1. Introduction
Since the development of the inverse scattering transformation (IST) method [1], the study of IST integrable systems has attracted high attention from the physics and mathematics communities. Thus, various wonderful and elegant properties of integrable systems were discovered. For example, IST integrable systems typically have both infinite symmetries and conservation laws, Painlevé property, Hirota's bilinear form and $\tau$ functions, Darboux and Bäcklund transformations and nonlinear superposition principle, bi-Hamiltonian structure, recursive operator and so on [2–12]. Continuous local classical integrable systems described by partial differential equations are also successfully extended to discrete integrable systems, nonlocal integrable models, quantum integrable models, Kuper- or super-integrable models, supersymmetric integrable systems, ren-integrable models and ren-symmetric integrable systems [13–18]. Simultaneously, integrable systems and their corresponding soliton theories have been widely applied in various physical branches such as condensed matter physics, particle and nuclear physics, field theory, cosmology, fluid mechanics, optics, plasma physics and other natural science fields [19–24].
Previously, the investigation on integrable systems has mainly focused on so-called lower dimensional integrable systems such as (1+1)-dimensional and/or (2+1)-dimensional integrable models. Because the real physical space-time is (3+1)-dimensional, various physicists and mathematicians have been trying to find some nontrivial higher dimensional integrable models [25–32]. The famous known (2+2)-dimensional integrable system is the so-called self-dual Yang-Mills (SDYM) field equation. All the (1+1)- and (2+1)-dimensional integrable models may be reduced from the SDYM equation due to the Ward conjecture [25]. In [27], Fokas proposed a method to obtain higher dimensional integrable systems by changing the real space-time to complex ones. In [28], Lou obtained some types of higher dimensional integrable models from lower dimensional ones by introducing the invariant Painlevé expansion and inner space variables. In the dispersionless case, some types of higher dimensional integrable models, heavenly equations, have also been obtained [30–32]. By using the Miura type transformations related to the KdV equation, the modified KdV equation and the Schwartz KdV equation, the (0+1)-dimensional Riccati equation had been deformed to (1+1)- and (2+1)-dimensional integrable sine-Gordon equations and Tzitzeica equations [33].
Recently, Lou, Hao and Jia [34] proposed an effective deformation algorithm such that any lower dimensional integrable systems can be deformed to higher dimensional ones by means of the conservation laws of the lower dimensional ones. By using this deformation algorithm, some lower dimensional integrable systems including the Korteweg-de Vries (KdV) equation, Ablowitz-Kaup-Newell-Segur [34], the nonlinear Schrödinger equation [35], the Camassa-Holm (CH) equation [36], the Burgers systems [37] and the Kaup-Newell systems [38] have been extended to higher dimensional ones. The usual integrable systems include only linear dispersion relations (like the KdV equation) and/or only nonlinear dispersion relations (like the CH equation). The high dimensional integrable systems obtained by the deformation algorithm possess both linear and nonlinear dispersion relations [34].
For a real physical system like the oceanic system and the atmospheric system, there are some different waves described by different equations. An interesting problem is whether we can find some significant models such that different types of nonlinear waves can be described by the same model. The deformation algorithm of [34] is just the key to answering this problem. In this paper, we are interested to reveal a new application of the deformation algorithm of [25] to couple a decoupled system by using combinations of the conservation laws. Firstly, there are some different types of dispersionless waves described by different dispersionless KdV equations. In section 2 of this paper, after reviewing the deformation algorithm of [25], the deformation algorithm is applied to couple a simple decoupled integrable dispersionless KdV system. Secondly, there may be some different KdV waves for the same physical system. Thus to couple two different KdV waves is an interesting topic. In section 3 , the deformation algorithm is used to study the interactions among some different KdV waves. In addition to the dispersion waves (KdV waves), there may be diffusion effects (Burgers equation). The direct summation of dispersion and diffusion effects leads to a nonintegrable system, the KdV-Burgers equation. Section 4 is devoted to coupling the KdV equation and the Burgers equation such that the resulting equation is still integrable. In section 5 , we investigate the interaction model of the linear dispersion wave (the KdV wave) and the nonlinear dispersion wave (the Harry-Dym (HD) wave). The last section includes a short summary and some discussions.
