- Research
- Open access
- Published:
Construction of invariant solutions and conservation laws to the \((2+1)\)-dimensional integrable coupling of the KdV equation
Boundary Value Problems volume 2020, Article number: 169 (2020)
Abstract
Under investigation in this paper is the \((2+1)\)-dimensional integrable coupling of the KdV equation which has applications in wave propagation on the surface of shallow water. Firstly, based on the Lie symmetry method, infinitesimal generators and an optimal system of the obtained symmetries are presented. At the same time, new analytical exact solutions are computed through the tanh method. In addition, based on Ibragimov’s approach, conservation laws are established. In the end, the objective figures of the solutions of the coupling of the KdV equation are performed.
1 Introduction
It is well known that the Korteweg–de Vries (KdV) equation describes the propagation of long waves on the surface of water with a small amplitude and is widely used to explain many complex science phenomena [1, 2]. Various forms of the expansion for the KdV equation have been proposed because of its importance, such as the KdV-Burgers equation [3], the KdV-BBM equation [4], the Rosenau–KdV equation [5], the modified KdV equation [6], KdV-hierarchy [7], and the \((2+1)\)-dimensional KdV equation [8]. In this research article, we consider the following \((2+1)\)-dimensional integrable coupling of the KdV equation which has the bi-Hamiltonian structure for the \((2+1)\)-dimensional perturbation equations of the KdV hierarchy [9]:
where \(u=u(x,y,t)\), \(v=v(x,y,t)\) are the unknown real functions, the subscripts denote the partial derivatives, and the variable y is called a slow variable. Equation (1) plays an important role in many analyses of physical phenomena such as stratified internal waves and lattice dynamics [10, 11], and it has aroused worldwide interest. The \((2+1)\)-dimensional hereditary recursion operators were examined in [12], its integrability was verified by using Painlevé in [13], some traveling wave solutions were established in [14], the auto-Bäcklund transformation, doubly periodic solutions and new non-traveling wave solutions were analyzed in [15].
As is well known, some methods have been used to explore exact solutions for models of nonlinear partial differential equations (PDEs) [16–18], the Lie group method is considered to be one of the most important methods to study the properties of solutions of PDEs [19, 20]. The main idea of the symmetry method is to construct an invariance condition and obtain reductions to differential equations [21–23]. Once reduction equations have been given, one can get a large number of corresponding exact solutions. In order to obtain the classification of all reduction equations, we require an optimal system of the one-dimensional subalgebra of the Lie algebra constructed by the Lie group method [19]. Using a symmetry analysis, we will get an optimal system of (1), from which the fascinating special solutions are inferred. Another important area is the conservation laws of PDEs which have an important impact on constructing solutions of PDEs [24–26]. We will obtain conservation laws of Eq. (1) by using Ibragimov’s approach [27].
The rest of this paper is organized as follows. In Sect. 2, symmetries of the \((2+1)\)-dimensional integrable coupling of the KdV equation are discussed; Sect. 3 considers the reduced equations by means of similar variables; in Sect. 4, some new explicit solutions are presented with the help of the tanh method, and some objective features of the solutions are presented; in Sect. 5, the nonlinearly self-adjointness of Eq. (1) is proved and its conservation laws are established by using Ibragimov’s method. Finally, concluding remarks are given at the end of the paper.
2 Lie point symmetry
In this section, we apply Lie’s theory of symmetries for Eq. (1), and get its infinitesimal generators, commutator of Lie algebra.
First, let us consider a Lie algebra of infinitesimal symmetries of Eq. (1) of the form
According to the invariance conditions for Eq. (1) with respect to the transformation (2), we have [19, 28]
where \(\operatorname{pr}^{(3)}X\) is the third-order prolongation of X [19, 28] and \(\Delta _{1}=u_{t}-u_{xxx}-6uu_{x}\), \(\Delta _{2}=v_{t}-v_{xxx}-3u_{xxy}-6(uv)_{x}-6uu_{y}\), on this condition,
where
and \(D_{x}\), \(D_{y}\), \(D_{t}\) stand for the operators of the total differentiation, for instance,
Next, we get a system of over-determined linear equations of \(\xi ^{1}\), \(\xi ^{2}\), \(\xi ^{3}\), ϕ and φ,
Solving these equations, one can get
where \(c_{1}\), \(c_{2}\), \(c_{3}\) are real constants, \(F(y)\) is an arbitrary function. To obtain physically crucial solutions, we take \(F_{1}(z)=c_{4}y+c_{5}\), then on substituting the above obtaining
Therefore, the Lie algebra \(L_{5}\) of the transformations of Eq. (1) is spanned by the following generators:
In order to classify all the group-invariant solutions, we need an optimal system of one-dimensional subalgebras. In this section, the optimal system of subgroups for Eq. (1) is constructed by only using the commutator table [29]. First, using the commutator \([X_{m}, X_{n}]=X_{m}X_{n}-X_{n}X_{m}\), we attained the commutation relations of \(X_{1}\), \(X_{2}\), \(X_{3}\), \(X_{4}\), \(X_{5}\) listed in Table 1.
