BT theorem. The types of bright and gray two front waves are shown by some figures. In the meanwhile, the nonlocal symmetry is obtained by the truncated Painlevé method and the Möbious invariant form. The initial value problem and an auto-BT are constructed by the localization procedure.


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## Body

## 1. Introduction

Finding soliton solutions of the nonlinear integrable systems is an important topic in nonlinear science. A large number of useful methods have been investigated, such as the inverse scattering transformation [1–3], Hirota’s bilinear method [4], symmetry reductions [5–7], the Darboux transformation [8], the Painlevé analysis method [9], the Bäcklund transformation (BT) [10], the separated variable method [11], etc [12, 13]. The similarity solutions can be found by the Lie point symmetry method [14, 15]. To describe the complex physical phenomena, the interactions among different nonlinear excitations are worth studying compared with soliton solutions [16–19]. These interaction excitations cannot be obtained by the direct Lie point symmetry method. Recently, different types of excitations are valid by using the symmetry reductions related to nonlocal symmetry and a consistent tanh expansion (CTE) method [20–28]. The CTE method can be investigated not only for various different types of excitations but also for an integrable property of the nonlinear systems, including the supersymmetric extension of the nonlinear systems [29–32]. Many dispersionless integrable systems have been developed in various applications, such as quantum field theories, string theory, conformal field theory, and condensed matter theory [33–38]. The geometry of the dispersionless equation is systematically studied by using the bishop frames and Darboux frames [39, 40]. Multisoliton solutions of two components of the CID equation are constructed by the Darboux transformation [41]. The nonlocal symmetry of two components of the CID equation is obtained by the BT method [42]. A coupled dispersionless integrable system and its generalizations based on nonabelian Lie groups have many potential applications in diverse areas [43, 44]. Here, we shall apply the CTE method to study three components of the coupled integrable dispersionless (CID) system. The features of the CTE solvability, interaction solutions, and nonlocal symmetry are considered which might be sufficient to explain the relevant physical processes.The layout of this paper is organized as follows: In Section2, the CID system does accept a CTE solvable system by using the CTE method. The nonauto-BT theorem which consists of the consistent condition is given by the CTE method. In Section 3, interactions between one soliton and multiple resonant soliton solutions, between one soliton and cnoidal periodic waves are given by means of the nonauto-BT theorem. In Section 4, the nonlocal symmetry of the CID equation is obtained by the truncated Painlevé analysis. The initial value problem related to the nonlocal symmetry is solved by the localization procedure. Section 5 is a simple summary and discussion.


## 2. CTE Solvability of CID System

The CID system reads(1)uxt+uxw+uwx=0,vxt−2uxv=0,wxt−2uxw=0,which is first given based on a point of view group theoretical [43]. The CID system (1) is solvable by the inverse scattering transform technique, the Lax pair, and the Painlevé property [43, 44]. The Darboux transformation of the CID system is studied based on a non-Abelian Lie group and expressed the matrix solutions in terms of quasideterminants [45]. The multisoliton solutions are constructed in terms of the Casorati determinants [46]. Based on the generalized Darboux transformation, the nth-order rogue wave solution of the CID equation is studied [47]. Lie symmetry analysis and group invariant solutions for the CID equation are given [48]. The multirotating loop soliton solutions are given by the perturbation technique and symbolic computation [49]. Traveling wave-guide channels of a CID system are investigated by using the fourth-order Runge-Kutta’s computational scheme [50] and Hirota’s bilinear method [51]. Algebraic structures of a general CID system are analyzed through the prolongation structure approach [52]. The multivalued loop soliton, chaotic soliton chain, and fractal pattern are studied by using the projective Riccati equation method [53].To apply the CTE method in the CID equation, the consistent tanh expansion is written as the following form based on the leading order analysis [21](2)u=u0+u1tanhf,v=v0+v1tanhf,w=w0+w1tanhf,where u0, u1, v0, v1, w0, w1 and f are arbitrary functions of t,x. By substituting equation (2) into the CID system (1) and vanishing the coefficients of powers of tanh3f and tanh2f, we obtain the solutions of u1,v1,w1 and u0,v0,w0, respectively(3)u1=−ft,v1=v1,w1=ft,and(4)u0=ftt2ft,v0=−v1t2ft−v1x2fx+v1fxt2fxft,w0=−ftt2ft.Vanishing the coefficients oftanh1f and tanh0f with equations (3 and 4), the functions of f and v1 satisfy(5a)f;tx−4ftfxt=0,(5b)v1xt−v1fxtftt+fxtftfxv1ftx=0,where f;t=∂/∂tftt/ft−1/2ftt/ft2 is Schwarzian derivative. Two functions for f and v1 satisfy equation (5). The expansion (2) is thus called a CTE and the CID system (1) is a CTE solvable system [20].Nonauto-BT theorem. If the solutionf and v1 satisfies the consistent condition (5), then the solution of u,v,and w for equation (6) is also a solution of the CID system (1)(6)u=−fttanhf+ftt2ft,v=v1tanhf−v1t2ft−v1x2fx+v1fxt2fxft,w=fttanhf−ftt2ft.The nonauto-BT theorem of the CID equation (1) can be constructed based on the aforementioned detail calculations. Some exact solutions including the interaction between soliton and other kinds of complicated waves can be obtained by means of aforementioned nonauto-BT theorem. In the next section, some concrete interesting examples are given via the nonauto-BT theorem.