2. Deformation algorithm and coupled integrable dispersionless KdV system
Deformation algorithm [34]. For a general (1+1)-dimensional integrable local evolution system
$\begin{eqnarray}\begin{array}{rcl}{u}_{t} & = & F(u,{u}_{x},\ldots ,{u}_{{xn}}),\\ {u}_{{xn}} & = & {\partial }_{x}^{n}u,u=({u}_{1},{u}_{2},\ldots ,{u}_{m}),\end{array}\end{eqnarray}$
if there exist some conservation laws $\begin{eqnarray}\begin{array}{rcl}{\rho }_{{it}} & = & {J}_{{ix}},i=1,2,\ldots ,D-1,\\ {\rho }_{i} & = & {\rho }_{i}(u),{J}_{i}={J}_{i}(u,{u}_{x},\ldots ,{u}_{{xN}}),\end{array}\end{eqnarray}$
where the conserved densities ρi are dependent only on the field u while the flows Ji can be field derivative dependent, then the deformed (D+1)-dimensional system $\begin{eqnarray}\hat{T}u=F(u,\hat{L}u,\ldots ,{\hat{L}}^{n}u)\end{eqnarray}$
is integrable with the deformation operators $\begin{eqnarray}\hat{L}\equiv {\partial }_{x}+\displaystyle \sum _{i=1}^{D-1}{\rho }_{i}{\partial }_{{x}_{i}},\quad \hat{T}\equiv {\partial }_{t}+\displaystyle \sum _{i=1}^{D-1}{\bar{J}}_{i}{\partial }_{{x}_{i}}\end{eqnarray}$
and the deformed flows $\begin{eqnarray*}{\bar{J}}_{i}={\left.{J}_{i}\right|}_{{u}_{{x}_{j}}\to {\hat{L}}^{j}u,j=1,2,\ldots ,N}.\end{eqnarray*}$
The algorithm has been strictly proved by Casati and Zhang of Ningbo University [39]. Using this deformation algorithm, any integrable lower dimensional integrable systems can be deformed to higher dimensional ones [33–37] which include the original models and their different reciprocal links related to different conservation laws.
In this section, we reveal a new application of the deformation algorithm such that a completely separated system can be intrinsically coupled. The simplest decoupled integrable system may possess the form5 ) is C-integrable (can be directly solved by integrations or can be directly linearized) and possesses infinitely many symmetries. A symmetry of (5 ) is defined as a solution of its linearized equation7 ) possesses infinitely many symmetries in the form
$\begin{eqnarray}{u}_{t}=6{{uu}}_{x},\end{eqnarray}$
$\begin{eqnarray}{v}_{t}=6{{vv}}_{x},\end{eqnarray}$
where the both fields u and v are solutions of the dispersionless KdV equations. The dispersionless KdV equation ( $\begin{eqnarray}{\sigma }_{t}=6\sigma {u}_{x}+6u{\sigma }_{x}.\end{eqnarray}$
By using the method of [40], one can prove that ( $\begin{eqnarray}\begin{array}{rcl}\sigma & = & S(u,{p}_{0},{p}_{1},\ldots ,{p}_{M}){u}_{x},\\ {p}_{0} & \equiv & {u}_{{xx}}{u}_{x}^{-3},\\ {p}_{m} & \equiv & {\left({u}_{x}^{-1}{\partial }_{x}\right)}^{m}{p}_{0},\\ m & = & 0,1,\ldots ,M,\end{array}\end{eqnarray}$
where S is an arbitrary function of the indicated variables and M is an arbitrary positive integer.It is known that symmetries may be closely related to conservation laws. If we restrict the symmetries of (8 ) in the form σ = S(u)ux, then one can directly find infinitely many conservation laws of the decoupled dispersionless KdV system (5 )-(6 )
$\begin{eqnarray}{\left({F}_{u}+{G}_{v}\right)}_{t}=6{\left({{uF}}_{u}+{{vG}}_{v}-F-G\right)}_{x},\end{eqnarray}$
where F ≡ F(u) and G ≡ G(v) are arbitrary functions of u and v, respectively. Thus, according to the deformation algorithm, one can introduce the deformation operators $\begin{eqnarray}\begin{array}{rcl}\hat{L} & = & {\partial }_{x}+\displaystyle \sum _{k=1}^{K-1}({F}_{{ku}}+{G}_{{kv}}){\partial }_{{x}_{k}},\\ \hat{T} & = & {\partial }_{t}+\displaystyle \sum _{k=1}^{K-1}({{uF}}_{{ku}}+{{vG}}_{{kv}}-{F}_{k}-{G}_{k}){\partial }_{{x}_{k}},\end{array}\end{eqnarray}$
where {Fk ≡ Fk(u), Gk ≡ Gk(v), k = 1, 2, …, K − 1} are arbitrary functions of the indicated variables.Applying the deformation algorithm to the decoupled dispersionless KdV system (5 )-(6 ), we have10 ). To see a more concrete form, we fix K = 2, F1 = u and G = v in (10 ), then (11 ) becomes a (2+1)-dimensional coupled dispersionless KdV system12 ) is guaranteed by the existence of infinitely many symmetries with arbitrary functions and higher order symmetries. Here are some simple examples (w ≡ u + v, X ≡ wuy + ux, Y ≡ wvy + vx)13 )-(17 ) satisfy the symmetry equation12 ). An implicit special shock wave solution of (12 ) is determined by
