New Lax Pairs and Darboux Transformation and Its Application to a Shallow Water Wave Model of Generalized KdV Type

Abstract

New Lax pairs of a shallow water wave model of generalized KdV equation type are presented. According to this Lax pair, we constructed a new spectral problem. By using this spectral problem, we constructed Darboux transformation with the help of a gauge transformation. Applying this Darboux transformation, some new exact solutions including double-soliton solution of the shallow water wave model of generalized KdV equation type are obtained. In order to visually show dynamical behaviors of these double soliton solutions, we plot graphs of profiles of them and discuss their dynamical properties.

1. Introduction

It is well known that the Lax pair and Darboux transformation can be employed to obtain multisoliton solution of nonlinear evolution equations. Darboux transformations provide us with purely algebraic, powerful method to construct solutions for systems of nonlinear equations. In recent years, more and more researchers used the Lax pair and Darboux transformation to investigate soliton solutions of classical nonlinear wave equations and some new soliton equations which were generated by new spectral problems; see [1–30] and references cited therein. In general, a systematical theory on such Darboux transformation even for 𝑛×𝑛 matrix spectral problem and the resulting zero curvature equation has a beautiful algebraic structure for associated evolution equations, which tells symmetry algebras of the obtained evolution equations; see [31, 32] and references cited therein. Sometimes, it is found that there are many infinity symmetries from the adopted zero curvature equation.

In this paper, we will investigate the Lax pairs, Darboux transformation, and double soliton solutions of the following famous shallow water wave model of generalized KdV equation type: 𝑢_𝑡−ℎ²₀3𝑢_𝑥𝑥𝑡+𝑐₀𝑢_𝑥+3𝛼2𝑢𝑢_𝑥−𝑐₀ℎ²₀6𝑢_𝑥𝑥𝑥=𝛼ℎ²₀3𝑢_𝑥𝑢_𝑥𝑥+𝛼ℎ²₀6𝑢𝑢_𝑥𝑥𝑥, (1) which appeared in [33], where 0<𝛼≪1 is a small-amplitude parameter. Only dropping the right-hand side of (1) gives BBM equation. Dropping the right-hand side of (1) and replacing the term 𝑢_𝑥𝑥𝑡 by the term −𝑐₀𝑢_𝑥𝑥𝑥 give KdV equation [34]. Thus, (1) can be seen as a BBM equation extended by retaining higher order terms in an asymptotic expansion in terms of the small-amplitude parameter 𝛼. Dropping the term −(𝑐₀ℎ²₀/6)𝑢_𝑥𝑥𝑥 from (1) and letting ℎ₀=√3,𝑐₀=2𝜅,𝛼=2, (1) becomes the celebrated Camassa-Holm equation [33] as follows: 𝑢_𝑡+2𝜅−𝑢_𝑥𝑥𝑡+3𝑢𝑢_𝑥=2𝑢_𝑥𝑢_𝑥𝑥+𝑢𝑢_𝑥𝑥𝑥, (2) where 𝑢 is the fluid velocity in the 𝑥 direction (or equivalent to the height of the water’s free surface above a flat bottom), 𝜅 is a constant related to the critical shallow water wave speed, and subscripts denote partial derivatives. Letting 𝑐₀=1,𝜈=−(1/3)ℎ²₀,𝛾=(3/2)𝛼,𝛽=−(1/6)ℎ²₀𝑐₀, (1) can be rewritten as 𝑢_𝑡+𝑢_𝑥+𝜈𝑢_𝑥𝑥𝑡+𝛽𝑢_𝑥𝑥𝑥+𝛾𝑢𝑢_𝑥+13𝛾𝜈𝑢𝑢_𝑥𝑥𝑥+2𝑢_𝑥𝑢_𝑥𝑥=0, (3) which comes from physical considerations via the methodology introduced by Fuchssteiner and Fokas in [35, 36]. The Lax pairs of (3) with 𝛾=1 are given by [37, 38] as follows: Ψ_𝑥𝑥=𝑘²()+𝑣1−𝑣4𝜈Ψ=0,𝜎𝑣=𝑢+𝜈𝑢_𝑥𝑥,𝜎=𝛽−𝜈,Ψ2𝜈_𝑡+−12𝛽+𝑐Ψ+𝑢+𝜈+2𝜎4𝜈𝑘²Ψ−1^𝑥=0, (4)(5) where 𝑐 is an arbitrary constant. It is a pity that the [37] is not a formal publication, and it is only preprint paper, so we cannot know its reality contents whether the authors have obtained soliton solutions of (3) by the Lax pairs (4) and (5). In fact, (3) has been studied by many authors in recent years; see the following brief introductions.