An arbitrary operator \(X\in L_{5}\) is given as
To establish the linear transformations of the vector \(l=(l_{1},l_{2},l_{3},l_{4},l_{5})\), we denote
where \(c_{ij}^{k}\) is constructed by the formula \([ X_{i},X_{j}]=c_{ij}^{k}X_{k}\). Based on Eq. (3) and Table 1, \(E_{1}\), \(E_{2}\), \(E_{3}\), \(E_{4}\), \(E_{5}\) can be written as
For \(E_{1}\), \(E_{2}\), \(E_{3}\), \(E_{4}\), \(E_{5}\), the Lie equations with parameters \(a_{1}\), \(a_{2}\), \(a_{3}\), \(a_{4}\), \(a_{5}\) and the initial condition \(\tilde{l}|_{a_{i}=0}=l\), \(i=1,2,3,4,5\) are given as
The solutions of the above equations are associated with the transformations
The establishment of the optimal system requires a simplification of the vector
by applying the transformations \(T_{1}\)–\(T_{5}\). Our task is to construct a simplest representative of each class of similar vectors (4). Two cases will be considered separately.
Case 2.1. \(l_{1}\neq 0\)
By making \(a_{2}=-\frac{l_{2}}{l_{1}}\) and \(a_{3}=- \frac{3l_{3}}{l_{1}}\) in \(T_{2}\) and \(T_{3}\), we enable \(\tilde{l_{2}},\tilde{l_{3}}=0\). The vector (4) becomes
2.1.1. \(l_{4}\neq 0\)
By making \(a_{5}=-\frac{l_{5}}{l_{4}}\) in \(T_{5}\), we can enable \(\tilde{l_{5}}=0\). The vector (5) is equivalent to
We get the following representatives:
2.1.2. \(l_{4}=0\)
The vector (5) is equivalent to
We get the following representatives:
Case 2.2. \(l_{1}= 0\)
The vector (4) becomes
2.2.1. \(l_{4}\neq 0\)
By making \(a_{5}=-\frac{l_{5}}{l_{4}}\) in \(T_{5}\), we can enable \(\tilde{l_{5}}=0\). The vector (9) is equivalent to
Making all the possible combinations, we get the following representatives:
2.2.2. \(l_{4}=0\)
The vector (9) becomes
We get the following representatives:
Finally, by gathering the operators (6, 8, 10 and 11), we obtain the following theorem.
Theorem 2.1
An optimal system of \(\{X_{1},X_{2},X_{3},X_{4},X_{5}\}\) is generated by
3 Similarity reductions of the \((2+1)\)-dimensional integrable coupling of the KdV equation
In this section, based on Theorem 2.1, we will find some reduced equations of Eq. (1) by using similarity variables.
Case 3.1. Reduction by \(X_{2}+X_{4}\).
Integrating the characteristic equation for \(X_{2}+X_{4}\), we get the invariance
and the invariant solution takes the form \(\tilde{u}=f(\tilde{x}, \tilde{t})\), \(\tilde{v}=g(\tilde{x},\tilde{t})\), that is, \(u=f(\tilde{x}, \tilde{y})\), \(v=\frac{g(\tilde{x},\tilde{t})}{y}\), Eq. (1) can be reduced to
Case 3.2. Reduction by \(X_{2}\).
Similarly, we have \(\tilde{x}=x\), \(\tilde{y}=y\), \(u=f(\tilde{x},\tilde{y})\), \(v=g( \tilde{x},\tilde{y})\). Equation (1) is reduced to
Case 3.3. For the generator \(X_{2}+X_{3}\), we have \(\tilde{y}=y\), \(\tilde{t}=x-t\), \(u=f(\tilde{y},\tilde{t})\), \(v=g(\tilde{y},\tilde{t})\). Equation (1) is reduced to
Case 3.4. For the generator \(X_{2}+X_{5}\), we have \(\tilde{x}=x\), \(\tilde{y}=y-t\), \(u=f(\tilde{x},\tilde{y})\), \(v=g(\tilde{x},\tilde{y})\). Equation (1) can be reduced to
Case 3.5. For the generator \(X_{2}+X_{3}+X_{5}\), we have \(\tilde{x}=x-y\), \(\tilde{t}=x-t\), \(u=f(\tilde{x},\tilde{t})\), \(v=g(\tilde{x}, \tilde{t})\). Equation (1) becomes
Case 3.6. For the generator \(X_{4}\), we have \(\tilde{x}=x\), \(\tilde{t}=t\), \(u=f(\tilde{x},\tilde{t})\), \(v= \frac{g(\tilde{x},\tilde{t})}{y}\). Equation (1) can be reduced to
Case 3.7. For the generator \(X_{3}+X_{5}\), we have \(\tilde{x}=x-y\), \(\tilde{t}=t\), \(u=f(\tilde{x},\tilde{t})\), \(v=g(\tilde{x},\tilde{t})\). Equation (1) becomes
Case 3.8. For the generator \(X_{1}\), we have \(\tilde{y}=y\), \(\tilde{t}=\frac{t}{x^{3}}\), \(u=\frac{f(\tilde{y},\tilde{t})}{x^{2}}\), \(v= \frac{g(\tilde{y},\tilde{t})}{x}\). Equation (1) can be reduced to
4 The exact solutions of reduced equations
In the previous section, we have dealt with the similarity reductions and derived the corresponding reduced equations. In this section, we use the tanh method on reduced equations, obtaining some exact solutions of Eq. (1). With the help of exact solutions, we can understand some motion rules of waves of the \((2+1)\)-dimensional integrable coupling of KdV equation.