$\begin{eqnarray}\left\{\begin{array}{l}\hat{T}u=6u\hat{L}u,\\ \hat{T}v=6v\hat{L}v,\end{array}\right.\end{eqnarray}$
where $\hat{L}$ and $\hat{T}$ are defined in ( $\begin{eqnarray}\left\{\begin{array}{l}{u}_{t}=6{{uu}}_{x}+3({u}^{2}-{v}^{2}+2{uv}){u}_{y},\\ {v}_{t}=6{{vv}}_{x}+3({v}^{2}-{u}^{2}+2{uv}){v}_{y}.\end{array}\right.\end{eqnarray}$
The symmetry integrability of the system ( $\begin{eqnarray}{\sigma }_{1}(f)=\left(\begin{array}{c}f(v){u}_{y}\\ f(v){v}_{y}-(u+v){f}_{v}{v}_{y}-{f}_{v}{v}_{x}\end{array}\right),\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{2}(g)=\left(\begin{array}{c}g(v){u}_{y}-(u+v){g}_{u}{u}_{y}-{g}_{u}{u}_{x}\\ g(u){v}_{y}\end{array}\right),\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{3}=\left(\begin{array}{c}\displaystyle \frac{{u}_{x}{u}_{{yy}}w}{{u}_{y}X}-\displaystyle \frac{{u}_{{xx}}}{X}+\left(\displaystyle \frac{{u}_{x}}{{u}_{y}}-w\right)\displaystyle \frac{{u}_{{xy}}}{X}+\displaystyle \frac{{u}_{y}^{2}{w}^{2}+{u}_{x}^{2}}{{Xw}}+\displaystyle \frac{2{u}_{x}{u}_{y}+{u}_{x}{v}_{y}-{u}_{y}{v}_{x}}{X}+{u}_{y}\mathrm{ln}\displaystyle \frac{{{Xv}}_{y}}{{{Yu}}_{y}}\\ \displaystyle \frac{{v}_{{xx}}}{Y}-\displaystyle \frac{{v}_{x}{v}_{{yy}}w}{{v}_{y}X}+\left(w-\displaystyle \frac{{v}_{x}}{{v}_{y}}\right)\displaystyle \frac{{v}_{{xy}}}{Y}+\displaystyle \frac{{u}_{x}{v}_{y}-{v}_{x}{u}_{y}}{Y}+{v}_{y}\mathrm{ln}\displaystyle \frac{{{Xv}}_{y}}{{{Yu}}_{y}}\end{array}\right),\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{4}=\left(\begin{array}{c}\displaystyle \frac{{{wu}}_{y}{\left({u}_{x}+{{wu}}_{y}\right)}^{2}-{{uu}}_{y}{\left({w}_{x}+{{ww}}_{y}\right)}^{2}}{2{X}^{2}Y}-\displaystyle \frac{u({u}_{{xx}}+2{{wu}}_{{xy}}+{w}^{2}{u}_{{yy}})}{2{X}^{2}}\\ \displaystyle \frac{v({w}^{2}{v}_{{yy}}+2{{wv}}_{{xy}}+{v}_{{xx}})}{2{Y}^{2}}+\displaystyle \frac{{{wv}}_{y}({u}_{x}+{{wu}}_{y})({{wv}}_{y}+{{ww}}_{y}+{v}_{x}+{w}_{x})-{{uv}}_{y}{\left({w}_{x}+{{ww}}_{y}\right)}^{2}}{2{Y}^{2}X}\end{array}\right),\end{eqnarray}$
$\begin{eqnarray}{\sigma }_{5}=\left(\begin{array}{c}\displaystyle \frac{{u}_{y}[({u}_{y}^{2}-{v}_{y}^{2}-2{u}_{y}{v}_{y}){w}^{2}+2w({u}_{x}{u}_{y}-{w}_{x}{v}_{y}-{v}_{x}{u}_{y})+{u}_{x}^{2}-2{u}_{x}{v}_{x}-{v}_{x}^{2}]}{2{X}^{2}Y}-\displaystyle \frac{{u}_{{xx}}+2{{wu}}_{{xy}}+{w}^{2}{u}_{{yy}}}{2{X}^{2}}\\ \displaystyle \frac{{v}_{y}[({u}_{y}^{2}+2{u}_{y}{v}_{y}-{v}_{y}^{2}){w}^{2}+2w({u}_{x}{w}_{y}+{v}_{x}{u}_{y}-{v}_{x}{v}_{y})+{u}_{x}^{2}+2{u}_{x}{v}_{x}-{v}_{x}^{2}]}{2{{XY}}^{2}}+\displaystyle \frac{{v}_{{xx}}+2{{wv}}_{{xy}}+{w}^{2}{v}_{{yy}}}{2{Y}^{2}}\end{array}\right),\end{eqnarray}$
where f ≡ f(v) and g ≡ g(u) are arbitrary functions of v and u respectively. One can directly verify that ( $\begin{eqnarray}\left\{\begin{array}{l}{\sigma }_{t}^{u}=6{\left(u{\sigma }^{u}\right)}_{x}+6[w{\sigma }^{u}+(u-v){\sigma }^{v}]{u}_{y}+3({u}^{2}-{v}^{2}+2{uv}){\sigma }_{y}^{u},\\ {\sigma }_{t}^{v}=6{\left(v{\sigma }^{v}\right)}_{x}+6[w{\sigma }^{v}+(v-u){\sigma }^{u}]{v}_{y}+3({v}^{2}-{u}^{2}+2{uv}){\sigma }_{y}^{v},\end{array}\right.\sigma \equiv \left(\begin{array}{c}{\sigma }^{u}\\ {\sigma }^{v}\end{array}\right)\end{eqnarray}$
of ( $\begin{eqnarray}\begin{array}{l}x+6{tu}+F(u)=0,\\ x+6t[v+(u-v){G}_{1}(v)]+{{yG}}_{2}(v)\\ \ +\ F(u){G}_{1}(v)+{G}_{3}(v)=0,\end{array}\end{eqnarray}$
where F ≡ F(u) and Gi ≡ Gi(v), v = 1, 2, 3 are arbitrary functions of the indicated variables.From this section, we know that the different nonlinear waves, say, the dispersionless KdV waves can be coupled by means of the deformation algorithm. It is known that for a real physical model, there may be not only dispersionless waves but also dispersion waves. In the next section, we study the higher dimensional coupled dispersion waves related to different KdV equations.