In [39], by using the bifurcation theory of dynamic system, some subsection-function and implicit function solutions such as compactons, solitary waves, smooth periodic waves, and nonsmooth periodic waves with peaks as well as the existence conditions have been presented by Bi. By using the same method, Li and Zhang [40] studied a generalization form of the modified KdV equation, which is more complex than (3). In [40], the existence of solitary wave, kink and antikink wave solutions, and uncountably many smooth and nonsmooth periodic wave solutions are discussed. By using the improved method named integral bifurcation method [41], Rui et al. [42] obtain all kinds of soliton-like or kink-like wave solutions, periodic wave solutions with loop or without loop, smooth compacton-like periodic wave solutions, and nonsmooth periodic cusp wave solutions for (3). In [43], Long and Chen discussed the existence of solitary wave, cusp wave, periodic wave, periodic cusp wave, and compactons were for (3). From the above references, (1) (i.e., (3)) is a very important water wave model.

The rest of this paper is organized as follows. In Section 2, we will derive new Lax pair and Darboux transformation of (1). In Section 3, by using this Darboux transformation, we will investigate soliton solutions of (1) and discuss the dynamic properties of these soliton solutions.

2. Lax Pair and Darboux Transformation of (1)

Through a series of tedious computation, we obtain Lax pairs of (1) as follows: 𝜙_𝑥𝑥=−𝑐₀+34𝜆𝛼4ℎ²₀−𝑢−(1/3)ℎ²₀𝑢_𝑥𝑥𝜙2𝜆𝜙,_𝑡𝑐=−⁰2+3𝜆𝛼2ℎ²₀+𝛼2𝑢𝜙_𝑥+𝛼4𝑢_𝑥𝜙. (6)(7) Obviously, the Lax pairs (6) and (7) are different from the Lax pairs (4) and (5) under 𝑐₀=1,𝜈=−(1/3)ℎ²₀,𝛾=(3/2)𝛼,𝛽=−(1/6)ℎ²₀𝑐₀. They are new Lax pairs which we obtained. By using the new Lax pairs (6) and (7), we will construct a Darboux transformation for obtaining soliton solutions of (1).

First, we consider the following spectral problems: 𝜙_𝑥=𝐌𝜙,𝜙_𝑡=𝐍𝜙, (8) with−𝑐𝐌=01₀+34𝜆𝛼4ℎ²₀−𝑢−(1/3)ℎ²₀𝑢_𝑥𝑥0,𝛼2𝜆𝐍=4𝑢_𝑥−𝑐₀2+3𝜆𝛼2ℎ²₀+𝛼2𝑢𝛼4𝑢_𝑥𝑥−𝑐₀2+3𝜆𝛼2ℎ²₀+𝛼2𝑢−𝑐₀+34𝜆𝛼4ℎ²₀−𝑢−(1/3)ℎ²₀𝑢_𝑥𝑥−𝛼2𝜆4𝑢_𝑥, (9)where 𝛼 is a constant, 𝜆 is a spectral parameter, and 𝑢 is a potential function. From compatibility, condition 𝜙_𝑥𝑥𝑡=𝜙_𝑡𝑥𝑥 yields a zero curvature equation 𝐌_𝑡−𝐍_𝑥+[𝐌,𝐍]=𝐎. Substituting 𝐌,𝐍 into the zero curvature equation, by a direct calculation, (1) is obtained successfully.

