Direct Similarity Reduction and New Exact Solutions for the Variable-Coefficient Kadomtsev–Petviashvili Equation

1 Introduction

In 1989, Clarkson and Kruskal [1–3] presented a new approach (CK method) for deriving the similarity reduction of nonlinear partial differential equations (PDEs). The unusual characteristic of this method was that it did not use group theory. Therefore, many authors have enlarged this method (see [4–12]).

The classical CK direct method is used to reduce an n-dimensional PDE to an (n–1)-dimensional PDE; otherwise, the method is quite complicated in practical calculations like other similarity reduction methods. But recently, Fan [13, 14] showed that there exists a close connection between the CK method and the homogeneous balance (HB) method. This connection simplifies the CK calculations a lot. Then Moussa et al. [15, 16] enlarged this connection to deal with PDE systems with variable coefficients. Also Moussa tried [17, 18] to make the CK method connected with the HB method reduce directly an n-dimensional PDE to only a one-dimensional nonlinear ordinary differential equation (ODE).

The Kadomtsev–Petviashvili (KP) equation is one of the most universal models in nonlinear theory. It is usually written as [19]

(1)(ut + 6uux + uxxx)x + 3λ2uyy = 0, (1)

where λ= ± 1. It is used to model water waves of long wavelength with weakly non-linear restoring forces and frequency dispersion. When λ= i, (1) is called the KPI, while when λ= 1, (1) is usually called KPII. The KP equation can be used as a model for two-dimensional shallow water waves [20–22] and ion-acoustic waves in plasmas [23]. Also, it has been obtained as a reduced model in ferromagnetic Bose–Einstein condensation and string theory [24–29]. Moreover, the KP equation has been extended to a (3+1)-dimensional equation as follows:

(2)(ut + 6uux + uxxx)x ± 3uyy ± 3uzz = 0, (2)

which describes three-dimensional solitons in weakly dispersive media, particularly in fluid dynamics and plasma physics [30, 31].

Currently, when inhomogeneous media and/or nonuniform boundaries are considered, nonlinear evolution equations with variable coefficients describe nonlinear phenomena of the complex world more realistically than their constant-coefficient forms so that much attention can been paid to it [32–36]. Therefore, a new generalized variable-coefficient (3+1)-dimensional KP equation is given as

(3)(ut + f(t)uux + g(t)uxxx + h(t)ux + q(t)uy + l(t)uz)x + m(t)uyy + n(t)uzz = 0, (3)

where f(t) ≠ 0, g(t) ≠ 0, h(t), q(t), l(t), m(t), and n(t) are arbitrary functions, representing the nonlinearity, dispersion perturbed effect and disturbed wave velocity along the y- and z-direction, respectively, while u has different meanings in various physical contexts [19–23]. In plasma physics, u represents the electrostatic wave potential, while in fluid dynamics, it represents the amplitude of the shallow-water wave and/or surface wave as seen in [19–23]. When h(t), q(t), l(t) are all zeros, then (3) becomes the variable-coefficient (3+1)-dimensional KP equation [37, 38]. If we add for the previous case n(t)=0, (3) becomes the (2+1)-dimensional KP equation with variable coefficients, which can be used to describe nonlinear waves with a weakly diffracted wave beam, internal waves propagating along the interface of two fluid layers [39], and internal solitary waves in Alboran Sea [40–42].

Recently, multiple soliton solutions and singular multiple soliton solutions for (3) have been obtained by a direct computational method in [43], and then Painlevé analysis, soliton collision, and Bäcklund transformation have been applied to this equation in [44] where (3) was not completely integrable.

In this paper, we will mainly study (3) by using two different direct CK methods: the classical CK method and the modified enlarged CK method connected with the HB method. A similarity reduction to a (2+1)-dimensional nonlinear PDE and a direct similarity reduction to a nonlinear ODE are obtained, respectively. The reduced ODE is solved by direct integration, and many solitary and periodic wave solutions are obtained. Finally, concluding remarks on the results will be given.

