The famous Kadomtsev–Petviashvili (KP) equation is written as 1 where the subscripts denote derivatives. It is firstly derived by Kadomtsev and Petviashvili to study the stability of soliton solutions of the Korteweg–de Vries (KdV) equation with respect to weak transverse perturbations.^[1] When δ = −1 and 1, the KP equation represents the KPI and KPII equations, respectively. As a extension of the KdV equation in two dimensions, both of the KPI equation and the KPII equation have arisen in various physical contexts, such as plasma physics, fluid mechanics, optics, condensed matter physics, and geophysics, etc.^[1–3]

Nowadays, the KP equation is one of the most important soliton equations, because the KP equation (1) is a universal completely-integrable (2+1)-dimensional nonlinear evolution equation. The KP equation is a member of the KP soliton hierarchy and it serves as a kernel model in the universal Sato’s theory.^[4,5] Many integrable properties of the KP equation have been researched in the past years, including lump solutions,^[6,7] mixed lump-kink solutions,^[8,9] line-soliton solutions,^[10] the Lax representation,^[11] multi-component Wronskian solution,^[12] Painlevé property,^[13] Darboux transformation,^[14,15] consistent tanh expansion,^[16] Bäcklund transformation,^[17] and similarity reductions.^[11]

In 2013, the theory of nonlocal residual symmetry was put forward.^[18] In order to localize the residual symmetries to the localized symmetries, the researched system should be extended to a extended system. The Lie point symmetries of the extended system are composed of the residual symmetries and the standard Lie point symmetries, which suggests that the residual symmetry method is a useful complement to the classical Lie group theory.^[18–21] The concepts of consistent Riccati expansion (CRE) and CRE solvability were proposed in 2015.^[22] A system having a CRE is then defined to be CRE solvable. The CRE solvability is demonstrated quite universal for various integrable systems. Especially, it is revealed that CRE can be applied to obtain interaction solution between solitons and cnoidal waves.

In Ref. [23], with the help of the Lax pair and the adjoint Lax pair of the KP equation, the authors researched the nonlocal symmetries of the KP equation related to the Darboux transformations. In this paper, we will research the nonlocal symmetries of the KP equation related to the Bäcklund transformations. To our knowledge, the CRE solvability of the KP equation has not been reported. So we focus our attention on the nonlocal symmetries and CRE of the KP equation in this paper.

This paper is organized as follows. In Section 2, truncated Painlevé expansion is applied to the KP equation, and a Bäcklund transformation of the KP equation is obtained. Section 3 is devoted to one-parameter group transformations and one-parameter subgroup invariant solutions. Bäcklund transformations related to nonlocal symmetries are discussed in Section 4. In Section 5, the CRE solvability of the KP equation is proved, and soliton–cnoidal wave interaction solutions of the KP equation are discussed. The final section is summary and discussion.

The truncated Painlevé expansion method is proved to be very useful in solving nonlinear partial differential equations (PDEs).^[24–26] For the KP equation (1), its truncated Painlevé expansion can be written as^[13] 2 where 3 4 5

The substitution of Eqs. (2)–(5) into the KP equation solves^[13] 6 where 7 with f being arbitrary function of {x,y,t}. S, K, and C are invariants under the Möbious transformation, then equation (6) can be called as Schwarzian KP equation. From the combination of Eqs. (2)–(7), we can obtain a Bäcklund transformation on the KP equation (1) and the Schwarzian KP equation (6).

Symmetry study is one of the most effective method to research PDEs.^[27–32] The symmetry determining equation of the KP equation is 9 where σ is the symmetry of u in the KP equation. It is easy to verify that σ = −2 f_xx satisfies Eq. (9) when u satisfies Eq. (8). From Eq. (2) and Eq. (4), we know that −2 f_xx is the residue of the truncated Painlevé expansion of the KP equation. The residue of the truncated Painlevé expansion is a symmetry of a PDE, so we call this symmetry as residual symmetry.