Next, we will construct a Darboux Transformation (DT) of the spectral problems (8). In fact, the DT is actually a gauge transformation 𝜙=𝐓𝜙 (10) of the spectral problems (8). It is required that 𝜙 also satisfies the same form of spectral problems 𝜙_𝑥=𝐌𝜙,𝐌=𝐓_𝑥+𝐓𝐌𝐓⁻¹,𝜙_𝑡=𝐍𝜙,𝐍=𝐓_𝑡+𝐓𝐍𝐓⁻¹. (11)(12)

It means that we have to find a matrix 𝐓 such that the old potential 𝑢 is replaced by the new one 𝑢.

Suppose (𝜆)=𝐴(𝜆)𝐵(𝜆)𝐶(𝜆)𝐷(𝜆)𝐓=𝐓, (13) where 𝐴(𝜆)=𝐴_𝑁𝜆^𝑁+^𝑁−1_𝑘=0𝐴_𝑘𝜆^𝑘,𝐵(𝜆)=𝐴_𝑁^𝑁−1_𝑘=0𝐵_𝑘𝜆^𝑘,𝐶(𝜆)=1𝐴_𝑁^𝑁−1_𝑘=0𝐶_𝑘𝜆^𝑘,𝐷(𝜆)=1𝐴_𝑁𝜆^𝑁+^𝑁−1_𝑘=0𝐷_𝑘𝜆^𝑘, (14)(15) and 𝐴_𝑁,𝐴_𝑘,𝐵_𝑘,𝐶_𝑘, and 𝐷_𝑘(0≤𝑘≤𝑁−1) are functions of 𝑥 and 𝑡.

Let 𝜙(𝜆_𝑗)=(𝜙₁(𝜆_𝑗),𝜙₂(𝜆_𝑗))^𝑇,𝜓(𝜆_𝑗)=(𝜓₁(𝜆_𝑗),𝜓₂(𝜆_𝑗))^𝑇 be two basic solutions of (8). From (10), there exist constants 𝑟_𝑗(0≤𝑗≤𝑁−1), which satisfy 𝐴𝜆_𝑗𝜙₁𝜆_𝑗+𝐵𝜆_𝑗𝜙₂𝜆_𝑗−𝑟_𝑗𝐴𝜆_𝑗𝜓₁𝜆_𝑗+𝐵𝜆_𝑗𝜓₂𝜆_𝑗𝐶𝜆=0,_𝑗𝜙₁𝜆_𝑗+𝐷𝜆_𝑗𝜙₂𝜆_𝑗−𝑟_𝑗𝐶𝜆_𝑗𝜓₁𝜆_𝑗+𝐷𝜆_𝑗𝜓₂𝜆_𝑗=0. (16) Further, (16) can be written as a linear algebraic system 𝐴𝜆_𝑗+𝛿_𝑗𝐵𝜆_𝑗=0,𝐶𝜆_𝑗+𝛿_𝑗𝐷𝜆_𝑗=0. (17) That is ^𝑁−1_𝑘=0𝐴_𝑘+𝛿_𝑗𝐵_𝑘𝜆^𝑘_𝑗=−𝜆^𝑁_𝑗,^𝑁−1_𝑘=0𝐶_𝑘+𝛿_𝑗𝐷_𝑘𝜆^𝑘_𝑗=−𝛿_𝑗𝜆^𝑁_𝑗, (18) where 𝛿_𝑗=𝜙₂𝜆_𝑗−𝑟_𝑗𝜓₂𝜆_𝑗𝜙₁𝜆_𝑗−𝑟_𝑗𝜓₁𝜆_𝑗,1≤𝑗≤2𝑁, (19) and the constants 𝜆_𝑗(𝜆_𝑘≠𝜆_𝑠 as  𝑘≠𝑠),𝑟_𝑗 are suitably chosen such that determinant of coefficients for (18) is nonzero. Therefore, 𝐴_𝑁,𝐴_𝑘,𝐵_𝑘,𝐶_𝑘, and 𝐷_𝑘(0≤𝑘≤𝑁−1) are uniquely determined by (18).

