Traveling Wave Solutions and Infinite-Dimensional Linear Spaces of Multiwave Solutions to Jimbo-Miwa Equation

and Applied Analysis 3 bounded solutions if and only if h + ≤ h < h − . To be exact, (10) defines a family of periodic solutions when h + < h < h − . When h = h + , (10) defines a bounded solution which approaches V + as ξ goes to infinity. Actually,


Introduction
Various nonlinear partial differential equations (NLPDEs) have been proposed to model different kinds of phenomena in natural and applied sciences such as fluid dynamics, plasma physics, solid-state physics, optical fibers, acoustics, mechanics, biology, and mathematical finance. Obviously, it is of significant importance to study the solutions of such NLPDEs from both theoretical and practical points of view. However, the solution spaces of nonlinear equations are infinite-dimensional and contain diverse solution structures, so it is usually a difficult job to determine the solutions to nonlinear NLPDEs.
A great idea to generate exact solutions of NLPDEs is to reduce the NLPDEs into some algebraic equations by assuming the solutions to have some special forms or satisfy some solvable simpler equations. This can be seen in, for example, the exp-function method [1], the tanh function method [2], the homogeneous balance method [3,4], the auxiliary function method [5,6], the sech-function method [7], the sine-cosine method [8,9], the tanh-coth method [10], the Jacobi elliptic function method [11], the -expansion method, and the extended -expansion method [12]. Normally, it is not an easy task to solve these nonlinear algebraic equations reduced from NLPDEs because they involve many parameters. However, the rapid development of symbolic computation makes it relatively easy to solve these algebraic equations [13]. At the same time one should be very careful when applying these methods as different methods might give the same solutions. For example, solutions obtained by using the sech-function method, the tanh-coth method, and the exp-function method are actually the same [14,15], because (sech 2 ) = (1 − tanh 2 ) = (4/( + − ) 2 ), for any function .
Recently, the planar dynamical system theorem has been employed to study the traveling wave solutions of NLPDEs [16,17]. The best advantage of this approach is that the boundedness, periodicity, the shapes of the solutions, and even the singular traveling wave solutions of NLPDEs can be recognized easily from the corresponding orbits of their phase portraits under various different parametric conditions. Also the exact solutions can be derived at the same time. It is worth pointing out that Hirota's bilinear method [18] is also an amazing method to find exact solutions to some NLPDEs. By some independent variable transformation, various nonlinear equations of mathematical physics are transformed into Hirota's bilinear equations [18,19], which possess some specific properties and thus might be applied to study the solution sets of nonlinear differential equations. Recently, even some programs have been designed and some algorithms have been proposed on searching for integrable bilinear equations [20][21][22][23][24][25][26]. Based on Hirota's bilinear form, soliton solutions were obtained by the Hirota perturbation technique [18], the multiple exp-function algorithm [27], and other methods [28][29][30][31]. Even some generalized bilinear form has been proposed recently by Ma [32].
Recently, this equation has attracted a great deal of attention which mainly focuses on its solutions, integrability properties and symmetries. Ma and Lee [34] proposed a direct approach to solve (1) by using rational function transformations. Li and Dai [35] applied the generalized Riccati equation method to look for its exact solutions. However, in [36], Kudryashov and Sinelshchikov pointed out some mistakes and proved that some solutions in [35] could be written in a uniform form and thus they are not new at all. The (3 + 1)-dimensional Jimbo-Miwa equation can be transformed into the Hirota bilinear equation through the dependent variable transformation = 2(ln ) .
In this paper, we firstly study the bounded traveling wave solutions of the Jimbo-Miwa equation (1) by investigating the bifurcation and phase portraits of a planar cubic polynomial ordinary differential equation by using the planar dynamical system theory [16,37]. We then explore the 1-wave and 2wave solutions and the infinite-dimensional linear spaces of multiwave solutions to Jimbo-Miwa equation by employing Hirota's bilinear method, thus confirming that Jimbo-Miwa equation possesses multiwave solutions of arbitrary order.

Traveling Wave Solutions to the (3 + 1)-Dimensional Jimbo-Miwa Equation
To investigate the traveling wave solutions to the (3 + 1)-dimensional Jimbo-Miwa equation, we make the traveling wave transformation ( , , , ) = ( ), = + + + , under which (1) is reduced to the nonlinear ordinary differential equation Integrating (1) once with respect to gives 3 3 where is an arbitrary constant. Let V = / ; then (1) becomes which is a second-order nonlinear ordinary differential equation. We will study the solutions of (5) by planar dynamical system method and thus derive the traveling wave solutions to the (3 + 1)-dimensional Jimbo-Miwa equation. (5). First, we rewrite (5) in a simpler and more general form, namely,