3. From decoupled KdV equations to integrable coupled KdV system
It is clear that the decoupled KdV equation system20 ), ai and bi are arbitrary constants and different aj denotes different linear dispersion effects.
$\begin{eqnarray}{u}_{{it}}={\left({a}_{i}{u}_{{ixx}}+3{b}_{i}{u}_{i}^{2}\right)}_{x},i=1,2,\ldots ,n,\end{eqnarray}$
is integrable. In (It is straightforward to find that the decoupled KdV system (20 ) possesses the conservation laws21 ) with (22 ) and (23 ) can be reconstructed as
$\begin{eqnarray}{\rho }_{{it}}={J}_{{it}},i=1,2,\ldots ,2n\end{eqnarray}$
with the conserved densities ρi $\begin{eqnarray}\begin{array}{rcl}{\rho }_{i} & = & {u}_{i},i=1,2,\ldots ,n,\\ {\rho }_{i} & = & {u}_{i}^{2},i=n+1,n+2,\ldots ,2n,\end{array}\end{eqnarray}$
and the conserved flows Ji $\begin{eqnarray}\begin{array}{rcl}{J}_{i} & = & {a}_{i}{u}_{{ixx}}+3{b}_{i}{u}_{i}^{2},i=1,2,\ldots ,n,\\ {J}_{i} & = & 2{{au}}_{i}{u}_{{ixx}}-{a}_{i}{u}_{{ix}}^{2}+4{b}_{i}{u}_{i}^{3},\\ i & = & n+1,n+2,\ldots ,2n.\end{array}\end{eqnarray}$
The conservation laws ( $\begin{eqnarray}{{ \mathcal P }}_{{kt}}={{ \mathcal J }}_{{kt}},k=1,2,\ldots ,K-1,\end{eqnarray}$
with the conserved densities ${{ \mathcal P }}_{k}$ $\begin{eqnarray}{{ \mathcal P }}_{k}=\displaystyle \sum _{i=1}^{2n}{c}_{{ik}}{\rho }_{i},k=1,2,\ldots ,K-1,\end{eqnarray}$
and the conserved flows Jk $\begin{eqnarray}{{ \mathcal J }}_{k}=\displaystyle \sum _{i=1}^{2n}{c}_{{ik}}{J}_{i},k=1,2,\ldots ,K-1\end{eqnarray}$
where K is an arbitrary positive integers and cik, i = 1, 2, …, 2n, k = 1, 2, …, K − 1 are arbitrary constants.Thus, applying the deformation algorithm, we can find the (K+1)-dimensional integrable coupled KdV system
$\begin{eqnarray}\begin{array}{rcl}\hat{T}{u}_{i} & = & \hat{L}({a}_{i}{\hat{L}}^{2}{u}_{i}+3{b}_{i}{u}_{i}^{2}),\\ i & = & 1,2,\ldots ,n,\end{array}\end{eqnarray}$
with the deformed operators $\begin{eqnarray}\begin{array}{rcl}\hat{L} & = & {\partial }_{x}+\displaystyle \sum _{k=1}^{K-1}{{ \mathcal P }}_{k}{\partial }_{{x}_{k}},\\ \hat{T} & = & {\partial }_{t}+\displaystyle \sum _{k=1}^{K-1}{\bar{{ \mathcal J }}}_{k}{\partial }_{{x}_{k}},\\ {\bar{{ \mathcal J }}}_{k} & = & \displaystyle \sum _{i=1}^{2n}{c}_{{ik}}{\bar{J}}_{i},\\ k & = & 1,2,\ldots ,K-1,\\ {\bar{J}}_{i} & = & {a}_{i}{\hat{L}}^{2}{u}_{i}+3{b}_{i}{u}_{i}^{2},\\ {\bar{J}}_{n+i} & = & 2{{au}}_{i}{\hat{L}}^{2}{u}_{i}-{a}_{i}{\left(\hat{L}{u}_{i}\right)}^{2}+4{b}_{i}{u}_{i}^{3},\\ i & = & 1,2,\ldots ,n.\end{array}\end{eqnarray}$
The integrable coupled KdV system (27 ) is quite complicated. Here we just write down a simplest special case for K = 2, a1 = a2 = b1 = 1, b2 = b, u1 ≡ u, u2 ≡ v and ${{ \mathcal P }}_{1}\equiv P\,=\,u\,+\,v$. In this special case, (27 ) is simplified to29 ) will reduce back to two separated (1+1)-dimensional KdV equations. However, when the model (29 ) is x-independent, it is still an integrable coupled system (P = u + v)31 ) to the (1+1)-dimensional integrable model (30 ), we have32 ) and (29 ) are completely same.
