The Multiple Exp-Function Method and the Linear Superposition Principle for Solving the (2+1)-Dimensional Calogero–Bogoyavlenskii–Schiff Equation

1 Introduction

Nonlinear phenomena appear in a wide variety of scientific applications such as plasma physics, solid-state physics, fluid dynamics, and so on. Partial differential equations (PDEs) have been the focus of many studies due to their frequent appearance in various applications in many fields, such as physics, biology, engineering, signal processing, control theory, and so on. Recently, a large amount of literature has been provided to construct the solutions of the PDEs. Several powerful methods have been proposed to obtain approximate and exact solutions of these equations, such as the inverse scattering transform [1], the Bäcklund transformation method [2], the Hirota bilinear method [3], the Adomian decomposition method [4, 5], the variational iteration method [6–8], the homotopy analysis method [9–12], the homotopy perturbation method [13–15], the Lagrange characteristic method [16], the fractional sub-equation method [17], the (G′/G)-expansion method [18, 19], the transformed rational function method [20], the multiple exp-function method [21, 22], the generalised Riccati equation method [23], the Frobenius decomposition technique [24], the local fractional differential equations method [25, 26], the local fractional variation iteration method [27], the multiple (G′/G)-expansion method [28], the cantor-type cylindrical coordinate method [29], the Riccati equation method combined with the (G′/G)-expansion method [30], the fractional complex transform method [31], the modified simple equation method [32–35], the first integral method [36–38], the linear superposition principle [39], and so on.

The objective of this article is to apply two interesting methods, namely, the multiple exp-function method and the linear superposition principle to construct the exact solutions for the following (2+1)-dimensional Calogero–Bogoyavlenskii–Schiff (CBS) equation [40–42]:

(1)uxt + uxxxz + 4uxuxz + 2uxxuz = 0, (1)

where u=u(x, z, t). The CBS equation was first constructed by Bogoyavlenskii and Schiff in different ways [41, 42]. Bogoyavlenskii used the modified Lax formalism, whereas Schiff derived the same equation by reducing the self-dual Yang–Mills equation. Equation (1) has been discussed in [43] by using the Hirota bilinear method.

2 Description of the Multiple Exp-Function Method

Consider the following nonlinear PDE:

(2)F(u, ut, ux, utt, uxx,…) = 0. (2)

We describe the main steps [21, 22] of the multiple exp-function method to solve (2) as follows:

Step 1: We introduce n-wave variables η_i=η_i (x, t), (i=1, 2, …, n), such that

(3)u(x, t) = U(ηi), ηi = cieξi, ξi = kix − λit, (3)

where k_i are the angular wave number and λ_i are the wave frequencies, while c_i are any constants, positive or negative.

Step 2: Assume that (2) has the rational solution

(4)u(x, t) = U(ηi)=p(η1, η2, …, ηn)q(η1, η2, …, ηn), (4)

where p and q are two polynomials of degree “n” and of the new variables η_i=η_i (x, t), (i=1, 2, …, n).

Step 3: Substituting (4) along with (3) into (2) yields a rational equation that can be written in the following form:

(5)Q(η1, η2, …, ηn)R(η1, η2, …, ηn) = 0, (5)

where R(η₁, η₂, …, η_n) ≠ 0.

Step 4: Setting the numerator of (5) to be zero yields a system of algebraic equations, which can be solved using the Maple to determine the two polynomials p and q and the exponents ξ_i, (i=1, 2, …, n). Consequently, we can get the exact explicit one-wave, two-wave, and three-wave solutions, which include one-soliton-, two-soliton-, and three-soliton-type solutions of (2).

3 One-Wave, Two-Wave, and Three-Wave Solutions to the Nonlinear CBS Equation

Let us apply the multiple exp-function method to the (2+1)-dimensional CBS equation (1). We analyze three cases of the two polynomials p and q for (1) to construct their multiple-wave solutions.

3.1 Case 1: One-Wave Solutions

With reference to [21, 22], let us try a pair of two polynomials p(η₁) and q(η₁) of degree one, such that

p(η1) = a0 + a1η1, q(η1) = b0 + b1η1,

where η1 = c1eξ1, ξ₁=k₁x + m₁z − λ₁t, while a₀, a₁, b₀, b₁, c₁, k₁, m₁, and λ₁ are constants to be determined later. By the multiple exp-function method, we assume that (1) has the rational solution

(6)u(x, z, t) = U(η1) = a0 + a1η1b0 + b1η1. (6)

Substituting (6) into (1), we have the polynomial equation

(7)c1η1 + c2η12 + c3η13 + c4η14 = 0, (7)

where

c1 = −k1(a1λ1b04 − a1k12m1b04 + b1k12m1a0b03 − b1λ1a0b03) = 0,c2 =−k1(−b12λ1a0b02 − 6k1m1a12b03 − 6k1m1b12a02b0 + a1λ1b03b1 + 12k1m1a1b02b1a0+11a1k12m1b03b1 − 11b12k12m1a0b02) = 0,c3 = −k1(−a1λ1b02b12 − 12k1m1a1b0b12a0 − 11a1k12m1b02b12 + b13λ1a0b0 + 11b13k12m1a0b0+6k1m1b13a02 + 6k1m1a12b02b1) = 0,c4 = −k1(−a1λ1b0b13 + b14λ1a0 − b14k12m1a0 + a1k12m1b0b13) = 0.

