Searching for higher-dimensional physical models is important and challenging in all branches of physics. In order to precisely describe the real (3+1)-dimensional physical space, many higher dimensional systems have been explored by various physicists and mathematicians. However, due to the complexity and difficulty, higher models are traditionally reduced to lower-dimensional ones to simplify the calculations and illustrate the general rules. For example, using the classical and nonclassical Lie approaches [1] to reduce a nonlinear partial differential equation (PDE) to some ordinary differential equation is one of the most effective ways of solving a nonlinear PDE.
Since the simplification of higher dimensional models to lower dimensional ones is popular and successful, it is interesting to consider the problem if it were possible to search for the higher dimensional systems from lower ones. This process is naturally more difficult. Fortunately, it is possible that some higher dimensional systems can be constructed from the lowers ones. For instance, some nontrivial (3+1)-dimensional integrable models can be constructed from the (2+1)-dimensional Kadomtsev-Petviashvili (KP), nonlinear Schrödinger (NLS), and Schwarz Korteweg–de Vries (KdV) equations [2] via Painlevé analysis. The famous (1+1)- and (2+1)-dimensional sine-Gordon equations and Tzitzeca equations can be obtained from the (0+1)-dimensional Riccati equation [3].
Recently, in order to obtain more high-dimensional integrable systems in some traditional integrable meanings, a deformation algorithm [4] is proposed, which enables arbitrary lower-dimensional integrable systems to be deformed into high-dimensional ones.
For a general (1+1)-dimensional integrable local evolution system
$\begin{eqnarray}\begin{array}{l}{u}_{t}=F(u,{u}_{x},\ldots ,{u}_{{xn}}),\qquad {u}_{{xn}}={\partial }_{x}^{n}u,\\ u=({u}_{1},{u}_{2},\ldots ,{u}_{m}),\end{array}\end{eqnarray}$
if there exist some conservation laws $\begin{eqnarray}\begin{array}{l}{\rho }_{{it}}={J}_{{ix}},i=1,2,\ldots ,D-1,\qquad {\rho }_{i}={\rho }_{i}(u),\\ {J}_{i}={J}_{i}(u,{u}_{x},\ldots ,{u}_{{xN}}),\end{array}\end{eqnarray}$
where the conserved densities ${\rho }_{i}$ are dependent only on the field u while the flows Ji can be field derivative dependent, then the deformed $D+1$-dimensional system $\begin{eqnarray}\hat{T}u=F(u,\hat{L}u,\ldots ,{\hat{L}}^{n}u)\ \end{eqnarray}$
may be integrable with the deformation operators $\begin{eqnarray}\hat{L}\equiv {\partial }_{x}+\displaystyle \sum _{i=1}^{D-1}{\rho }_{i}{\partial }_{{x}_{i}},\qquad \hat{T}\equiv {\partial }_{t}+\displaystyle \sum _{i=1}^{D-1}{\bar{J}}_{i}{\partial }_{{x}_{i}}\end{eqnarray}$
and the deformed flows $\begin{eqnarray}\bar{{J}_{i}}={\left.{J}_{i}\right|}_{{u}_{{xj}}\to {\hat{L}}^{j}u,j=1,2,\ldots ,N}.\end{eqnarray}$
The correctness of the deformation algorithm (or the deformation conjecture) has been checked by the authors of [4] for almost all the known local integrable evolution systems. It is found that the new high dimensional integrable systems obtained from the deformation algorithm possess completely different structures and properties from the traditional integrable systems. Although the new models possess elegant properties, such as the existence of the Lax pair and the infinitely many symmetries, the traditional research methods of integrable systems can no longer be successfully applied because the original models and their several reciprocal systems are included in the same models.
The (1+1)-dimensional Boussinesq equation, considered as the first model for nonlinear dispersive wave propagation, was obtained by Boussinesq in his study of long water waves [5]. Later, the Boussinesq equation was independently considered by Zakharov [6] in studying one-dimensional chains of nonlinear oscillators, Kruskal and Zabusky in their study of one Fermi-Pasta-Ulam problem [7], and Ablowitz and Haberman in higher order matrix eigenvalue problem [8]. The well-known KdV equation can be derived from the Boussinesq equation with the wave motion being restricted to be unidirectional. The Boussinesq equation is proved to be Lax integrable and solvable via the inverse scattering method, possessing the bilinear form and N-soliton solutions [9]. Recently, the Boussinesq equation is extended to the different types of nonlocal Boussinesq systems by [10] via the parity or time-reversal symmetry reduction, [11, 12]. It is interesting that the nonlocal Boussinesq systems exhibit different properties for the prohibitions [10], symmetry-breaking solutions and bright and dark soliton solutions [11] caused by nonlocalities. Furthermore, the Boussinesq equation in fractional type is also derived in [13] and exact solutions are found to describe fractal domains.
