The investigation in finding the exact solutions of nonlinear evolution equations (NLEEs) is important for the understanding of nonlinear phenomena in various fields of science, especially in physics. For example, the nonlinear wave phenomena observed in fluid dynamics, plasma, and optical fibers are often modelled by the bell-shaped sech solutions and the kink-shaped tanh solutions.^[1] Up to now, many powerful and systematic methods have been developed to construct explicit solitary wave solutions for the NLEEs, such as the Darboux transformation (DT),^[2–3] the Bäcklund transformation (BT),^[4] and the function expansion method.^[5] Recently, inspired from the nice results via nonlocal symmetry method,^[6–7] Lou proposed the consistent Riccati expansion (CRE) method,^[8] which is a lucid and systematic method to construct interaction solutions between different types of nonlinear excitations. Many interaction solutions for NLEEs, for instance, the Boussinesq–Burgers equation,^[9] Boiti–Leon–Pempinelli equation,^[10] modified KP equation,^[11] Sharma–Tasso–Olver equation,^[12] breaking soliton equation,^[13] Bogoyavlenskii coupled KdV system,^[14] higher-order Broer–Kaup system,^[15] KdV-mKdV equation,^[16] modified Bogoyavlenskii–Schiff equation,^[17] coupled mKdV–BLMP system,^[18] are discussed in detail.

In this paper, we focus on the following extended (2+1)-dimensional shallow water wave equation

The purpose of this work is to employ the truncated Painlevé expansion, consistent Riccati expansion (CRE) and symbolic computation to study the nonlocal symmetry, BT, consistent Riccati solvability and interaction solutions for Eq. (2). In Sec. 2, the CRE method is used to prove the CRE solvability of Eq. (2). In Sec. 3, we will give some interaction solutions of Eq. (2), including soliton-cnoidal wave interaction solutions and solitoff-typed solutions. In Sec. 4, the nonlocal symmetry related to truncated Painlevé expansion is studied. Section 5 will be our conclusions.

As is well-known, investigation of integrability for NLEEs can be regarded as a pre-test and first step of its exact solvability. Among the methods employed to study the integrability of NLEEs, the Painlevé analysis not only can be used to verify the integrability, but also can be used to find exact solutions and other significant properties such as Lax pair and BT.^[25–26] Recently, Lou defined the consistent Riccati solvability for NLEEs if it has a consistent Riccati expansion.^[8] It has been shown that many more integrable systems are CRE solvable which indicates that the CRE solvable systems are integrable.^[8–18]

In the following, we will apply the CRE to Eq. (2), and prove it is consistent Riccati solvable.

For Eq. (2), we give the possible CRE solution in the form of

By substituting the expansion (3) with Eq. (4) into Eq. (2) and vanishing different powers of R yields

It is clear that if w is a solution of Eq. (6), then expression (3) is also a solution of Eq. (2). Hereby, we have the CRE solvable theorem for Eq. (2):

It is easy to see that the expression (8) can also be regarded as the BT of Eq. (2). As is well-known, the Riccati equation (4) has the following tanh function solution

It is obvious that once the solutions of w satisfies Eq. (6) are obtained, then we can get the interaction solutions of Eq. (1). Here are some interesting examples.

Substituting Eq. (14) with Eq. (15) into the w Eq. (6), we can obtain

It is well known that the explicit solutions of Eq. (15) can be expressed by Jacobi elliptic functions, here, we just take a simple solution of Eq. (15) as

From Fig. 1, we can find that a single soliton moving on a cnoidal wave background instead of on the plane continuous wave background. This kind of interactions can be used to display many interesting physical phenomena, such as the Fermionic quantum plasma.^[27]

Fig. 1

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	Fig. 1 Plots of the single soliton on a cnoidal wave background expressed by Eq. (20) at t = 0 with the parameter selections {α, n, ι1, ι2, ω1, ω2, κ2} = {1, 3, 1, 1, 1.5, 1.5, 1} of Eq. (1): (a) Two-dimensional image for v at t = 0; (b)–(d) One-dimensional image for v at y = 0, x = 0 and its density plot, respectively.

Fig. 2

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	Fig. 2 Plots of the solitoff-typed solutions expressed by Eq. (23) at t = 0 with the parameter selections {α, ι1, ι2, κ2, ω2} = {1, 2, 2, 2, 2} of Eq. (1): (a) Two-dimensional image for v at t = 0; (b)–(d) One-dimensional image for v at y = 0, x = 0 and its density plot, respectively.

One knows that the symmetry method^[30–31] is very effective for constructing explicit solutions of NLEEs. In Refs. [6–7], the nonlocal symmetries related to the BT and DT are used successfully to construct the interaction solutions among different types of nonlinear excitation, such as the soliton-cnoidal wave solutions. Recently, it is found that the Painlevé analysis can be also used to find nonlocal symmetries of NLEEs.^[32–33] In this section, with the help of the truncated Painlevé expansion method, we will first give a non-auto BT of Eq. (2), then write down the nonlocal symmetry related to the BT, and localize the nonlocal symmetry by introducing an enlarged system.

According to the Painlevé analysis,^[25] one can know that Eq. (2) exists the following truncated Painlevé expansion

Substituting Eq. (25) into Eq. (2), and vanishing all the coefficients of different powers of 1/f, we have

Here, , , are the Schwarzian variables shown in Eq. (7), which function w replaced with function f. It is known that the Schwarzian variables keep the Möbius invariance property

From the Theorem 2, on the one hand, one can construct various exact solutions of Eq. (2), on the other hand, one can also obtain the nonlocal symmetry. As is known, under the invariant property

By solving f from Eq. (30), one knows that the residual symmetry (33) of u is nonlocal, therefore, Eq. (33) is a nonlocal symmetry of Eq. (2). To find out the symmetry group of the nonlocal symmetry (33), we need to extend the original system such that nonlocal symmetry becomes the local Lie point symmetries of an enlarged system. Therefore, we introduce a new dependent variable f₁ satisfies

Based on the Lie’s first theorem, the corresponding initial value problem of Eq. (35) reads

By solving the initial value problem (36), we have the following finite transformation theorem:

Theorem 3 provides a way to obtain new solution of Eq. (2) from the old ones.

In this paper, we investigate the extended (2+1)-dimensional shallow water wave equation by means of CRE, truncated Painlevé expansion and symbolic computation. On the one hand, by using the CRE method, Eq. (2) is proved CRE solvable. It is interesting that the CRE solvable can be regarded as the pre-test for finding the exact interaction solutions of NLEEs. And on this basis we obtain the soliton-cnoidal wave interaction solutions (20) and solitoff-typed solutions (23) for Eq. (1). On the other hand, with the help of truncated Painlevé expansion (25) of Eq. (2), a non-auto BT (30) is obtained, and from which, we present the nonlocal symmetry (33) of Eq. (2). Furthermore, the nonlocal symmetry (33) is localized by prolonging the original system (2) to the enlarged system (2), (30), and (34). The finite transformation (37) of the nonlocal system are also obtained by solving the standard Lie’s initial value problem (36).

It has been shown that Eq. (2) is CRE solvable and possesses nonlocal symmetry (33), therefore, some other interesting problems such as the nonlocal symmetry reduction, rational solutions, and so on^[34–35] are worthy of further study, which will be discussed in our future work.