Abstract
A consistent Riccati expansion (CRE) method is proposed for obtaining interaction solutions to the modified Korteweg-de Vries (mKdV) equation. Using the CRE method, it is shown that interaction solutions such as the soliton-tangent (or soliton-cotangent) wave cannot be constructed for the mKdV equation. More importantly, exact soliton-cnoidal periodic wave interaction solutions are presented. While soliton-cnoidal interaction solutions were found to degenerate to special resonant soliton solutions for the values of modulus (n) closer to one (upper bound of modulus) in the Jacobi elliptic function, a normal kink-shaped soliton was observed for values of n closer to zero (lower bound).
1. Introduction
The direct study of exact solutions to nonlinear evolution equations (NLEEs) has received much attention from many mathematicians and physicists due to the fact that new strides in nonlinear science, which were made possible by a substantial increase in computational platforms such as Mathematica, Maple, and MATLAB, have enabled improvements in the performance of complicated and tedious numerical computational methods. Indeed, several powerful methods such as the Tanh-function method [1–3], F-expansion method [4], Jacobian elliptic function method [5], and variational approach [6, 7] have been proposed for constructing exact solutions to NLEEs. Despite the successful implementation of such methods, it is still challenging to obtain solutions for interactions among different types of nonlinear excitations such as the soliton-soliton interaction.
Recently, some new soliton structure solutions were obtained for nonlinear systems. Chen et al. studied the vortex solitons in Bose-Einstein condensates with spin-orbit coupling and Gaussian optical lattices, based on the analytical and numerical method [8]. Milan et al. found exact fundamental soliton solutions in the spiraling guiding structures by the modified Petviashvilis iteration method [9]. Cheng et al. investigated the formation and propagation of a multipole soliton in a cold atomic gas with a parity-time symmetric potential using the modified square operator method [10]. Liu et al. obtained the three-soliton solutions for high-order nonlinear Schrodinger equation by Hirotas bilinear method [11].
Specially, Lou [12] proposed a consistent Riccati expansion (CRE) method, which is a more generalized yet simpler method to find interaction solutions for various NLEEs [13–17]. The core concept of CRE is the construction of interaction solutions based on the usual Riccati equation method and the consistent equation or the -equation [12]. The CRE method is critical to finding more new solutions to the -equation.
In this study, the CRE method is used to construct several types of interaction solutions for the focusing real modified Korteweg-de Vries (mKdV) equation [18] shown in where and are arbitrary constants. The mKdV equation plays an important role in describing some physical phenomena, such as optical cycles [19, 20], soliton propagation in plasma [21] and lattices [22], the Schottky barrier transmission lines [23], and fluid mechanics [24].
To provide better insights into these physical phenomena, finding and analyzing exact solutions to the mKdV equation is important. Previously, many powerful methods have been proposed for constructing exact solutions to the mKdV equation. For instance, in 1972, Hirota obtained an exact solution to the mKdV equation for the case of multiple collisions of solitons with different amplitudes [25]. Subsequently, he also derived the exact envelope soliton solution to the mKdV equation [26]. In 1973, Ablowitz et al. obtained exact solutions to the mKdV equation by using the inverse scattering technique [27]. In 1988, Akhmediev et al. used the Darboux transformation scheme to obtain second-order periodic solutions to the mKdV equation [28]. In 2004, Kevrekidis et al. derived some classes of periodic solutions to the mKdV equation by using direct methods [29]. In 2015, Jiao and Lou constructed a new soliton-cnoidal periodic wave interaction solution by using the CRE method [30]. However, they did not investigate how the soliton-cnoidal interaction solutions may be used to derive soliton-soliton or soliton-periodic wave interaction solutions among other types of solutions. Moreover, new interaction solutions to the mKdV equation involving different types of nonlinear waves must be investigated in depth.
The present article is structured as follows. Section 2 introduces the CRE solvability of the mKdV equation. Section 3 describes new explicit interaction solutions such as soliton-soliton, multiple resonant soliton, soliton-cosine wave, and soliton-cnoidal wave solutions to the mKdV equation obtained using the CRE method. Furthermore, it is demonstrated that interaction solutions such as the soliton-tangent wave solution cannot be constructed for the mKdV equation. The last section presents a summary and discussion.
2. CRE Solvability of the mKdV Equation
Consider the following NLEE, shown in (2), with independent variables and a dependent variable where is a polynomial function of some arguments with the subscripts denoting partial derivatives. We assume that the solution to (2) is the following possible truncated expansion form where is determined from the leading order analysis of (2). All the expansion coefficient functions () are determined by substituting (3) into (2) and then vanishing all the coefficients for a given power of . Further, and are functions of and satisfies the following simple Riccati equation shown below: which includes the following five special solutions [31]. For ,For 0,For = 0,
Definition. If the equation for and , obtained by vanishing all the coefficients of each power in after the substitution of (3) into (2), is either consistent or not overdetermined, then the expansion in (3) is considered a CRE and the nonlinear system in (2) is said to be CRE solvable [8].
According to the CRE method defined above, one can obtain the following form based on the leading order analysis of the mKdV equation in (1) where , , and are functions of and satisfies the Riccati equation (see (4) above).
Substituting (8) and (4) into (1) and vanishing all the coefficients of different powers of , one obtains
Based on the definition above, (11) is the consistent equation of the mKdV equation (or the mKdV -equation). If is a solution to the MDWW -equation in (11), the mKdV equation in (1) is CRE solvable. In this study, we set and . Thus, the solutions to the mKdV equation are expressed as follows.