## 3. Interaction Solutions for CID System

A quite trivial straight line solution of equation (5) has the form(7)f=kx+ωt,v1=kx+ωt,where k and ω are the free constants. Substituting the trivial solution equations (7) into (6), the exact solution of the CID system yields(8)u=−ωtanhkx+ωt,v=−1+kx+ωttanhkx+ωt,w=ωtanhkx+ωt.The nontrivial solution (8) of the CID equation is given from a quite trivial solution of equation (7).To find the interaction between one soliton and other nonlinear waves, we can assume the solutions as one straight line (7) plus undetermined waves. The interaction between one soliton and multiple resonant soliton solutions f of equation (6) is given as(9)f=k0x+ω0t+a0ln1+∑i=1nexpkix+ωit,where k0,ki and ωi are arbitrary constants while a0 and ω0 are determined by the relations(10)a0=±12,ω0=∓ωn2.The solutionv1 of equation (6) has the following form(11)v1=Aft,with the arbitrary constant A. Equations (5a) and (5b) will become one equation by substituting the relation (11) to (5). It demonstrates that equations (5a) and (5b) transform into a consistent system by means of the relation (11). The solution of the CID equation can be obtained by substituting equations (9) and (11) into (6). By selecting n=1,a0=1/2,ω0=−ω1/2, the solution of the CID equation reads(12)u=ω1tanhk0x−ω1/2t+1/2lnk1x+ω1t−expk1x+ω1t21+expk1x+ω1t,v=−ω1Atanhk0x−ω1/2t+1/2lnk1x+ω1t+expk1x+ω1t21+expk1x+ω1t,and w=−u from the nonauto-BT theorem. Figures 1 and 2 display the special interaction solution with the parameters selected as k0=1/8,k1=1/3,A=10. The parameter ω1 is chosen as 1/2 and −1/2 in Figures 1 and 2, respectively. The amplitude of v can be controlled with the parameter A. The amplitudes in Figures 1(b) and 2(b) are larger than Figures 1(a) and 2(a) due to A=10. Figures 1(a) and 2(a) are the types of bright ω1>0 and gray ω1<0 two front waves for u, and vice versa for v.Figure 1
The propagation foru (bright) and v (gray) is plotted in (a) and (b), respectively.
(a)(b)Figure 2
The propagation foru (gray) and v (bright) is plotted in (a) and (b), respectively.
(a)(b)Similar to the form of equation (9), the interaction between one soliton and cnoidal periodic waves reads as(13)f=k0x+ω0t+FX,X=kx+ωt,where k0, ω0, k and ω are all the free constants. Substituting the expression (13) into (5), we obtain the standard elliptic function equation of F1X(14)F1X2−4F14+a1F13+a2F12+a3F1+a4=0,F1=FX,with(15)a1=8C2ω−8ω0ω,a2=16ω0C2ω2−4ω02ω2−4C22ω2−C1ω24,a3=8ω02C2ω3−8ω0C22ω3−C1ωω02,a4=−C1ω02−16ω02C22ω4,and C1 and C2 are arbitrary constants. The interaction between one soliton and cnoidal periodic waves f in equation (5) can be written as [22, 23].(16)f=k0x+ω0t+a0EllipticPiJacobiSNkx+ωt,m,n,m,where k0,ω0,a0,k,ω,n and m are constants and EllipticPi is the third incomplete elliptic integrals. The interaction between a soliton and cnoidal periodic waves f satisfies (5) with the parameter n=0. Besides the solution v1 of equation (11), one can also get other solutions of the field v1 by the transformation v1=V1X. The ordinary differential equation (ODE) is obtained from equation (5)(17)V1XX−a02a0kωF1+kω0+k0ωa0ωF1+ω0a0kF1+k0F1XV1X+a02ω2a0kωF1+kω0+k0ωa0ωF1+ω02a0kF1+k0F1X2−a0ωa0ωF1+ω0F1XXV1=0.The solution ofV1 can be obtained directly by solving ODE (17)(18)v1=V1X=C3a0ωF1+ω0+C4a0ωF1+ω0∫a0kF1+k0a0ωF1+ω0dX.The interaction between solitons and cnoidal periodic waves can happen in the ocean [22]. The results are useful for explaining ocean phenomena.