Equations (14) and (15) show that the det𝐓(𝜆) is a 2𝑁th-order polynomial in 𝜆, and det𝐓𝜆_𝑗=𝐴𝜆_𝑗𝐷𝜆_𝑗−𝐵𝜆_𝑗𝐶𝜆_𝑗. (20) On the other hand, from (17), we have 𝐴(𝜆_𝑗)=−𝛿_𝑗𝐵(𝜆_𝑗),𝐶(𝜆_𝑗)=−𝛿_𝑗𝐷(𝜆_𝑗). Thus we have det𝐓(𝜆)=𝛽^2𝑁−1_𝑗=1𝜆−𝜆_𝑗, (21) where 𝛽 is independent of 𝜆. Equation (21) implies that 𝜆_𝑗(1≤𝑗≤2𝑁) are 2𝑁 roots of det𝐓(𝜆).

Second, we prove the following theory of Darboux transformation for special variable.

Theorem 1. Let 𝐴_𝑁 satisfy 𝐴²_𝑁=1. (22) Then the matrix 𝐌 determined by (11) has the same form as 𝐌; that is, −𝑐𝐌=01₀+34𝜆𝛼4ℎ²₀−𝑢−(1/3)ℎ²₀𝑢_𝑥𝑥02𝜆, (23) where the transformation from the old potential 𝑢 into new one 𝑢 is given by 6𝑢=𝑢+ℎ²₀𝐴_𝑁−1,𝐴_{𝑁−1,𝑥}=𝐶_𝑁−1−34ℎ²₀𝐵_𝑁−1,𝐵_{𝑁−1,𝑥}=𝐷_𝑁−1−𝐴_𝑁−1,𝐶_{𝑁−1,𝑥}=34ℎ²₀𝐴_𝑁−1−𝐷_𝑁−1+𝑐₀+()ℎ4𝛼𝑢−1/3²₀𝑢_𝑥𝑥2,𝐷_{𝑁−1,𝑥}=34ℎ²₀𝐵_𝑁−1−𝐶_𝑁−1. (24)(25)

Proof. Let 𝐓⁻¹=𝐓^∗/det𝐓 and 𝐓_𝑥+𝐓𝐌𝐓^∗=𝑓₁₁(𝜆)𝑓₁₂(𝜆)𝑓₂₁(𝜆)𝑓₂₂(𝜆). (26) It is easy to see that 𝑓₁₁(𝜆) and 𝑓₂₂(𝜆) are 2𝑁th-order polynomials in 𝜆, 𝑓₁₂(𝜆) and 𝑓₂₁(𝜆) are (2𝑁−1)th-order polynomials in 𝜆. From (19) and (8), we find 𝛿_𝑗𝑥𝑐=−⁰+34𝜆𝛼4ℎ²₀−𝑢−(1/3)ℎ²₀𝑢_𝑥𝑥2𝜆−𝛿²_𝑗. (27) Through direct calculation, all 𝜆_𝑗(0≤𝑗≤2𝑁) are roots of 𝑓_𝑛𝑠(𝑛,𝑠=1,2). Together with (21) and (26), we get 𝐓_𝑥+𝐓𝐌𝐓^∗=(det𝐓)𝑃(𝜆), (28) with 𝑃(𝜆)=𝑝⁽⁰⁾₁₁𝑝⁽⁰⁾₁₂𝑝⁽⁰⁾₂₁𝑝⁽⁰⁾₂₂, (29) where 𝑝⁽⁰⁾_𝑛𝑠(𝑛,𝑠=1,2) are independent of spectral parameter 𝜆. Indeed, (28) can be written as 𝐓_𝑥+𝐓𝐌=𝑃(𝜆)𝐓. (30)
Comparing the coefficients of 𝜆^𝑁 in (30), we find 𝑝⁽⁰⁾₁₁=−𝑝⁽⁰⁾₂₂=𝜕_𝑥ln𝐴_𝑁,𝑝⁽⁰⁾₁₂=𝐴²_𝑁,𝑝⁽⁰⁾₂₁=1𝐴²_𝑁34ℎ²₀. (31) Substituting (22) into (31) yields 𝑝⁽⁰⁾₁₁=−𝑝⁽⁰⁾₂₂=0,𝑝⁽⁰⁾₁₂=𝐴²_𝑁=1. (32) From (22), (24), (25), and (31) and noticing 𝑢 in (23), we get 𝑝⁽⁰⁾₂₁𝑐=−⁰+34𝜆𝛼4ℎ²₀−𝑢−(1/3)ℎ²₀𝑢_𝑥𝑥.2𝜆 (33) Thus 𝑃(𝜆)=𝐌. The proof of Theorem 1 is completed.