Bounded Solutions of
. We now study the bifurcation and exact solutions of (6). Let V = ; then (6) is equivalent to the dynamical system which has the Hamiltonian Clearly, the phase orbits defined by the vector fields of system (7) determine all solutions of (6). The bounded solutions of (6) correspond to the bounded phase orbits of system (7), which we now investigate. Along the orbit corresponding to (V, ) = ℎ, Consequently, the general formula of the solutions of (6) can be expressed as However, it is not easy to know the properties and the shapes of (10) which actually is determined by the parameters , , , and ℎ. Clearly, the abscissas of equilibrium points of system (7) are the zeros of V 2 + V + = 0. Obviously, the system has no bounded orbits when 2 − 4 < 0. We suppose that 2 − 4 > 0 in order to study the bounded orbits of system (7). Denoting V ± = (− ± √ 2 − 4 )/2 , then + (V + , 0) and − (V − , 0) are two equilibrium points of system (7). By the theory of planar dynamical system, we know that + is a saddle point and − is a center. Denote ℎ ± = (V ± , 0), and, by careful computation, we get Obviously, ℎ − < ℎ < ℎ + . (V, ) = ℎ + corresponds to homoclinic orbits, and (V, ) = ℎ − corresponds to the center − and (V, ) = ℎ, where ℎ + < ℎ < ℎ − corresponds to a family of closed orbits surrounding the center − , which are surrounded by a homoclinic orbit. That is to say, (10) defines Abstract and Applied Analysis 3 bounded solutions if and only if ℎ + ≤ ℎ < ℎ − . To be exact, (10) defines a family of periodic solutions when ℎ + < ℎ < ℎ − . When ℎ = ℎ + , (10) defines a bounded solution which approaches V + as goes to infinity. Actually, where V 0 = −( + 2 √ 2 − 4 )/2 , so (10) can be reduced to from which we can get the exact solution as By further simplification, (14) becomes which is an exact bounded solution of (5). Thus, we have the following lemma.

Lemma 1. The general second-order ODE (6) has bounded solutions if and only if
The bounded solutions can be expressed as (10) in an implicit form. In fact, provided ℎ − < ℎ < ℎ + , (10) defines a family of bounded periodic solutions and ℎ = ℎ + defines a bounded solution which approaches V + as goes to infinity and can be expressed explicitly as (15), where V + = (− + √ 2 − 4 )/2 and ℎ ± is defined by (11).

Bounded Traveling Wave Solutions to the (3 + 1)-Dimensional Jimbo-Miwa Equation.
According to the analysis and results in Section 2.1, we know that (5) has only two kinds of bounded solutions, among which one is a family of periodic solutions and another is a family of solutions approaching a fixed number as goes to infinity. Note that what we aim to study is the bounded traveling wave solutions to the (3 + 1)dimensional Jimbo-Miwa equation which are determined by V = / and V satisfies (5). So we have to investigate how we can get the bounded solutions to (3) from the bounded solution of (5).
Clearly, ( ) = ∫ 0 V( ) , and V( ) can be expressed implicitly as (10). By the geometry meaning of the integral and the properties of the solutions of (5), we get the traveling wave solutions to the (3+1)-dimensional Jimbo-Miwa equation. To get the bounded solution, we choose the integral constant to be 0; that is, = 0 in (15), and so that is, which is a family of exact bounded kink traveling wave solutions to the (3 + 1)-dimensional Jimbo-Miwa equation, where is an arbitrary constant. However, we may not get bounded solutions from the family of periodic solutions of (5). It is easy to see that if V( ) is a periodic solution of (5) By the theory of planar dynamical system, when ℎ − < ℎ < ℎ + , where Clearly, V 0 > 0 when > 0 and 0 > > −4√ /3. So V( ) > 0 and thus ( , , , ℎ) > 0 when > 0 and 0 > > −4√ /3. However, V( ) < V + < 0 and thus ( , , , ℎ) < 0 when > 0, > 0, and > 0. There must exist at least one zero of ( , , , ℎ) = 0 in the region > 0 of the parameter space since ( , , , ℎ) is a continuous function of , , , and ℎ. The same happens in the region < 0 of the parameter space. So, we know that there exist at least two families of bounded periodic traveling wave solutions to the (3 + 1)-dimensional Jimbo-Miwa equation. Thus we have the following theorem.
where 0 and are two arbitrary constants.
It follows from Theorem 3 that is an -wave solution to the Hirota bilinear equation (2) where = / 1 , 1, = 1, − 1,1 and k i = k i − k 1 . Obviously, the function above is an To get other -wave solutions to the (3 + 1)-dimensional Jimbo-Miwa equation, we study the linear independent solution sets to (20) which are required to satisfy the dispersion relations. Actually, (20) is a system possessing ( + 1)/2 coupled equations (plus the dispersion conditions). It is usually not so easy to get the solutions of (20). Fortunately, the number of the equations is 1 when = 1 and it is 3 when = 2, which might make it easy to get the solutions in these two cases. Clearly, 1-wave solution is the traveling wave solution. Let us check what kinds of traveling wave solutions we can obtain by the Hirota bilinear method first.

-Wave Solutions to the (3 + 1)-Dimensional Jimbo-Miwa Equation.
To get the -wave solutions to the (3 + 1)-dimensional Jimbo-Miwa equation, we need to investigate the independent solution set {k 1 , k 2 , . . . , k N } of (20), which satisfies the dispersion relations. Generally, it is very difficult to get the solution if is greater than 3 in which case the number of (20) is ( + 1)/2. However, it might be possible to solve these equations by assuming some special relations between these parameters [30,31].
Consequently, besides the family of -wave solution (40), the (3+1)-dimensional Jimbo-Miwa equation (1) has another group of -wave solution for any arbitrary positive integer and so we have the following theorem.