Cnoidal wave, snoidal wave, and soliton solutions of the D(m,n) equation

This paper studies the D(m, n) equation, which is the generalized version of the Drinfeld–Sokolov equation. The traveling wave hypothesis and exp-function method are applied to integrate this equation. The mapping method and the Weierstrass elliptic function method also display an additional set of solutions. The kink, soliton, shock waves, singular soliton solution, cnoidal and snoidal wave solutions are all obtained by these varieties of integration tools. Mathematics Subject Classification 37K10 · 35Q51 · 35Q55

One of the NLEEs that is very commonly studied is the Korteweg-de Vries (KdV) equation, which is studied in the context of shallow water waves on lakes and canals. This equation was generalized to the K (m, n) equation about a decade ago. The soliton solutions as well as the compacton solutions, which are solitons with compact support, were obtained. The alternate model to describe the dynamics of shallow water waves is the Boussinesq equation, which was also generalized to the B(m, n) equation. The soliton solutions and other aspects of this equation were subsequently studied. The third NLEE that is studied is the Drinfeld-Sokolov (DS) equation which appears often in the context of untwisted affine Lie algebra. This paper will shine light on the generalized version of the DS equation that is known as the D(m, n) equation. The integrability aspect of this equation will be the main focus of this work.
The most important part of the history of the D(m, n) equation is when it was solved earlier by the ansatz method where a soliton solution was obtained [3]. This equation was then studied with generalized evolution in the same paper during 2011 [3]. During the same year, the semi-inverse variational principle was applied to obtain the solitary wave solution analytically to the DS equation [13]. In addition, the bifurcation analysis of the traveling wave solutions to the D(m, n) equation was addressed in 2010 [12]. Besides these, as far as it is known, there were no further studies that were carried out with the D(m, n) equation.
This paper reports the research results of the D(m, n) equation that are obtained by the aid of various mathematical techniques. The traveling wave hypothesis will first be applied to obtain the solitary wave solution to the D(m, n) equation for arbitrary m and n. There are a few constraint conditions that will fall out during the course of derivation of the soliton solution. Subsequently, the exp-function method will be applied in order to extract a few additional solutions. Finally, the mapping method and the Weierstrass elliptic function method will reveal several other solutions that are in terms of Weierstrass elliptic function, cnoidal waves, and snoidal waves. The limiting cases of these solutions, namely the topological solitons, also known as kinks or shock waves, and singular solitons will be given.

Traveling wave hypothesis
The dimensionless form of the D(m, n) equation is given by where a, b, c, k, m, and n are all real valued constants and the dependent variables are q(x, t) and r (x, t).
Taking n to be 1, the D(m, 1) equation is Using the traveling wave assumption that (1) and (2) reduce to the coupled system of ordinary differential equations as Integrating (4) and taking the constant of integration to be zero, since the search is for soliton solutions, leads to Inserting (6) into (5) yields Integrating (7) and again taking the constant of integration to be zero, since the search in this section is for soliton solutions, gives Multiplying (8) by h and a third time taking the constant of integration to be zero yields After separating variables and integrating, (9) leads to Solving (10) for h(s) and using (3) gives the exact traveling wave solution to (1) and (2) as and respectively. These soliton solutions (11) and (12) can be re-written as and where the amplitudes (A 1 and A 2 ) of the solitary waves in (13) and (14) are given by and while the inverse width (B) of the soliton is given by Thus Relation (17) implies that solitary waves will exist provided

Exponential function method
The exp-function method was first proposed by He and Wu and systematically studied for solving a class of nonlinear partial differential equations. We consider the general nonlinear partial differential equation of the type Using the transformation where k and w in (19) are constants, we can rewrite Eq. (18) as the following nonlinear ODE: According to the exp-function method, as developed by He and Wu, we assume that the wave solutions can be expressed in the following form: where p, q, d and c are positive integers which are known to be further determined, and a n and b m are unknown constants. We can rewrite Eq. (20) in the following equivalent form: This equivalent formulation plays an important and fundamental part in finding the analytic solution of problems. To determine the values of c and p, we balance the linear term of highest order of Eq. (21) with the highest order nonlinear term. Similarly, to determine the values of d and q, we balance the linear term of lowest order of Eq. (20) with the lowest order nonlinear term.

Application to D(m, n) equation
The dimensionless form of the D(m, n) equation is given by , c, k, m, and n are all real constants and the dependent variables are q(x, t) and r (x, t). In particular, m and n are natural numbers. Taking n to be 1, the D(m, 1) equation is Using the traveling wave assumption and (23) reduce to the coupled system of ordinary differential equations as Integrating (24) and taking the constant of integration to be zero, since the search is for soliton solutions, leads to Integrating (27) and again taking the constant of integration to be zero, since the search in this section is for soliton solutions, gives Multiplying (28) by h and a third time taking the constant of integration to be zero yields Introducing the transformation According to the Exp-function method, we assume that the solution of Eq. (30) can be expressed in the form By balancing the highest order of linear term (W ) 2 with the highest order nonlinear term W 4 we get we have 4 p = 2c + 2 p, which leads to the result p = c. Similarly, balancing the lowest order of linear term (W ) 2 with the highest order nonlinear term W 4 we obtain and Therefore, we can obtain the following relation −4d = −2d − 2q, resulting in d = q. We can freely choose the values of c and d, but the final solution does not strongly depend upon the choice of values of c and d. For simplicity, we set p = c = 1 and d = q = 1, then Eq. (31) becomes Substituting Eqs. (32), (33), and (34) into Eq. (30), and equating to zero the coefficients of all powers of exp(nξ) yields a set of algebraic equations for a 0 , b 0 , a 1 , a −1 , b −1 and v. Solving the system of algebraic equations by the help of Maple, we obtain Case 1: Substituting Eq. (35) into Eq. (34) yields Thus, the exact solution to (22) and (23) is where a 0 and b −1 are arbitrary constants.