Finally, by using same way to Theorem 1, we prove that 𝐍 in (12) has the same form as 𝐍 under the transformation (10) and (24); see the following theory and its proof.

Theorem 2. The matrix 𝐍 defined by (12) has the same type as 𝐍, in which the old potential 𝑢 is mapped into 𝑢 via the same DT (24).

Proof. Let 𝐓⁻¹=𝐓^∗/det𝐓 and 𝐓_𝑡+𝐓𝐍𝐓^∗=𝑔₁₁(𝜆)𝑔₁₂(𝜆)𝑔₂₁(𝜆)𝑔₂₂(𝜆). (34) It is easy to see that 𝑔₁₁(𝜆) and 𝑔₂₂(𝜆) are 2𝑁th-order polynomials in 𝜆, 𝑔₁₂(𝜆) and 𝑔₂₁(𝜆) are (2𝑁+1)th-order polynomials in 𝜆. By using (19) and (8), we obtain 𝛿_𝑗𝑡=𝛼4𝑢_𝑥𝑥−𝑐₀2+3𝜆𝛼2ℎ²₀+𝛼2𝑢×−𝑐₀+34𝜆𝛼4ℎ²₀−𝑢−(1/3)ℎ²₀𝑢_𝑥𝑥−𝛼2𝜆2𝑢_𝑥𝛿_𝑗+𝑐₀2+3𝜆𝛼2ℎ²₀+𝛼2𝑢𝛿²_𝑗. (35) Through direct calculation, all 𝜆_𝑗(0≤𝑗≤2𝑁) are roots of 𝑔_𝑛𝑠(𝑛,𝑠=1,2). Together with (21) and (34), we get 𝐓_𝑡+𝐓𝐍𝐓^∗=(det𝐓)𝑄(𝜆), (36) with 𝑄(𝜆)=𝑞⁽⁰⁾₁₁𝑞⁽¹⁾₁₂𝜆+𝑞⁽⁰⁾₁₂𝑞⁽¹⁾₂₁𝜆+𝑞⁽⁰⁾₂₁𝑞⁽⁰⁾₂₂, (37) where 𝑞^(𝑙)_𝑛𝑠(𝑛,𝑠=1,2,𝑙=0,1) are independent of spectral parameter 𝜆. Equation (37) can be written as 𝐓_𝑡+𝐓𝐍=𝑄(𝜆)𝐓. (38)
Comparing the coefficients of 𝜆^𝑁+1 and 𝜆^𝑁 in (38) leads to 