## 4. Nonlocal Symmetry and BT for CID System

Based on the truncated Painlevé analysis of the CID equation, the Laurent series expansion ofu,v and w reads [9](19)u=u1ϕ+u0,v=v1ϕ+v0,w=w1ϕ+w0,where ϕ, u0,u1,v0,v1,w0, and w1 are functions in a neighborhood of the noncharacteristic singular manifold. By substituting the expansion (19) into (1) and vanishing the coefficients of ϕ independently, one obtains(20)u1=−ϕt,v1=Aϕt,w1=ϕt.and(21)u0=ϕtt2ϕt,v0=−Aϕtt2ϕt,w0=−ϕtt2ϕt,with an arbitrary constant A. The field ϕ satisfies the following Schwarzian CID form(22)ϕ;tx=0,where ϕ;t is the Schwarzian derivative.By the definition of residual symmetry [21], the nonlocal symmetry of the CID equation is given by the truncated Painlevé analysis (19).(23)σu=−ϕt,σv=Aϕt,σw=ϕt.The nonlocal symmetry (23) can be obtained by the Möbious transformation [54–56]. The Schwarzian form (22) is invariant under the Möbious transformation [9](24)ϕ⟶aϕ+bcϕ+d,ad≠bc.By selectingd=1,b=0,c=−ϵ in equations (24) and (22) possesses the symmetry(25)σϕ=aϕ+aϕ2.The nonlocal symmetry (23) will be given by substituting the Möbious transformation symmetry (25) into the symmetry equation of (21).For Lie’s first principle, the initial value problem related to the nonlocal symmetry (23) is(26)du¯dϵ=−ϕt,u¯|ϵ=0=u,dv¯dϵ=Aϕt,v¯|ϵ=0=v,dw¯dϵ=ϕt,w¯|ϵ=0=w.The initial value problem of (26) exists in the function ϕ and its differentiation so it is difficult to solve [21]. To solve the initial value problem (26), one introduces a new field to remove the differentiation of ϕ. To eliminate the differentiation, the new field satisfies the following relation(27)ϕt=g.The symmetry for the prolonged systems (1), (21), and (27) gives(28)σu=−g,σv=Ag,σw=g,σϕ=−ϕ2,σg=−2ϕg.The nonlocal symmetry (23) in the original space x,t,u,v,w is localized to a Lie point symmetry (28) in the prolonged space x,t,u,v,w,ϕ,g. The corresponding vector form is(29)V=−g∂∂u+Ag∂∂v+g∂∂w−ϕ2∂∂ϕ−2ϕg∂∂g.For the prolonged CID systems (1), (21), and (27), the initial value problem reads as(30)du¯dϵ=−g,u¯|ϵ=0=u,dv¯dϵ=Ag,v¯|ϵ=0=v,dw¯dϵ=g,w¯|ϵ=0=w,dϕ¯dϵ=−ϕ2,ϕ¯|ϵ=0=ϕ,dg¯dϵ=−2ϕg,g¯|ϵ=0=g.The auto-BT theorem of the enlarged CID systems is constructed by solving the aforementioned initial value problem (30).Auto-BT Theorem. If u,w,v,ϕ and g are a solution to the enlarged CID systems, then u¯,v¯,w¯,ϕ¯ and g¯ are also a solution to the enlarged CID systems(31)u¯=u−ϵgβϵϕ+1,v¯=v+ϵAgβϵϕ+1,w¯=w+ϵgβϵϕ+1,ϕ¯=f1+ϵϕ,g¯=g1+ϵϕ2,with an arbitrary group parameter ϵ. The new solution can be obtained by means of the old solution and the aforementioned auto-BT theorem.


## 5. Conclusions

In summary, the CTE method has been successfully applied to the CID equation. A nonauto-BT theorem is derived by using the CTE method. It demonstrates that the CID system possesses the CTE solvability property. Abundant interaction between solitons and other types of solitary waves for the CID system is studied by means of the nonauto-BT theorem. One obtains two relations of the solutions betweenv1 and f for equations (11) and (18). The types of bright and gray two front waves for u and v are plotted in Figures 1 and 2. Based on the Painlevé analysis, the nonlocal symmetry is constructed. The initial value problem and the auto-BT related by the nonlocal symmetry are obtained by introducing a new field. In the meanwhile, interactions between solitons and other types of waves can be constructed by the symmetry reductions related to nonlocal symmetry. Besides the classical integrable systems, the study of nonlocal symmetry systems has become an important subject in nonlinear science [57, 58]. These aspects of the nonlocal CID system are worthy of study in the future.



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