Though many higher dimensional Boussinesq equations have been constructed by various authors, such as the (2+1)-dimensional Boussinesq equations describing the propagation of gravity waves on the surface of water [14–16] and the (3+1)-dimensional Boussinesq equations in [17, 18], the obtained higher dimensional Boussinesq equation is generally related to the known KP equation up to the scaling and exchange of y and t [19]. Some of the (2+1)-dimensional Boussinesq equations are not in the classes of completely integrable equations [14, 18, 20].
Because the Boussinesq equation is widely used in fluid dynamics and possesses properties different from other integrable systems in nonlinear science, it is necessary to extend the lower dimensional Boussinesq equation to the higher dimensional models different from the known integrable systems. In this manuscript, we devote our search to finding various higher dimensional Boussinesq equations using the deformation algorithm.
We start from the (1+1)-dimensional Boussinesq equation written in NLS type [19], say6 ) is Lax integrable with the Lax pair7 ) leads to the Boussinesq equation (6 ).
$\begin{eqnarray}{\alpha }_{t}={\alpha }_{{xx}}+{\left(\alpha +\beta \right)}^{2},\qquad -{\beta }_{t}={\beta }_{{xx}}+{\left(\alpha +\beta \right)}^{2}.\end{eqnarray}$
It is easy to check the (1+1)-dimensional Boussinesq equation ( $\begin{eqnarray}\begin{array}{l}{\psi }_{{xxx}}+(\alpha +\beta ){\psi }_{x}-\displaystyle \frac{3+{\rm{i}}\sqrt{3}}{12}\\ \times \left(\sqrt{3}{\rm{i}}{\beta }_{x}-2{\alpha }_{x}-{\beta }_{x}+\lambda \right)\psi =0,\\ \sqrt{3}{\rm{i}}{\psi }_{t}-3{\psi }_{{xx}}-2\psi (\alpha +\beta )=0,\end{array}\end{eqnarray}$
where ${\rm{i}}\equiv \sqrt{-1}$ and λ is the spectral parameter. The compatibility condition for ψxxxt = ψtxxx of equation (From equation (6 ), it is not difficult to find the Boussinesq equation possesses a type of conservation law6 ) with the deformation operators
$\begin{eqnarray}{(\alpha +\beta )}_{t}={({\alpha }_{x}-{\beta }_{x})}_{x},\end{eqnarray}$
with the conserved density ρ = α + β and conserved flow J = αx − βx. Thus, by applying the deformation algorithm to the Boussinesq equation ( $\begin{eqnarray}\hat{L}={\partial }_{x}+(\alpha +\beta ){\partial }_{y},\qquad \hat{T}={\partial }_{t}+{\bar{J}}_{1}{\partial }_{y},\end{eqnarray}$
where the deformed flow is defined as $\begin{eqnarray}{\bar{J}}_{1}=(\hat{L}\alpha -\hat{L}\beta ),\end{eqnarray}$
a particular (2+1)-dimensional Boussinesq equation is successfully derived $\begin{eqnarray}\hat{T}\alpha ={\hat{L}}^{2}\alpha +{\left(\alpha +\beta \right)}^{2},\qquad \hat{T}\beta =-{\hat{L}}^{2}v-{\left(\alpha +\beta \right)}^{2},\end{eqnarray}$
i.e., $\begin{eqnarray}\begin{array}{rcl}{\alpha }_{t} & = & {\alpha }_{{xx}}+{\left(\alpha +\beta \right)}^{2}{\alpha }_{{yy}}+2(\alpha +\beta ){\alpha }_{{xy}}\\ & & +2{\alpha }_{y}{\beta }_{x}+2{\alpha }_{y}{\beta }_{y}(\alpha +\beta )+{\left(\alpha +\beta \right)}^{2},\\ {\beta }_{t} & = & -{\beta }_{{xx}}-{\left(\alpha +\beta \right)}^{2}{\beta }_{{yy}}-2(\alpha +\beta ){\beta }_{{xy}}\\ & & -2{\alpha }_{x}{\beta }_{y}-2{\alpha }_{y}{\beta }_{y}(\alpha +\beta )-{\left(\alpha +\beta \right)}^{2}.\end{array}\end{eqnarray}$
The integrability of the (2+1)-dimensional Boussinesq equation (12 ) is guaranteed by the Lax pair, say,13 ) for12 ).