𝑞⁽¹⁾₁₂=−3𝛼2ℎ²₀𝐴²_𝑁,𝑞⁽¹⁾₂₁1=−𝐴²_𝑁3𝛼2ℎ²₀34ℎ²₀,𝑞⁽⁰⁾₁₁=𝜕_𝑡ln𝐴_𝑁+𝛼4𝑢_𝑥−3𝛼2ℎ²₀34ℎ²₀𝐵_𝑁−1+3𝛼2ℎ²₀𝐶_𝑁−1,𝑞⁽⁰⁾₁₂𝑐=−⁰2+𝛼2𝑢𝐴²_𝑁−3𝛼2ℎ²₀𝐴²_𝑁𝐴_𝑁−1+3𝛼2ℎ²₀𝐴²_𝑁𝐷_𝑁−1,𝑞⁽⁰⁾₂₁=1𝐴²_𝑁𝛼4𝑢_𝑥𝑥−1𝐴²_𝑁34ℎ²₀𝑐₀2+𝛼2𝑢−1𝐴²_𝑁3𝛼2ℎ²₀34ℎ²₀𝐷_𝑁−1+1𝐴²_𝑁3𝛼2ℎ²₀𝑐₀+()ℎ4𝛼𝑢−1/3²₀𝑢_𝑥𝑥2+𝐴_𝑁−1𝐴²_𝑁3𝛼2ℎ²₀34ℎ²₀,𝑞⁽⁰⁾₂₂=−𝜕_𝑡ln𝐴_𝑁−3𝛼2ℎ²₀𝐶_𝑁−1−𝛼4𝑢_𝑥+3𝛼2ℎ²₀34ℎ²₀𝐵_𝑁−1. (39)(40)(41)(42)(43)
Substituting (22), (24), and (25) into (40) to (43), we can get 𝑞⁽⁰⁾₁₁=𝛼4𝑢_𝑥−3𝛼2ℎ²₀34ℎ²₀𝐵_𝑁−1+3𝛼2ℎ²₀𝐶_𝑁−1=𝛼4𝑢_𝑥−3𝛼2ℎ²₀𝐴_{𝑁−1,𝑥}=𝛼4𝑢_𝑥,𝑞⁽¹⁾₁₂𝜆+𝑞⁽⁰⁾₁₂=−3𝛼2ℎ²₀𝑐𝜆−⁰2+𝛼2𝑢−3𝛼2ℎ²₀𝐴_𝑁−1+3𝛼2ℎ²₀𝐷_𝑁−1=−3𝛼2ℎ²₀𝑐𝜆−⁰2+𝛼2𝑢−3𝛼ℎ²₀+3𝛼2ℎ²₀𝐴_𝑁−1+𝐷_𝑁−1𝑐=−⁰2+3𝜆𝛼2ℎ²₀+𝛼2𝑢,𝑞⁽⁰⁾₂₂=−3𝛼2ℎ²₀𝐶_𝑁−1−𝛼4𝑢_𝑥+3𝛼2ℎ²₀34ℎ²₀𝐵_𝑁−1𝛼=−4𝑢_𝑥−3𝛼2ℎ²₀𝐴_{𝑁−1,𝑥}𝛼=−4𝑢_𝑥,𝑞⁽¹⁾₂₁𝜆+𝑞⁽⁰⁾₂₁=−3𝛼2ℎ²₀34ℎ²₀𝛼𝜆+4𝑢_𝑥𝑥−34ℎ²₀𝑐₀2+𝛼2𝑢−3𝛼2ℎ²₀34ℎ²₀𝐷_𝑁−1+3𝛼2ℎ²₀𝑐₀+()ℎ4𝛼𝑢−1/3²₀𝑢_𝑥𝑥2+3𝛼2ℎ²₀34ℎ²₀𝐴_𝑁−1=𝛼4𝑢_𝑥𝑥−34ℎ²₀𝑐₀2+3𝜆𝛼2ℎ²₀+𝛼2𝑢+3𝛼2ℎ²₀𝐶_{𝑁−1,𝑥}. (44) On the other hand, from (24), (25), and (33), we have 𝛼4𝑢_𝑥𝑥−𝑐₀2+3𝜆𝛼2ℎ²₀+𝛼2𝑢×−𝑐₀+34𝜆𝛼4ℎ²₀−𝑢−(1/3)ℎ²₀𝑢_𝑥𝑥=𝛼2𝜆4𝑢_𝑥𝑥−𝑐₀2+3𝜆𝛼2ℎ²₀+𝛼2𝑢34ℎ²₀=𝛼4𝑢_𝑥𝑥+3𝛼2ℎ²₀𝐴_{𝑁−1,𝑥𝑥}−𝑐₀2+3𝜆𝛼2ℎ²₀+𝛼2𝑢34ℎ²₀−6𝛼2ℎ²₀34ℎ²₀𝐴_𝑁−1=𝛼4𝑢_𝑥𝑥−𝑐₀2+3𝜆𝛼2ℎ²₀+𝛼2𝑢34ℎ²₀+3𝛼2ℎ²₀𝐶_{𝑁−1,𝑥}. (45) From (44) and (45), obviously we have 𝑞⁽¹⁾₂₁𝜆+𝑞⁽⁰⁾₂₁=𝛼4𝑢_𝑥𝑥−𝑐₀2+3𝜆𝛼2ℎ²₀+𝛼2𝑢×−𝑐₀+34𝜆𝛼4ℎ²₀−𝑢−(1/3)ℎ²₀𝑢_𝑥𝑥.2𝜆 (46) Thus 𝑄(𝜆)=𝐍, and this completes the proof of Theorem 2.