$\begin{eqnarray}\begin{array}{rcl}{u}_{t} & = & {u}_{{xxx}}+3{{Pu}}_{{xxy}}+3{P}^{2}{u}_{{xyy}}\\ & & +{P}^{3}{u}_{{yyy}}+3({{Pu}}_{{yy}}+{u}_{{xy}})({{PP}}_{y}+{P}_{x})\\ & & +6{{uu}}_{x}+3({u}^{2}-{{bv}}^{2}+2{uv}){u}_{y},\\ {v}_{t} & = & {v}_{{xxx}}+3{{Pv}}_{{xxy}}+3{P}^{2}{v}_{{xyy}}\\ & & +{P}^{3}{v}_{{yyy}}+3({{Pv}}_{{yy}}+{v}_{{xy}})({{PP}}_{y}+{P}_{x})\\ & & +6{{bvv}}_{x}+3({{bv}}^{2}-{u}^{2}+2{buv}){v}_{y}.\end{array}\end{eqnarray}$
When u and v are y-independent, the (2+1)-dimensional coupled KdV system ( $\begin{eqnarray}\begin{array}{rcl}{u}_{t} & = & {P}^{3}{u}_{{yyy}}+3{P}^{2}{u}_{{yy}}{P}_{y}+3({u}^{2}-{{bv}}^{2}+2{uv}){u}_{y},\\ {v}_{t} & = & {P}^{3}{v}_{{yyy}}+3{P}^{2}{v}_{{yy}}{P}_{y}+3({{bv}}^{2}-{u}^{2}+2{buv}){v}_{y}.\end{array}\end{eqnarray}$
It is easy to check that the (2+1)-dimensional integrable system possesses the conservation law $\begin{eqnarray}{\left({P}^{-1}\right)}_{t}=-{\left[{{PP}}_{{yy}}+{P}_{y}^{2}+3({{bv}}^{2}+{u}^{2}){P}^{-1}\right]}_{y},\end{eqnarray}$
Applying the deformation algorithm with the conservation law ( $\begin{eqnarray}\begin{array}{rcl}{\hat{T}}_{1}u & = & {P}^{3}{\hat{L}}_{1}^{3}u+({\hat{L}}_{1}{P}^{3}){\hat{L}}_{1}^{2}u+3({u}^{2}-{{bv}}^{2}+2{uv}){\hat{L}}_{1}u,\\ {\hat{T}}_{1}v & = & {P}^{3}{\hat{L}}_{1}^{3}v+({\hat{L}}_{1}{P}^{3}){\hat{L}}_{1}^{2}v+3({{bv}}^{2}-{u}^{2}+2{buv}){\hat{L}}_{1}v,\end{array}\end{eqnarray}$
with $\begin{eqnarray}\begin{array}{l}{\hat{L}}_{1}={\partial }_{y}+{P}^{-1}{\partial }_{x},\\ {\hat{T}}_{1}=-{\partial }_{t}-\left[P{\hat{L}}_{1}^{2}P+{\left({\hat{L}}_{1}P\right)}^{2}+3({{bv}}^{2}+{u}^{2}){P}^{-1}\right]{\partial }_{x}.\end{array}\end{eqnarray}$
It is straightforward to check that the systems (In this section, by using the deformation algorithm the decoupled dispersion waves related to the decoupled KdV system are coupled in higher dimensions. In fact, for a real physical system, there are not only dispersion effects (related to the KdV equation) but also the diffusion effects (related to the Burgers equation). In the next section, we investigate coupled KdV-Burgers systems by considering both the dispersion effects and the diffusion effects.