$\begin{eqnarray}\begin{array}{l}{\hat{L}}^{3}\psi +(\alpha +\beta )\hat{L}\psi -\displaystyle \frac{3+{\rm{i}}\sqrt{3}}{12}\\ \times (\sqrt{3}{\rm{i}}\hat{L}\beta -2\hat{L}\alpha -\hat{L}\beta +\lambda )\psi =0,\\ \sqrt{3}{\rm{i}}\hat{T}\psi -3{\hat{L}}^{2}\psi -2\psi (\alpha +\beta )=0,\end{array}\end{eqnarray}$
which can be directly obtained from the Lax pair of the (1+1)-dimensional Boussinesq equation by applying the deformation algorithm. The compatibility condition of equation ( $\begin{eqnarray}{\psi }_{{xxxt}}={\psi }_{{txxx}}\end{eqnarray}$
yields the (2+1)-dimensional Boussinesq equation (To investigate the properties of the obtained integrable system equation (12 ), we seek the travelling wave solution structures for this model. One can easily check the (2+1)-dimensional Boussinesq equation (12 ) has the following travelling wave solution
$\begin{eqnarray}\begin{array}{l}\alpha =A(X)\equiv A,\qquad \beta =U(X)-A(X)\equiv U-A,\\ X\equiv {kx}+{py}+\omega t,\end{array}\end{eqnarray}$
$\begin{eqnarray}{A}_{X}=\displaystyle \frac{1}{2}{U}_{X}-\displaystyle \frac{1}{2}\displaystyle \frac{\omega U}{k({pU}+k)},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{k}^{2}{\left({pU}+k\right)}^{2}{U}_{{XX}}+{{pk}}^{2}({pU}+k){C}_{X}^{2}\\ +U(2{k}^{2}U-{\omega }^{2})=0,\end{array}\end{eqnarray}$
where the constants k, p and ω are arbitrary constants.It is known the solution of equation (17 ) can be expressed by the following elliptic integral18 ) is implicitly obtained by
$\begin{eqnarray}\begin{array}{l}{\displaystyle \int }^{U}\displaystyle \frac{({pf}+k){\rm{d}}f}{\sqrt{-12{\left(f-{a}_{0}\right)}^{3}+9{a}_{1}f+{a}_{2}}}=\pm (X-{X}_{0}),\\ {a}_{0}=\displaystyle \frac{{\omega }^{2}}{4{k}^{2}},\quad {a}_{1}=\displaystyle \frac{9{\omega }^{4}}{4{k}^{4}},\quad {a}_{2}=-\displaystyle \frac{3{\omega }^{6}}{16{k}^{6}}+{c}_{2},\end{array}\end{eqnarray}$
where c2 and X0 are two arbitrary integral constants. With suitable parameters selections, the travelling wave solution for equation ( $\begin{eqnarray}U=\displaystyle \frac{3{\omega }^{2}}{4{k}^{2}}{{\rm{sech}} }^{2}\left(\displaystyle \frac{p\omega }{4{k}^{3}}\sqrt{9{\omega }^{2}-12{k}^{2}U}+\displaystyle \frac{\omega X}{2{k}^{2}}\right).\end{eqnarray}$
Thus, the exact travelling wave solution for the new (2+1)-dimensional Boussinesq equation (12 ) is16 ) and U is implicitly given by equation (19 ).
$\begin{eqnarray}\alpha =A,\qquad \beta =U-A,\end{eqnarray}$
where A and U are constrained by equation (Though the travelling wave solution equation (20 ) can not be expressed analytically and explicitly, we can really find the solution is different from the traditional integrable systems before. The special solution U is displayed in figure 1 with the same parameters selections for k and ω and various values for p. It is discovered as p increases, the soliton shape will deform to an asymmetric soliton and a foldon or folded solitary wave (multi-valued soliton or solitary wave [21]).
Figure 1. The implicit solution U given by equation ( |
In conclusion, we establish a (2+1)-dimensional Boussinesq equation from the (1+1)-dimensional Boussinesq equation in NLS type via a deformation algorithm. This type of Boussinesq equation is discussed here because it has the special conservation law with the conserved density ρ dependent on α and β and the flow J dependent on αx and βx. By applying the algorithm to the Lax pair of the (1+1)-dimensional Boussinesq equation, the Lax pair of the newly obtained (2+1)-dimensional Boussinesq equation is constructed. The travelling wave solutions of the (2+1)-dimensional Boussinesq equation are expressed by a complicated elliptic integral equation (18 ) and a constraint ordinary differential equation (16 ). Because of the effects of the deformation, it is very difficult to find analytical explicit solutions of the obtained (2+1)-dimensional Boussinesq equation.
Though the deformation algorithm has been applied to many known integrable systems and has been successfully used to find higher-dimensional ones, it is still difficult to search for higher-dimensional discrete integrable systems. New methods and progresses should be explored for the discrete integrable systems.