3. Exact Soliton Solution of (1) and Its Dynamic Properties

In this section, we will construct the explicit solutions of the integrable shallow water wave (1) by using the Darboux transformation (24).

Choosing 𝑢=𝑢₀ (𝑢₀ is an arbitrary constant) as a seed solution of (1), and substituting 𝑢=𝑢₀ into the spectral problems (8). Then we get two basic solutions of (8) as follows: 𝜙𝜆_𝑗=cosh𝜇_𝑗−𝑐₀4𝜆_𝑗𝛼+34ℎ²₀−𝑢₀2𝜆_𝑗sinh𝜇_𝑗,𝜓𝜆_𝑗=sinh𝜇_𝑗−𝑐₀4𝜆_𝑗𝛼+34ℎ²₀−𝑢₀2𝜆_𝑗cosh𝜇_𝑗 (47) with 𝜇_𝑗=−𝑐₀4𝜆_𝑗𝛼+34ℎ²₀−𝑢₀2𝜆_𝑗𝑐𝑥−⁰2+3𝜆_𝑗𝛼2ℎ²₀+𝑢₀𝛼2𝑡, (48) where 𝜆_𝑗 is a nonzero constant and 0<𝑗≤2𝑁.

According to (19), we have 𝛿_𝑗=−𝑐₀4𝜆_𝑗𝛼+34ℎ²₀−𝑢₀2𝜆_𝑗tanh𝜇_𝑗−𝑟_𝑗1−𝑟_𝑗tanh𝜇_𝑗, (49) where 𝑟_𝑗(0<𝑗≤2𝑁) are nonzero constants.

Using the Cramer rule to solve the linear algebraic system (18), we obtain 𝐴_𝑁−1=Δ_𝐴𝑁−1Δ, (50) whereΔ=1𝛿₁𝜆₁𝛿₁𝜆₁⋯𝜆^𝑘₁𝛿₁𝜆^𝑘₁⋯𝜆₁^𝑁−1𝛿₁𝜆₁^𝑁−11𝛿₂𝜆₂𝛿₂𝜆₂⋯𝜆^𝑘₂𝛿₂𝜆^𝑘₂⋯𝜆₂^𝑁−1𝛿₂𝜆₂^𝑁−1⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮1𝛿_2𝑁𝜆_2𝑁𝛿_2𝑁𝜆_2𝑁⋯𝜆^𝑘_2𝑁𝛿_2𝑁𝜆^𝑘_2𝑁⋯𝜆^𝑁−1_2𝑁𝛿_2𝑁𝜆^𝑁−1_2𝑁,Δ_𝐴𝑁−1=1𝛿₁𝜆₁𝛿₁𝜆₁⋯𝜆^𝑘₁𝛿₁𝜆^𝑘₁⋯−𝜆^𝑁₁𝛿₁𝜆₁^𝑁−11𝛿₂𝜆₂𝛿₂𝜆₂⋯𝜆^𝑘₂𝛿₂𝜆^𝑘₂⋯−𝜆^𝑁₂𝛿₂𝜆₂^𝑁−1⋮⋮⋮⋮⋮⋮⋮⋮⋮⋮1𝛿_2𝑁𝜆_2𝑁𝛿_2𝑁𝜆_2𝑁⋯𝜆^𝑘_2𝑁𝛿_2𝑁𝜆^𝑘_2𝑁⋯−𝜆^𝑁_2𝑁𝛿_2𝑁𝜆^𝑁−1_2𝑁. (51)