4. From decoupled KdV and Burgers equations to integrable coupled KdV-Burgers system
In this section, we apply the deformation algorithm to the decoupled KdV and Burgers equations34 ) explicitly shows n + m conservation laws
$\begin{eqnarray}\begin{array}{rcl}{u}_{{it}} & = & {\left({a}_{i}{u}_{{ixx}}+3{b}_{i}{u}_{i}^{2}\right)}_{x},i=1,2,\ldots ,n,\\ {v}_{{jt}} & = & {\left({c}_{j}{v}_{{jx}}+{d}_{j}{v}_{j}^{2}\right)}_{x},j=1,2,\ldots ,m,\end{array}\end{eqnarray}$
where ai, bi, cj and dj are arbitrary constants. Different {ai, i = 1, 2, …, n} are related to different linear dispersions and different {ci, i = 1, 2, …, m} denote different diffusions. The system ( $\begin{eqnarray}\begin{array}{rcl}{\rho }_{{it}} & = & {J}_{{ix}},i=1,2,\ldots ,n+m,\\ {\rho }_{i} & = & {u}_{i},i=1,2,\ldots ,n,\\ {\rho }_{n+j} & = & {v}_{j},j=1,2,\ldots ,m,\\ {J}_{i} & = & {a}_{i}{u}_{{ixx}}+3{b}_{i}{u}_{i}^{2},i=1,2,\ldots ,n,\\ {J}_{n+j} & = & {c}_{j}{v}_{{jx}}+{d}_{j}{v}_{j}^{2},j=1,2,\ldots ,m.\end{array}\end{eqnarray}$
Similar to the last section, the conservation laws (35 ) can be reconstructed as
$\begin{eqnarray}\begin{array}{rcl}{{ \mathcal P }}_{{kt}} & = & {{ \mathcal J }}_{{kx}},k=1,2,\ldots ,K-1,\\ {{ \mathcal P }}_{k} & = & \displaystyle \sum _{i=1}^{n+m}{c}_{{ik}}{\rho }_{i},\\ {{ \mathcal J }}_{k} & = & \displaystyle \sum _{i=1}^{n+m}{c}_{{ik}}{J}_{i},k=1,2,\ldots ,K-1\end{array}\end{eqnarray}$
with arbitrary constants cik and arbitrary integer K.Thus, applying the deformation algorithm with the conservation laws (36 ) to (34 ), we have37 ) reads (p = u + v)39 ) is reduced to a simple (1+1)-dimensional integrable coupled system
$\begin{eqnarray}\begin{array}{rcl}\hat{{ \mathcal T }}{u}_{i} & = & \hat{{ \mathcal L }}({a}_{i}{\hat{{ \mathcal L }}}^{2}{u}_{i}+3{b}_{i}{u}_{i}^{2}),i=1,2,\ldots ,n,\\ \hat{{ \mathcal T }}{v}_{j} & = & \hat{{ \mathcal L }}({c}_{j}\hat{{ \mathcal L }}{v}_{j}+{d}_{j}{v}_{j}^{2}),j=1,2,\ldots ,m,\end{array}\end{eqnarray}$
where the deformed operators $\hat{{ \mathcal L }}$ and $\hat{{ \mathcal T }}$ read $\begin{eqnarray}\begin{array}{rcl}\hat{{ \mathcal L }} & = & {\partial }_{x}+\displaystyle \sum _{k=1}^{K-1}{{ \mathcal P }}_{k},\quad \hat{{ \mathcal T }}={\partial }_{t}+\displaystyle \sum _{k=1}^{K-1}{\bar{{ \mathcal J }}}_{k},\\ {\bar{{ \mathcal J }}}_{k} & = & \displaystyle \sum _{i=1}^{n+m}{c}_{{ik}}{\bar{J}}_{i},\\ {\bar{J}}_{i} & = & {a}_{i}{\hat{{ \mathcal L }}}^{2}{u}_{i}+3{b}_{i}{u}_{i}^{2},i=1,2,\ldots ,n,\\ {\bar{J}}_{n+j} & = & {c}_{j}\hat{{ \mathcal L }}{v}_{j}+{d}_{j}{v}_{j}^{2},j=1,2,\ldots ,m.\end{array}\end{eqnarray}$
A simplest nontrivial example of ( $\begin{eqnarray}\begin{array}{rcl}{u}_{t} & = & {u}_{{xxx}}+6{{uu}}_{x}+3{{pu}}_{{xxy}}+3{p}^{2}{u}_{{xyy}}\\ & & +{p}^{3}{u}_{{yyy}}+3({{pu}}_{{yy}}+{u}_{{xy}})({{pp}}_{y}+{p}_{x})\\ & & +[{v}_{{xx}}+2{{pv}}_{{xy}}+{p}^{2}{v}_{{yy}}-{v}_{x}\\ & & +({p}_{x}+{{pp}}_{y}-p){v}_{y}+3{p}^{2}-4{v}^{2}]{u}_{y},\\ {v}_{t} & = & {v}_{{xx}}+2{{vv}}_{x}+2{{pv}}_{{xy}}+{p}^{2}{v}_{{yy}}\\ & & -[{u}_{{xx}}+2{{pu}}_{{xy}}+{p}^{2}{u}_{{yy}}+4{u}^{2}-{p}^{2}\\ & & -{u}_{x}+({{pp}}_{y}+{p}_{x}-p){u}_{y}]{v}_{y}.\end{array}\end{eqnarray}$