On solving the above algebraic equations using the Maple, we get the following result:

(8)a0 = −b0(2b1k1 − a1)b1, λ1 = k12m1, a1= a1, m1 = m1, b1 = b1, b0 = b0, k1 = k1. (8)

Now, the corresponding one-wave solution of (1) can be written in the new form

(9)u(x, z, t) = −b0(2b1k1 − a1) + b1a1c1eξ1b1(b0 + b1c1eξ1), (9)

where ξ1 = k1x + m1z − k12m1t, where c₁, a₁, m₁, b₀, b₁, and k₁ are arbitrary constants. On comparing our new result (9) with the well-known solution (25) obtained in [43] using the Hirota method, we deduce that they are equivalent in the special case b₀=b₁=c₁=1, a₁=2k₁.

3.2 Case 2: Two-Wave Solutions

With reference to [21, 22], let us try a particular pair of two polynomials p(η₁, η₂) and q(η₁, η₂) of degree two, such that

p(η1, η2) = 2[k1η1 + k2η2 + a12(k1 + k2)η1η2], q(η1, η2) = 1 + η1 + η2 + a12η1η2,

where ηi = cieξi, ξi = kix + miz − λit, while c_i, k_i, m_i, λ_i (i= 1, 2), and a₁₂ are constants to be determined. By the multiple exp-function method, we assume that (1) has the rational solution

(10)u(x, z, t) = U(η1, η2) = p(η1, η2)q(η1, η2). (10)

Substituting (10) into (1), we have the polynomial equation

(11)c1η1 + c2η2 + c3η12 + c4η22 + c5η1η2 + c6η12η2 + c7η1η22 + c8η12η22 + c9η1η23 + c10η2η13 + c11η12η23 + c12η13η22 = 0, (11)

where the coefficients c_i (i=1, 2, …, 12) of the polynomial equation (11) have been determined in terms of η₁, η₂, which are omitted here for simplicity. On setting these coefficients to be zero, we have a system of algebraic equations, which can be solved by the Maple, to get the following result:

(12)k1 = k1, k2 = k2, m1 = m1, m2 = m2, λ1 = k12m1, λ2 = k22m2, a12 = (k1 − k2)2(k1 + k2)2. (12)

Now, the corresponding two-wave solution of (1) has the new form

(13)u(x, z, t) = 2[k1η1 + k2η2 + a12(k1 + k2)η1η2]1 + η1 + η2 + a12η1η2, (13)

where a₁₂=(k₁ − k₂)²/(k₁ + k₂)², ηi = cieξi,ξi = kix + miz − ki2mit (i=1, 2).

One specific solution of two-wave solution is plotted in Figure 1.

Figure 1:

Two-wave solution (13) with m₁=k₁=c₁=1, m₂=k₂=−2, c₂=2, t=0, −50 ≤ x ≤ 50, −100 ≤ z ≤ 100.

3.3 Case 3: Three-Wave Solutions

With reference to [21, 22], let us try a particular pair of two polynomials p(η₁, η₂, η₃) and q(η₁, η₂, η₃) of degree three, such that

p(η1, η2, η3) = 2[k1η1 + k2η2 + k3η3 + a12(k1 + k2)η1η2 + a13(k1 + k3)η1η3 + a23(k2 + k3)η2η3+a12a13a23(k1 + k2 + k3)η1η2η3],and q(η1, η2, η3) = 1 + η1 + η2 + η3 + a12η1η2 + a13η1η3 + a23η2η3 + a12a13a23η1η2η3,

where ηi = cieξi, ξi = kix + miz − λit, (i=1, 2, 3), while c_i, k_i, m_i, λ_i and a₁₂, a₁₃, a₂₃ are constants to be determined. By the multiple exp-function method, we assume that (1) has the rational solution

(14)u(x, z, t) = U(η1, η2, η3) = p(η1, η2, η3)q(η1, η2, η3). (14)