Case 2:
Substituting Eq. (36) into Eq. (34) yields where s = x − vt. Thus, the exact solution to (22) and (23) is where z is the root of the following equation.
and a 0 and a −1 are arbitrary constants.

Case 4:
Substituting Eq. (38) into Eq. (34) yields Thus, the exact solution to (22) and (23) is where a −1 and b 0 are arbitrary constants. The results of this section are all in terms of exponential functions and are all reducible to soliton solutions or pure exponential functions by a proper choice of the arbitrary constants that appear.

Mapping methods and Weierstrass elliptic function method
Now, we solve the coupled Eqs. (1) and (2) where and prime indicates differentiation with respect to s.

Mapping method
Here, we assume that Eq. (39) has a solution in the form where Equation (40) is the mapping relation between the solution to Eq. (41) and that of Eq. (39). We substitute Eq. (40) into Eq. (39), use Eq. (41) and equate the coefficients of like powers of f to zero so that we arrive at the set of equations From Eqs. (42) and (43), we obtain . Equation (44) gives rise to the constraint relation has two solutions f (s) = sn 2 (s) and f (s) = cd 2 (s). Therefore, we obtain the PWSs of Eqs. (1) and (2) as and As m → 1, Eq. (46) gives rise to the SWS has the solution f (s) = cn 2 (s). Therefore, the PWS of Eqs. (1) and (2) will be As m → 1, Eq. (48) will lead us to the same SWS (47).
Here, the PWS of Eqs. (1) and (2) will be As m → 1, Eq. (54) will give rise to the same singular solution (51). It is evident from the constraint relation (45) that Q 2 − 4P R should always be positive with our choices of P, Q and R for real solutions to exist. In all the cases considered, we can see that Q 2 − 4P R is equal to 16m 4 − 16m 2 + 16 which is always positive for 0 < m < 1. Thus, all our solutions are valid with the constraint relation (45).
The solutions derived in this section are all in terms of doubly periodic functions and are typically referred to as cnoidal waves, snoidal waves or dnoidal waves.

Modified mapping method
In this case, we assume that Eq. (39) has a solution in the form where f satisfies Eq. (41). Equation (55) is the mapping relation between the solution to Eq. (41) and that of Eq. (39). We substitute Eq. (55) into Eq. (39), use Eq. (41) and equate the coefficients of like powers of f to zero so that we will obtain a set of equations giving rise to the solutions and the constraint relation  1) and (2) are and When m → 1, Eq. (57) will degenerate to the solution Equation (58)  In this case, the PWS of Eqs. (1) and (2) is When m → 1, Eq. (59) will give rise to the SWS (47).  (2) is When m → 1, Eq. (60) will lead to the singular solution (51).
The expression Q 2 + 16P R reduces to 16m 4 + 224m 2 + 16 in case 1 and to 16m 4 − 256m 2 + 256 in cases 2 and 3. Both of them are positive for 0 ≤ m ≤ 1. Thus all our solutions are valid with the constraint relation (56).

Weierstrass elliptic function method
Here, we will present the solutions in terms of WEFs. For this purpose, we multiply Eq. (39) by r and rewrite it as We assume that is a solution to Eq. (61), where λ and μ are constants to be determined. Here, ℘ (s) is the WEF which satisfies the equation where g 2 and g 3 are the invariants of the WEF which satisfies the inequality Substituting Eq. (62) into Eq. (61), and equating the coefficients of powers of ℘ (s) to zero, we obtain It may be observed that g 2 has to be positive to satisfy the inequality (63) and it is indeed so from Eq. (68). Also, Eq. (67) is automatically satisfied by the values of λ, μ and g 2 .
Thus the solution of Eq. (61) is, bk .
Now we can write the corresponding periodic wave solutions in terms of JEFs in three different forms as follows: Here, e 1 , e 2 , e 3 satisfy the cubic equation with e 1 > e 2 > e 3 and m, the modulus of the JEF is given by In the infinite period limit, as m → 1, e 1 → e 2 and sn(s) → tanh(s), cn(s) → sech(s), dn(s) → sech(s). From the cubic equation (72), we have e 1 + e 2 + e 3 = 0, which leads to e 3 = −2e 1 .
Hence we can deduce the SWS from Eq. (69) as bk , and the singular soliton solution from Eqs. (70) and (71) as bk .

Conclusions
This paper addressed the D(m, n) equation by the aid of a few integration tools. These techniques of integration displayed a plethora of solutions to the equation. The traveling wave hypothesis obtained the soliton solutions along with a few constraints that must hold in order for the solitons to exist. The exp-function approach also yielded a list of several other forms of solutions to the D(m, n) equation. These solutions are several mixtures and rational combinations of the hyperbolic functions, although they are written in terms of exponential functions. The special values of the coefficients of these exponential functions will give the hyperbolic functions. Subsequently, the mapping method lead to the cnoidal and snoidal wave solutions to the equation of study. These solutions in the limiting cases were periodic solutions, singular periodic solutions, kinks or shock waves, singular solitons or solitary waves. This paper therefore listed a profound and stupendous list of results that are being reported for the first time in the context of this equation. In the future, there are several other aspects of this equation that will be touched upon. These include the stochasticity aspects of the equation. For example, the D(m, n) equation will be studied when one or more of the coefficients are random, rather than being deterministic. Soliton perturbation theory will also be addressed where perturbation terms will be taken into account. The conservation laws will also be determined to this equation. Additionally, the numerical integration technique will be applied in order to address this equation. These form only the tip of the iceberg.
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