2 The Classical CK Direct Method

The classical CK direct reduction method [1–12] is used to reduce the n-dimensional PDE

(4)P(xi, u(xi), u(xi)x1, u(xi)x1x2,...) = 0,  where ​​ i = 1, 2, ..., n, (4)

to an (n–1)-dimensional PDE. Assume that

(5)u(xi) = α(xi) + β(xi)U(ζj),  where i = 1, 2,  ...,  n, j = 1, 2,  ...,  n − 1, (5)

with α, β, and ζ_j being functions in x_i.

According to the CK direct method, one requires that the function U(ζ_j) satisfy only one reduction. Therefore, (4) should be the (n–1)-dimensional PDE of U w.r.t. ζ_j (where j= 1, …, n–1) only. So, the ratios of U must be functions of ζ_j only, say Ω(ζ_j). To determine those Ω(ζ_j) and the unfixed functions α, β, and ζ_j we can use the following rules to simplify the calculations without loss of generality.

• Rule 1: If α has the form α(x_i)=β(x_i) Ω(ζ_j) + α₀(x_i), then we can take Ω(ζ_j) ≡ 0 (by substituting U(ζ_j) → U(ζ_j) – Ω(ζ_j)).

• Rule 2: If β has the form β(x_i)=β₀(x_i) Ω(ζ_j), then we can take Ω(ζ_j) ≡ C= constant (by substituting U(ζ_j) → U(ζ_j)C/Ω(ζ_j)).

• Rule 3: If ζ(x_i)=ζ(ζ₀(x_i), ζ_{j– 1}), then we can take ζ≡ ζ₀ (by substituting U(ζ(ζ₀,ζ_{j– 1}), ζ_{j– 1}) → U(ζ_0,ζ_{j – 1}).

• Rule 4: If ζ_j(x_i) has the form Ω(ζ_j)=ζ₀(x_i) for fixed j, where Ω is an invertible function, then we can take Ω(ζ_j)=ζ_j(x_i) for fixed j.

3 (2+1)-Dimensional Similarity Reduction for the Variable- Coefficient Kadomtsev– Petviashvili Equation

The variable-coefficient (3+1)-dimensional Kadomtsev–Petviashvili equation (VCKPE) is reduced to the (2+1)- dimensional form using the classical CK method according to the previous section as follows: Assumed that

(6)u(x, y, z, t) = α(x, y, z, t) + β(x, y, z, t)U(ζ, η, τ), (6)

where ζ= ζ(x, y, z, t), η= η(x, y, z, t), and τ= τ(x, y, z, t). With the aid of Maple, substituting (6) into (3), we get a PDE in U with respect to ζ, η, τ; then we equate the various coefficients of derivatives and powers of U with Ω(ζ, η, τ) to be functions in ζ, η, τ only. By using Rules 1–4 in the previous section, we obtain (the calculations are very long and complicated to be put in the paper, so we conclude here only the results)

(7)α = − 1k2f(t)(n(t)b(t)2 + m(t)a(t)2 + kq(t)a(t) + k2h(t) + k(a′(t)y+b′(t)z+c′(t)+kl(t)b(t)))β = k2, ζ = η = τ = kx + a(t)y + b(t)z + c(t), (7)

with the condition

(8)g(t) = Af(t), (8)

where A and k are constants, and also a(t), b(t), and c(t) are arbitrary functions of t. Moreover, the reduced (2+1)-dimensional PDE is as follows:

(9)A(Uττττ + Uζζζζ + Uηηηη) + 4A(Uζτττ + Uζηηη + Uητττ + Uηζζζ + Uτηηη + Uτζζζ)+ 6A(Uζζηη + Uζζττ + Uηηττ) + 12A(Uζηττ + Uζζητ + Uζηητ) + Uζ2 + Uη2 + Uτ2+ (Uττ + Uζζ + Uηη)U + 2(UζUη + UζUτ + UηUτ) + 2(Uζτ + Uητ + Uζη)U=0. (9)

4 The CK Direct Method Connected with the HB Method

The modified enlarged CK direct reduction method connected with the HB method for high-dimensional nonlinear PDEs is given in the following in detail [15–18].