The residual symmetry can be combined the classical Lie symmetries, and the full Lie point symmetries can be obtained. Then we can establish an extended system, which include the KP equation, the Schwarzian KP equation and the Bäcklund transformations between the two equations. The extended system can be written as 10a 10b 10c 10d 10e For the extended KP system, the symmetry σ should be extended to four symmetry components {σ^u,σ^f,σ^g,σ^h}, which satisfy the symmetry determining equations in the form of 11a 11b 11c 11d 11e

From the above equations, we can obtain the subvectors in the form of 12 where F₁, F₂, and F₃ are functions of t. The generalized vector is 13 where is related to residual symmetry, is the scaling transformation, is translation transformation, and the others denote Galilean translation transformations.

From the vector fields, one can obtain one-parameter invariant subgroups. The partial operator ∂_t in shows that time t is variable, while the other terms on F₁ are functions of t, which make it too complicated to obtain a one-parameter invariant subgroup from . Only when special function of F₁ is given, we can obtain some special one-parameter invariant subgroups. From , , , , and , five one-parameter invariant subgroups in the following form can be obtained: 14 By means of one-parameter subgroups, the exact solutions dependent on a one-parameter can be obtained from a known exact solutions. Then, the following Bäcklund transformation theorem can be obtained.

Symmetry method is a very powerful method to research PDEs. From the symmetry components, we can further obtain reduction equations and the corresponding similarity solutions. The substitution of the similarity solutions into the extended KP system will solve symmetry reduction equations. Six types of nontrivial reduction cases are obtained.

In the first case, we will discuss the most general condition. In this case, we do not suppose any concrete form for F₁, F₂, and F₃. The group invariants are 16a 16b Because the special form of F₁, F₂, and F₃ are not given, all integral terms on F₁, F₂, and F₃ cannot be simplified. Then, the reduction equations and the similarity solutions are very lengthy, and we will not list them in this case. For simplicity, we will assume some simple concrete forms for F₁, F₂, and F₃ in the following cases.

In this case, the group invariants are simplified to 17a 17b We take the parameter for simplicity. The similarity solution of {u,f,g,h} is 18a 18b 18c 18d where {U ≡ U(ξ, η), F ≡ F(ξ,η), G ≡ G(ξ,η), H ≡ H(ξ,η)}, which satisfy the reduction equations 19a 19b 19c 19d The substitution of Eqs. (19a)–(19c) into Eq. (18a) leads to an exact solution of the KP equation.

Substituting F₁ = C₅, F₂ = C₆ t + C₇, F₃ = C₈ t + C₉, and C₁ = 0 into symmetry components, we will find that the group invariants are in the form of 25a 25b and the similarity solution is 26a 26b 26c 26d where U = U(ξ,η), F = F(ξ,η), G = G(ξ,η), and H = H(ξ,η). The substitution of the similarity solution Eqs. (26a)–(26d) into the extended KP system Eqs. (10a)–(10e) will solve the reduction equations in the form of 27a 27b 27c 27d Substituting Eq. (27a) into Eq. (26a) leads to an exact solution of u for the KP equation, which can be expressed by the follow theorem.

On this condition, we can obtain the traveling transformation, and the similarity solution is 29 where group invariants are 30a 30b The corresponding reduction equations are 31a 31b 31c 31d The combination of Eq. (29) and Eq. (31a) makes an exact traveling wave solution of the KP equation.

CRE is an important method to obtain some interaction wave solutions for PDEs. CRE solvability method is a way to judge whether the equation is integrable by means of consistent Riccati expansion. The Riccati equation is in the form of 33 with a₀, a₁, and a₂ being arbitrary constants. The authors in Ref. [33] systematically presented the general solution to the Riccati equation. One exact solution of the Riccati equation is 34 where 35

A system 36 can be expanded as 37 where R(w) is a solution of the Riccati equation. Plugging formula (37) into the system (36), and vanishing all the coefficients on Rⁱ(w), the following system will be obtain 38 If the system (38) is consistent, then the expansion (37) is a CRE and the nonlinear system (36) is CRE-solvable.^[22]

To our knowledge, CRE of the KP equation has not been researched. In this section, we will discuss the CRE of the KP equation, then obtain some interaction wave solutions of the KP equation. u in Eq. (1) can be expanded as 39 with q₀, q₁, q₂, and w being functions of {x,y,t}, and R(w) being a solution of the Riccati equation.