As examples, we will investigate exact solutions of (1) in two simple cases 𝑁=1 and 𝑁=2. When 𝑁=1, solving the linear algebraic system (18) leads to 𝐴₀=𝛿₁𝜆₂−𝛿₂𝜆₁𝛿₂−𝛿₁. (52) Substituting (49) and (52) into (24), a singular double-soliton solution of (1) is obtained as follows: 𝑢[1]=6ℎ²₀𝑓−𝑐₀4𝜆₁𝛼+34ℎ²₀−𝑢₀2𝜆₁tanh𝜇₁−𝑟₁1−𝑟₁tanh𝜇₁𝜆₂−−𝑐₀4𝜆₂𝛼+34ℎ²₀−𝑢₀2𝜆₂tanh𝜇₂−𝑟₂1−𝑟₂tanh𝜇₂𝜆₁, (53) where 𝑓=−𝑐₀4𝜆₂𝛼+34ℎ²₀−𝑢₀2𝜆₂tanh𝜇₂−𝑟₂1−𝑟₂tanh𝜇₂−−𝑐₀4𝜆₁𝛼+34ℎ²₀−𝑢₀2𝜆₁tanh𝜇₁−𝑟₁1−𝑟₁tanh𝜇₁. (54) By using program of computer, it is easy to verify that the solution (53) satisfies (1), and this shows that the Darboux transformation (24) which we obtained is correct. In order to show the properties of the above singular double-soliton solutions visually, as an example, we plot the 3-D graphs of solution (53) for some fixed parameters, which are shown in Figures 1 and 2.

(a) 0.3<ℎ0<0.5,𝜆1<0,𝜆2<0

(b) 0.2<ℎ0<0.3,𝜆1<0,𝜆2<0

Figure 1 

The 3-D graphs of profiles of the singular double-soliton solution (53) for fixed parameters 𝑐0=1.5, 𝛼=0.2, 𝜆1=−1, 𝜆2=−0.2, 𝑟1=−8.0, 𝑟2=2.0, and 𝑎=0.1.

(a)  0<ℎ0<0.2,𝜆1<0,𝜆2<0

(b) ℎ0>1,𝜆1>0,𝜆2>0

Figure 2 

The 3-D graphs of profiles of the singular double-soliton solution (53) for parameters: (a) ℎ0=0.15, 𝑐0=1.5, 𝛼=0.2, 𝜆1=−1, 𝜆2=−0.2, 𝑟1=−8.0, 𝑟2=2.0, and 𝑎=0.1; (b) ℎ0=3, 𝑐0=−1.5, 𝛼=0.2, 𝜆1=1, 𝜆2=0.2, 𝑟1=8, 𝑟2=2.0, and 𝑎=−2.

When 𝑁=2, using the Cramer rule to solve the linear algebraic system (18), we obtain 𝐴₁=Δ_𝐴1Δ, (55) with Δ=1𝛿₁𝜆₁𝛿₁𝜆₁1𝛿₂𝜆₂𝛿₂𝜆₂1𝛿₃𝜆₃𝛿₃𝜆₃1𝛿₄𝜆₄𝛿₄𝜆₄,Δ_𝐴1=1𝛿₁−𝜆²₁𝛿₁𝜆₁1𝛿₂−𝜆²₂𝛿₂𝜆₂1𝛿₃−𝜆²₃𝛿₃𝜆₃1𝛿₄−𝜆²₄𝛿₄𝜆₄, (56) where 𝛿_𝑗(𝑗=1,2,3,4) are given by (49). From (24), an explicit solution of (1) is obtained by the following: 𝑢[2]6=𝑢+ℎ²₀𝐴₁, (57) where 𝐴₁ is given by (55). Equation (57) is a very complex solution, and it is not soliton solution. In order to show the properties of solution (57), under the fixed parameters 𝜆₁=−0.2, 𝜆₂=−0.3, 𝜆₃=−0.4, 𝜆₄=−0.1, 𝑎=0, 𝑐₀=15, 𝛼=1/2, ℎ₀=0.8, 𝑟₁=−0.2, 𝑟₂=−0.3, 𝑟₃=−0.4, 𝑟₄=−0.5, 𝑡=0.1, we plot its 2-D profile, which is shown in Figure 3.

Figure 3 

The 2-D graph of profile of the exact soliton solution (57) for fixed parameters.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 11361023), the Natural Science Foundation of Chongqing Normal University (no. 13XLR20), the Scientific Foundation of Education of Yunnan Province (no. 2012C199), and the Program Foundation of Chongqing Innovation Team Project in University under Grant no. KJTD201308.