3. Interaction Solutions to the mKdV Equation
Upon the determination of solutions to (11) by using (12), the corresponding solutions to the mKdV equation in (1) can be obtained. In this section, we construct interaction solutions to the mKdV equation by using different types of trivial solutions to (11).
3.1. Soliton-Soliton Interaction Solutions to the mKdV Equation
To obtain soliton-soliton interaction solutions to the mKdV equation, we consider the following form in (13) as the trial solution to (11): where satisfies the following Riccati equation in where is an arbitrary constant. This equation has special solutions similar to those in (5a), (5b), (6a), (6b), and (7). By vanishing all the coefficients for each power of after the substitution of (13) and (14) into the mKdV -equation in (11), one can obtain
From (15), it can be seen that both and r are less than 0 when 0. Based on (5a), (5b), (6a), and (6b), Eqs. (4) and (14) have only solitary solutions (viz., (5a) and (5b)) but not tangent or cotangent solutions such as the ones described in (6a) and (6b). This shows that interaction solutions such as soliton-tangent (or soliton-cotangent) wave cannot be constructed for the mKdV equation.
Under condition (15), from (5a), (5b), (12a), and (13), the soliton-soliton interaction solutions of the mKdV equation are expressed as
3.2. Interaction Solution between and Trigonometric Periodic Wave for the mKdV Equation
To investigate the interaction between a soliton and a periodic wave in the interaction solution to the mKdV equation, we consider a solution of the following form (11): where is determined constant. Substituting (17) into (11) and then vanishing all coefficients of powers , we can obtain
Then, based on (5a), (17), and (12a), one can obtain a soliton-trigonometric periodic wave interaction solution as where .
3.3. Interaction Solutions between Soliton and Cnoidal Wave for the mKdV Equation
In [26], Jiao and Lou constructed a solution of the following form for (11) where is the usual Jacobi elliptic sine function and is the third type of incomplete elliptic integral. Jiao and Lou used the following parameters in (21) to obtain a special soliton-cnoidal wave interaction solution to the mKdV equation: As seen from (21), Jiao and Lou chose the modulus () of Jacobi elliptic function to be 1.5, which is outside the allowed range [13].
In this study, we will further investigate how soliton-cnoidal interaction solutions can be used to derive soliton-soliton and soliton-periodic wave interaction solutions among other types of solutions. To this end, we performed all the substitutions and evaluations by using the Mathematica software.
Consider a trial solution of the following form for solving (11) where satisfies the following elliptic equation: Substituting (22) and (24) into (11), one obtains From the analysis of (24), we assume the solution of (24) in the following form Substituting (26) into (24) and setting the coefficient of , cn, equal to zero, one obtains Based on (25) and (27), one can find a group solution where , and . Under the substitution of (26), (22), and (5a) into (12a), under the conditions imposed by (28), one obtains where
To investigate how the soliton-cnoidal interaction solutions could be used to derive soliton-soliton interaction or other types of solutions, we illustrate the following two cases corresponding to the soliton-cnoidal wave interaction solution described in (29) by selecting different sets of parameters. For the first case, the parameters are chosen as While Figure 1 shows two-dimensional views for interaction solution at and . Figure 2 displays three-dimensional plots for the evolution of soliton-cnoidal wave interaction solution with different values for the modulus in the Jacobian elliptic function, viz., , 0.5, and 0.99999. While exhibited a particular periodic-kink soliton wave interaction, the extreme values of (a value close to the lower modulus limit or 0) showed a normal kink-shaped soliton and (a value close to the upper modulus limit or 1) displayed an interaction between a periodic wave and another periodic wave.
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For the second case, the parameters were altered as shown in (32) by changing the angular frequency
Similar to the first case, we illustrate the structures of the soliton-cnoidal wave interaction solution for different values of , 0.5, and 0.99999. Clearly, as shown in Figures 3 and 4, wavenumbers and the amplitudes in the range of x(-40,40) and t(-40,40) are less than that of the first case (cf. Figures 1 and 2 ). While there is still a normal kink soliton in the x-u plot for n = 0.00001, an incomplete kink soliton is observed in the t-u plot in contrast to the first case shown in Figure 1(b). Building on the above two cases, soliton and soliton-soliton wave interaction solutions are derived from the soliton-cnoidal wave interaction solution by making the limit of the modulus approach either 0 or 1.
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4. Summary and Discussion
In this study, we investigated the focusing mKdV equation by using the CRE method. This nonlinear equation was shown to be CRE solvable and interaction solutions; namely, soliton-soliton, soliton-trigonometric periodic waves, and soliton-cnoidal periodic wave for the mKdV equation were explicitly provided by choosing different trial solutions for the mKdV w-equation shown in (11). In addition, analytical solutions for interactions between soliton and cnoidal wave were provided and their properties were discussed graphically. According to the presented analysis, soliton and soliton-soliton wave interaction solutions can be derived from the soliton-cnoidal wave interaction solution by making the limit of the modulus approach either 0 or 1 (i.e., lower or upper bounds for the modulus in the Jacobi elliptical function).
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant No. 11864007 and No. 11564006), the Doctor Foundation of Guizhou Normal University (2015), and the Science and Technology Planning Project of Guizhou Province (Grant No. 20185769).