Suppose we have a nonlinear PDE for a function u(t, x_i):

(10)P(xi, t, ut, uxi, uxit, ...)=0, where i = 1, 2, ..., n − 1. (10)

The modified enlarged method proceeds in the following steps:

Step 1: We consider the solution of (10) in the form

(11)u = ∂r∂xrF(w) + v, (11)

where F= F(w) and w= w(t, x_i) are undetermined functions, and u and v are two solutions of (10).

According to the HB method, r can be determined by balancing the linear term of the highest order derivative and the highest nonlinear terms of u in (10).

Step 2: Substituting (11) into (10) and collecting all terms of F with the same derivative and power.

Step 3: Then to make the associated equation be an ordinary equation of F and w, requiring ratios of their coefficients being functions of w, we obtain a set of undetermined equations for w, v and other undetermined functions, and by using the following two freedoms, without loss of generality.

1. If v has the form v=v¯(t, xi) + ∂r∂xrΩ, then we can assume that Ω=0 (make the transformation f(w) → f(w) – Ω.

2. If w(t, x_i) is defined by an equation of the form Ω(w)=w₀(x_i, t), we can also assume that Ω=w (make the transformation w→ Ω^–1(w)).

5 One-Dimensional Similarity Reduction for the VCKPE

In this section, we apply the modified enlarged CK method connected with the HB method to the VCPKE as follows.

Balancing the highest order derivative term and the nonlinear term of (3), we can suppose that the solution of the VCKPE takes the form

(12)u(t, x, y, z)=∂2∂x2F + v, (12)

where F= F(w), w= w(t, x, y, z), and v= v(t, x, y, z) are functions to be determined later. Substituting (12) into (3), and by collecting the same coefficients of F and its derivatives together; then, to make it as an ODE of F for w, the ratios of coefficients of different derivatives and powers of F have to be functions of w say Γ_i(i= 1, 2, …, 14) and by using the coefficient of F′″² as the normalized coefficient (see the Appendix). Also by using the two freedoms given in Section 4, we obtain

(13)Γ1(w) = A, Γ2(w) = Γ3(w) = Γ4(w) = Γ5(w) = Γ6(w) = Γ7(w) = Γ9(w) = Γ10(w) = 0Γ11(w) = Γ13(w) = Γ14(w) = 0,  Γ8(w) = 1,  Γ12(w) = B, (13)

(14)v(t, x, y, z) = − a(t)′kf(t)y − b(t)′kf(t)z + c(t), (14)

and the similarity reduction transformation is given by

(15)w(t, x, y, z) = kx + a(t)y + b(t)z − ∫(kf(t)c(t) − k3Bf(t) + m(t)a(t)2k + n(t)b(t)2k + kh(t) + a(t)q(t) + l(t)b(t))dt, (15)

where a(t), b(t), c(t), a(t)′ = ddta(t), and b(t)′ = ddtb(t) are arbitrary functions of t, and A, B, and k are arbitrary constants. Equation (3) is reduced to the following ODE:

(16)AF″″″ + BF″″ + F″F″″ + F′″2 = 0, (16)

under the same condition as given in (8). Putting R(w)=F″ in (16), we obtain

(17)AR″″ + (RR′)′ + BR″ = 0. (17)

Therefore, substituting (14) into (12). (3) has a similarity solution in the following form:

(18)u(t, x, y, z) = k2R(w) − a(t)′kf(t)y − b(t)′kf(t)z + c(t). (18)

6 New Exact Solutions for the VCKPE

Integrating (17) twice, we get

(19)AR″ + 12R2 + BR = c1w + c2, (19)

where c₁ and c₂ are integration constants. To be able to integrate (19) again we assume that c₁=0, and multiply it by R′; then we obtain

(20)R′2 = 1A(2c2R − 13R3 − BR2 + c3), (20)

where c₃ is an integration constant. Equation (20) is a Riccati equation and it has many solutions by direct integration as follows:

(21)R1 = − 3B sech2(− B4Aw)  if BA ≺ 0R2 = − 3B sec2(B4Aw)       if BA ≻ 0R3 = − 12Aw2,  where c2 = c3 = B = 0R4 = Weierstrass P(− 112Aw, 24c2, 12c3), where B = 0, A ≺ 0, (21)

where solution R₄ is of Weierstrass elliptic doubly periodic type. By substitution of (21) into (18), the following new solitary, periodic, and singular wave solutions for the VCKPE are obtained:

(22)u1(t, x, y, z) = − 3Bk2 sech2(− B4Aw) − a(t)′kf(t)y − b(t)′kf(t)z + c(t)  ifBA ≺ 0u2(t, x, y, z) = − 3Bk2 sec2(B4Aw)  − a(t)′kf(t)y − b(t)′kf(t)z + c(t)      ifBA ≻ 0u3(t, x, y, z) = − 12Ak2w2 − a(t)′kf(t)y − b(t)′kf(t)z + c(t),  where c2 = c3 = B = 0u4(t, x, y, z)=k2Weierstrass P(− 112Aw, 24c2, 12c3) − a(t)′kf(t)y − b(t)′kf(t)z + c(t), where B = 0, A ≺ 0. (22)

In the following, some figures for the soliton solution u₁ and the periodic wave solution u₂ at the same constants and variables’ values illustrate the difference between the two solutions and the effect of the variable coefficients on the propagation of both solutions.

7 Concluding Remarks

In this work, first, we have applied the classical CK direct method to the VCKPE, and only one direct similarity reduction in the form of a (2+1)-dimensional nonlinear PDE has been obtained. Secondly, the CK direct reduction method connected with the HB method has been used to reduce the VCKPE to only a one-dimensional nonlinear ODE. Moreover, we have obtained new solutions from that reduction in the form of solitary, periodic, and singular solutions by solving the reduced ODE. Finally, from the comparison between the applications of the two methods on the VCKPE, we have some important remarks.

Remark 1: The connection between the CK and the HB method has simplified the calculations of the direct CK method a lot and reduced the number of freedoms from 4 to 2.

Remark 2: The classical CK method reduces the n-dimensional PDE to the (n–1)-dimensional PDE, and we need to perform more reductions to be able to solve the PDE under study; otherwise, the connection between the CK method and the HB method, with its generalized form given by Moussa in [17], makes us able to reduce the n-dimensional PDE to only the one-dimensional ODE.

Remark 3: If we try to reduce (9) to an ODE using the travelling wave transformation U(ζ, η, τ)=R(ω), where w = 13(ζ + η + τ), we get AR″″ + (RR′)′=0, which is a special case of the reduced ODE in (17) under the same integrability condition (8).

Remark 4: From Figures 1–6 we have seen that how the soliton and periodic wave propagation for both solutions u1 and u2 have been affected with different values of the variable coefficients.

From the above remarks, we conclude that the enlarged CK method connected with the HB method is more powerful, direct, and simple than the classical CK method.

Figure 1:

The soliton solution propagation u₁ with B=-1, k=A=y=z=1, f(t)=1 and all other variable functions have the same value t.

Figure 2:

The periodic wave solution u₂ with B=1, k=A=y=z=1, f(t)=1 and all other variable functions have the same value t.

Figure 3:

The soliton wave solution u₁ with B=-1, k=A=y=z=1, a(t)=b(t)=c(t)=1 and all other variable functions have the same value sin(t).

Figure 4:

The periodic wave propagation for solution u₂ with B=1, k=A=y=z=1, a(t)=b(t)=c(t)=1 and all other variable functions have the same value sin(t).

Figure 5:

The soliton wave solution u₁ with B=-1, k=A=y=z=1, a(t)=b(t)=t₂,c(t)=1, h(t)=cos(5t) and all other variable functions have the same value t.

Figure 6:

The periodic wave propagation for solution u₂ with B=1, k=A=y=z=1, a(t)=b(t)=t₂, c(t)=1, h(t)=cos(5t) and all other variable functions have the same value t.