Substituting (14) into (1), we have the equation

(15)c1η1 + c2η2 + c3η3 + c4η12 + c5η22 + c6η32 + c7η1η2 + c8η1η3 + c9η2η3 + c10η1η22 + c11η1η32 + c12η2η32 + c13η12η2 + c14η12η3 + c15η22η3 + c16η12η22 + c17η12η32 + c18η22η32 + c19η1η23 + c20η1η33 + c21η2η13 + c22η2η33 + c23η3η13 + c24η3η23 + c25η12η23 + c26η12η33 + c27η22η13 + c28η22η33 + c29η32η13 + c30η32η23 + c31η1η2η3 + c32η1η2η32 + c33η1η22η3 + c34η12η2η3 + c35η1η22η32 + c37η12η2η32 + c38η12η22η3 + c39η12η22η32 + c40η1η2η33 + c41η1η23η3 + c42η13η2η3 + c43η1η22η33 + c44η1η23η32 + c45η13η2η32 + c46η13η22η3 + c47η12η2η33 + c48η12η23η3 + c49η1η23η33 + c50η13η2η33 + c51η13η23η3 + c52η12η22η33 + c53η12η23η32 + c54η13η22η32 + c55η12η23η33 + c56η13η22η33 + c57η13η23η32 = 0, (15)

where the coefficients c_i (i=1 − 57) of the polynomial equation (15) have been determined in terms of η₁, η₂, η₃, which are omitted here for simplicity. On setting these coefficients to zero, we get a system of algebraic equations, which can be solved using the Maple, to get the following result:

(16)ki = ki, mi = mi, λi = ki2mi, aij = (ki − kj)2(ki + kj)2, 1 ≤ i < j ≤ 3. (16)

Now, the corresponding three-wave solution of (1) can be written in the new form

(17)u(x, z, t) = U(η1, η2, η3) = p(η1, η2, η3)q(η1, η2, η3), (17)

where

p(η1, η2, η3) = 2[k1η1 + k2η2 + k3η3 + a12(k1 + k2)η1η2 + a13(k1 + k3)η1η3 + a23(k2 + k3)η2η3+a12a13a23(k1 + k2 + k3)η1η2η3],q(η1, η2, η3) = 1 + η1 + η2 + η3 + a12η1η2 + a13η1η3 + a23η2η3 + a12a13a23η1η2η3,

and ηi = cieξi,ξi = kix + miz − ki2mit,a_ij=(k_i − k_j)²/(k_i + k_j)², 1 ≤ i ≤ j ≤ 3.

The specific solution of three-wave solution is plotted in Figure 2.

Figure 2:

Three-wave solution (17) with m₁=k₂=c₂=2, k₁=m₃=−1, k₃=m₂=c₃=3, c₁=−1, t=0, −150 ≤ x ≤ 150, −100 ≤ z ≤ 100.

4 The Construction of the n-Wave Solution of (1)

In this section, we apply the linear superposition principle [39] to find the n-wave solution of (1).

Through the dependent variable transformation u(x, z, t)=2 [ln (1 + f(x, z, t)]_x, (1) can be written in the Hirota bilinear form

(18)(Dx3Dz + DxDt)f⋅f = 0, (18)

which is equivalent to

(19)(fxxxz + fxt)f − fxxxfz − 3fxxzfx + 3fxzfxx − fxft = 0. (19)

Let us introduce the n-wave variables

(20)ξi = kix + miz − λit, i = (1, 2, …, n), (20)

and n-exponential wave functions

(21)fi = eξi, (21)

where k_i, m_i, and λ_i (i =1, 2, …, n) are constants to be determined.

Then, the n-wave solution condition (2.8) of [39] becomes

(22)ki3mi − ki3mj − kj3mi + kj3mj − 3ki2kjmi + 3ki2kjmj + 3kikj2mi−3kikj2mj − kiλi + kiλj + kjλi − kjλj = 0,  1 ≤ i ≠ j ≤ n. (22)

By inspection, a solution to this equation is given by

(23)ki = k,  λi = k2mi, (i = 1, 2, …, n), (23)

where k is an arbitrary constant.

Therefore, by the linear superposition principle in theorem 2.1 of [39], the nonlinear CBS equation (1) has the following n-wave solution:

(24)u(x, z, t) = 2[ln(1 + f(x, z, t))]x,  f(x, z, t) = ∑i = 1nεifi = ∑i = 1nεiekx + miz − k2mit, (24)

where ε_i are arbitrary constants.

5 Conclusions

The multiple exp-function method and the linear superposition principle are applied successfully for solving the (2+1)-dimensional CBS equation (1). The first method gives one-wave, two-wave, and three-wave solutions including one-soliton-, two-soliton-, and three-soliton-type solutions of (1). It is our guess that higher-wave solutions to the (2+1)-dimensional CBS equation (1) could be presented in a parallel manner. But the required computation is pretty complicated, even in the case of four-wave solutions. The second method gives a specific subclass of n-soliton solutions formed by linear combinations of exponential traveling waves. Finally to our knowledge, the obtained solutions (9), (13), and (17) of (1) using the multiple exp-function method and the obtained solution (24) of (1) using the linear superposition principle are all new and not published elsewhere.