For ux = vx = 0, the model ( $\begin{eqnarray}\begin{array}{rcl}{u}_{t} & = & {p}^{3}{u}_{{yyy}}+3{p}^{2}{u}_{{yy}}{p}_{y}\\ & & +[{p}^{2}{v}_{{yy}}+p({p}_{y}-1){v}_{y}+3{p}^{2}-4{v}^{2}]{u}_{y},\\ {v}_{t} & = & {p}^{2}{v}_{{yy}}-\left[{p}^{2}{u}_{{yy}}+4{u}^{2}-{p}^{2}\right.\\ & & \left.+p({p}_{y}-1){u}_{y}\right]{v}_{y}.\end{array}\end{eqnarray}$
The higher dimensional integrable systems obtained in this section can be used to describe both the dispersion effects and diffusion effects. In the next section, we study the higher dimensional coupled integrable models obtained from the decoupled linear dispersion equations (KdV equations) and the nonlinear dispersion equations (HD equations).5. From decoupled KdV equations and HD equations to integrable coupled systems
It is known that the HD equation is a reciprocal link of the KdV equation. A variant form of the HD equation possesses the form [41]
$\begin{eqnarray}{v}_{t}={\left({{cv}}^{3}{v}_{{xx}}+{{dv}}^{3}\right)}_{x}.\end{eqnarray}$
In [25], it is pointed out that KdV equation and the HD equation can be combined to the same model by using the deformation algorithm to the KdV equation. In this section, we combine the separated KdV equations and the HD equations to be coupled models in a different way by reconstructing the conservation laws of the KdV and HD equations.We take the decoupled KdV-HD equation system in the form
$\begin{eqnarray}\begin{array}{rcl}{u}_{{it}} & = & {\left({a}_{i}{u}_{{ixx}}+3{b}_{i}{u}_{i}^{2}\right)}_{x},i=1,2,\ldots ,n,\\ {v}_{{jt}} & = & {\left({c}_{j}{v}_{j}^{3}{v}_{{jxx}}+{d}_{j}{v}_{j}^{3}\right)}_{x},j=1,2,\ldots ,m\end{array}\end{eqnarray}$
with arbitrary constants ai, bi, cj, dj and arbitrary positive integers n and m.From the decoupled KdV-HD equation system (42 ), it is readily to find the following conservation laws43 ) can be reconstructed to
$\begin{eqnarray}\begin{array}{rcl}{\rho }_{{it}} & = & {J}_{{ix}},i=1,2,\ldots ,n+m,\\ {\rho }_{i} & = & {u}_{i},{\rho }_{n+i}={u}_{i}^{2},i=1,2,\ldots ,n,\\ {\rho }_{2n+j} & = & {v}_{j},{\rho }_{2n+m+j}={v}_{j}^{-1},j=1,2,\ldots ,m,\\ {J}_{i} & = & {a}_{i}{u}_{{ixx}}+3{b}_{i}{u}_{i}^{2},\\ {J}_{n+i} & = & 2{a}_{i}{u}_{i}{u}_{{ixx}}-{a}_{i}{u}_{{ix}}^{2}+4{b}_{i}{u}_{i}^{3},\\ i & = & 1,2,\ldots ,n,\\ {J}_{2n+j} & = & {c}_{j}{v}_{j}^{3}{v}_{{jxx}}+{d}_{j}{v}_{j}^{3},\\ {J}_{2n+m+j} & = & -{c}_{j}{v}_{j}{v}_{{jxx}}-{c}_{j}{v}_{{jx}}^{2}-3{d}_{j}{v}_{j},\\ j & = & 1,2,\ldots ,m.\end{array}\end{eqnarray}$
Similarly, the conservation laws ( $\begin{eqnarray}\begin{array}{rcl}{{ \mathcal D }}_{{kt}} & = & {{ \mathcal F }}_{{kx}},k=1,2,\ldots ,K-1,\\ {{ \mathcal D }}_{k} & = & \displaystyle \sum _{i=1}^{2n+2m}{c}_{{ik}}{\rho }_{i},\\ {{ \mathcal F }}_{k} & = & \displaystyle \sum _{i=1}^{2n+2m}{c}_{{ik}}{J}_{i},k=1,2,\ldots ,K-1\end{array}\end{eqnarray}$
with arbitrary constants cik and arbitrary integer K.The application of the deformation algorithm and the reconstructed conservation laws (44 ) yields the integrable coupled KdV-HD system45 ) reads ($p=u+v,q=1-{v}^{3},\hat{L}={\partial }_{x}+p{\partial }_{y}$)47 ) is x-independent, we have a (1+1)-dimensional coupled integrable system48 ) will reduce to (41 ) with c = d = 1 by replacing {u, y} → {v, x}.