All differential coefficients on R(w) of the combination of Eqs. (1), (33), and (39) show that 40a 40b 40c with w satisfying 41 According to the definition on CRE and CRE solvable, the KP equation is CRE-solvable. The combination of Eqs. (34), (39), and Eqs. (40a)–(40c) shows that an exact solution of the KP equation can be expressed as the following formula 42 where w is solved by Eq. (41). Then, the concrete form of the exact solution u can be proposed if w is solved. We will try to solve Eq. (41) in the following paragraphs.

In Eq. (42), w can be supposed to have the form of 43 where k₁, l₁, ω₁, k₂, l₂, ω₂, a₃, n, and m are parameters to be determined, and E_π is the third type of incomplete elliptic integral. Substituting Eq. (43) into Eq. (41), and collecting the coefficients of different powers on sn(k₂x + l₂ y + ω₂ t, μ), one will find the relationships of the parameters. The five types of parameter restrictions can lead to five types of nontrivial solutions of w and u, i.e., 44 45 46 47 48 where a₄ = l₁ k₂ + l₂ k₁, a₅ = k₁ + a₃ k₂.

The substitution of Eq. (43) into Eq. (42) makes the solution of u in the form of 49 where 50 with the parameters satisfying one of formulas (44)–(48).

The evolution of u with x and t at y = 1 is demonstrated in Fig. 1(a), where the parameters satisfy Eq. (46) and the free parameters being {δ = 1, μ = 0.8, ν = 0.2, k₁ = 2, ω₁ = −2,k₂ = 3,l₂ = 8}. Figure 1(b) shows the density of u in Fig. 1(a). Figure 1 clearly shows the interactions of cnoidal waves and solitary waves.

Fig. 1.

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	Fig. 1. The solution and the density of u expressed by Eq. (49) with formula (46), respectively. The free parameters are δ = 1, μ = 0.8, ν = 0.2, k1 = 2, ω1 = −2, k2 = 3, and l2 = 8.

The solution of u satisfying formula (47) is demonstrated in Fig. 2, with the free parameters being selected as follows: δ = 1, μ = 0.9, a₃ = 0.4, k₁ = 2, k₂ = −2, ω₁ = 4, and l₂ = 6. Figure 2(a) displays the evolution of u with x and y, which shows the cnoidal waves reside on solitary waves. The evolution of the shifted periodic wave u with x and t is displayed in Fig. 2(b). Figure 2(b) demonstrates that the exact solution is rapidly approached the periodic waves on both sides of the solitons.

Fig. 2.

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	Fig. 2. Evolution of u with space and time. The parameters are constrained by Eq. (47), and the free parameters are δ = 1, μ = 0.9, a3 = 0.4, k1 = 2, k2 = −2, ω1 = 4, and l2 = 6. Panel (a) is the evolution of u with x and y, and panel (b) is the evolution of u with x and t.

Figures 3(a) and 3(b) demonstrate the density of u in Figs. 2(a) and 2(b), respectively. We can see that figure 3 clearly displays the interaction between solitons and cnoidal waves.

Fig. 3.

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	Fig. 3. The density plots for the corresponding Fig. 2.

A Bäcklund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painlevé expansion. By means of the truncated Painlevé expansion, nonlocal residual symmetries of the KP equation are studied. One-parameter group transformation and one-parameter subgroup-invariant solutions are obtained. Several Bäcklund transformations related to the nonlocal symmetries are proposed. The CRE method is applied to study the KP equation and the CRE solvability of the KP equation is proved by CRE. With the help of CRE, the interaction solutions between solitons and cnoidal waves are obtained.