$\begin{eqnarray}\begin{array}{rcl}\hat{T}{u}_{i} & = & \hat{L}({a}_{i}{\hat{L}}^{2}{u}_{i}+3{b}_{i}{u}_{i}^{2}),i=1,2,\ldots ,n,\\ \hat{T}{v}_{j} & = & \hat{L}({c}_{j}{v}_{j}^{3}{\hat{L}}^{2}{v}_{j}+{d}_{j}{v}_{j}^{3}),j=1,2,\ldots ,m\end{array}\end{eqnarray}$
where the deformation operators $\hat{L}$ and $\hat{T}$ are defined as $\begin{eqnarray}\begin{array}{rcl}\hat{L} & = & {\partial }_{x}+\displaystyle \sum _{k=1}^{K-1}{{ \mathcal D }}_{k}{\partial }_{{x}_{k}},\quad \hat{T}={\partial }_{t}+\displaystyle \sum _{k=1}^{K-1}{\bar{{ \mathcal F }}}_{k}{\partial }_{{x}_{k}},\\ {\bar{{ \mathcal F }}}_{k} & = & \displaystyle \sum _{i=1}^{2n+2m}{c}_{{ik}}{\bar{J}}_{i},\quad {\bar{J}}_{i}={a}_{i}{\hat{L}}^{2}{u}_{i}+3{b}_{i}{u}_{i}^{2},\\ {\bar{J}}_{n+i} & = & 2{a}_{i}{u}_{i}{\hat{L}}^{2}{u}_{i}-{a}_{i}{\left(\hat{L}{u}_{i}\right)}^{2}+4{b}_{i}{u}_{i}^{3},\\ i & = & 1,2,\ldots ,n,\quad {\bar{J}}_{2n+j}={c}_{j}{v}_{j}^{3}{\hat{L}}^{2}{v}_{j}+{d}_{j}{v}_{j}^{3},\\ {\bar{J}}_{2n+m+j} & = & -{c}_{j}{v}_{j}{\hat{L}}^{2}{v}_{j}-{c}_{j}{\left(\hat{L}{v}_{j}\right)}^{2}-3{d}_{j}{v}_{j},\\ j & = & 1,2,\ldots ,m.\end{array}\end{eqnarray}$
One of the simplest two-component form of ( $\begin{eqnarray}\begin{array}{rcl}{u}_{t} & = & {u}_{{xxx}}+3p\hat{L}{u}_{{xy}}+{p}^{3}{u}_{{yyy}}\\ & & +3(\hat{L}{u}_{y})\hat{L}p+{{qu}}_{y}(p\hat{L}{v}_{y}+\hat{L}{v}_{x})\\ & & +6{{uu}}_{x}+({{qv}}_{y}\hat{L}p-{v}^{3}+3{u}^{2}+6{uv}){u}_{y},\\ {v}_{t} & = & -\hat{L}\left[{v}_{y}(p\hat{L}{u}_{y}+\hat{L}{u}_{x})+{v}^{3}({v}_{y}-1)\right.\\ & & \times ({p}^{2}{v}_{{yy}}+{{pp}}_{y}{v}_{y}+2{{pv}}_{{xy}}+{p}_{x}{v}_{y}+{v}_{{xx}}+1)\\ & & \left.+{{pv}}_{y}{u}_{y}^{2}+{v}_{y}{u}_{y}\hat{L}p+3{u}^{2}{v}_{y}\right].\end{array}\end{eqnarray}$
For the model ( $\begin{eqnarray}\begin{array}{rcl}{u}_{t} & = & {p}^{3}{u}_{{yyy}}+3{p}^{2}{u}_{{yy}}{p}_{y}+{{qp}}^{2}{u}_{y}{v}_{{yy}}\\ & & +({{pqv}}_{y}{p}_{y}-{v}^{3}+3{u}^{2}+6{uv}){u}_{y},\\ {v}_{t} & = & -p\left[{v}_{y}{p}^{2}{u}_{{yy}}+{v}^{3}({v}_{y}-1)\right.\\ & & \times ({p}^{2}{v}_{{yy}}+{{pp}}_{y}{v}_{y}+1)\\ & & {\left.+{{pv}}_{y}{u}_{y}^{2}+{{pv}}_{y}{u}_{y}{p}_{y}+3{u}^{2}{v}_{y}\right]}_{y}.\end{array}\end{eqnarray}$
When v = 0, the system (It is known that the deformation algorithm of [34] will combine linear dispersion effects and nonlinear dispersion effects to the same higher dimensional integrable system. In this section, the deformation algorithm is applied in an alternative way to couple both the linear and nonlinear dispersion effects to a same model.
6. Summary and discussions
In summary, by applying the deformation algorithm and the reconstructed conservation laws of decoupled integrable systems, one can find some coupled integrable systems in any dimensions. In this paper, the dispersionless KdV system (5 )-(6 ) is deformed to arbitrary dimensions because of the existence of infinitely many conservation laws with some arbitrary functions of fields. The (K+1)-dimensional integrable coupled dispersionless KdV system (11 ) is integrable because of the existence of higher order general symmetries. The coupled dispersionless KdV system can be used to describe the interactions among different dispersionless waves. The method can be used to study the interactions between more complicated dispersionless waves, say, the dispersionless waves ut = F(u)ux and vt = G(v)vx for arbitrary F and G.
In a complicated real physical system like the oceanic system and the atmospheric system, there are many physical effects such as the linear dispersion effects (described by KdV equations), the diffusion effects (described by Burgers equations) and the nonlinear dispersion effects (described by HD equations). In this paper, multiple separated KdV equations, Burgers equations and HD equations are used to find their coupled integrable systems including the coupled KdV systems, the coupled KdV-Burgers systems and the coupled KdV-HD systems. These models may be applied to study the interactions among different nonlinear waves with different physical effects. In fact, applying a similar procedure to more separated integrable systems, one can find more generalized coupled integrable systems.
Though the models we obtained are integrable, it is still very difficult to find their exact solutions because the original models and their reciprocal links are included in the same systems. In this paper, only a special shock wave solution with four arbitrary functions for the special (2+1)-dimensional two-component integrable coupled dispersionless KdV system (12 ) are given in (19 ). The more about the deformation algorithm and the exact solutions of the higher dimensional integrable systems should be further investigated.