The main steps of the tanh method [24, 25] are expressed as follows:
1. Consider the following nonlinear differential equations:
where \(F_{1}\), \(F_{2}\) are polynomials of the u, v and their derivatives.
2. By using the wave transformations
where \(\xi =lx+ky+ct\), and l, k, c are unknown constants, and substituting (21) into Eq. (20), we obtain the following nonlinear ordinary differential equations:
3. Next, we introduce the independent variable
which leads to the following changes:
4. We assume that the solution of Eq. (22) is written as the following form:
where n, m are positive integers, which are decided by balancing the highest order nonlinear terms with the derivative terms in the resulting equations. After deciding n, m, taking (23) and (24) into (22), we obtain a polynomial concerning \(Y^{i}\) (\(i=0,1,2,\ldots \)). Then we gather all terms of \(Y^{i}\) (\(i=0,1,2,\ldots \)) and make all them equal to zero. Solving these algebraic equations, we get the values of the unknown numbers \(a_{i}\), \(b_{i}\) (\(i=0,1,\ldots \)), l, k and c. Then, putting these values into the equations, we get exact solutions of equations.
Case 4.1. For Eq. (12), substituting Eq. (21) into (12), we get the following equations:
Concerning (25), balancing \(\Phi ^{(3)}\) with \(\Phi \Phi '\), we have
balancing \(\Phi ^{(3)}\) with \(\Phi '\Psi \), we have
Hence, according to Eq. (24), the solution of Eq. (12) is assumed to be
Then, substituting Eq. (23) and Eq. (26) into Eq. (25), we collect all terms of \(Y^{i}\) and obtain the algebraic equations including unknown numbers \(a_{i}\), \(b_{i}\) (\(i=0,1,2\)), l and k. By solving these equations, we have the following solutions:
Putting (27) into Eq. (12), we obtain the exact solution as follows:
where \(l\neq 0\), k are arbitrary constants.
Figures 1 and 2 depict the exact solution of Eq. (12), which is obtained by taking \(l=1\), \(k=1\) at \(t=1\).
Case 4.2. For Eq. (13), similarly, substituting Eq. (21) into (13), we have the following ordinary differential equations:
Then, balancing \(\Phi ^{(3)}\) and \(\Phi \Phi '\), \(\Phi ^{(3)}\) and \(\Phi '\Psi '\) for (29), we have \(n=m=2\).
Therefore, on the basis of Eq. (24), the solution of Eq. (13) can be assumed to be
Next, substituting Eq. (23) and Eq. (30) into Eq. (29), we make all coefficients of \(Y^{i}\) vanish and obtain the algebraic equations including the unknown numbers \(a_{i}\), \(b_{i}\) (\(i=0,1,2\)), l and k. Solving these equations, we have the following solutions:
So, the exact solution of Eq. (13) is
where l, k are arbitrary constants. This solution is a static solution of Eq. (1).
When we take \(l=1\), \(k=1\), the values of u, v are illustrated in Figs. 3 and 4.
Case 4.3. For Eq. (14), equally, substituting Eq. (21) into (14), we get the following ordinary differential equations:
Furthermore, balancing \(\Phi ^{(3)}\) and \(\Phi \Phi '\), \(\Phi ^{(3)}\) and \(\Phi '\Psi '\) for (33), we have \(n=m=2\).
Therefore, based on Eq. (24), the solution of Eq. (14) can be assumed to be
Next, substituting Eq. (23) and Eq. (34) into Eq. (33), we make all coefficients of \(Y^{i}\) vanish and obtain the algebraic equations including unknown numbers \(a_{i}\), \(b_{i}\) (\(i=0,1,2\)), k and c. Solving these equations, we have the following solutions:
So, the exact solution of Eq. (14) is
where \(c\neq 0\) and k are arbitrary constants.
Figures 5 and 6 portray the solution of Eq. (14), which is obtained by taking \(k=-1\), \(c=1\) at \(t=1\).
Case 4.4. For Eq. (15), in the same way, substituting Eq. (21) into (15), we have the following ordinary differential equations:
Then, balancing \(\Phi ^{(3)}\) and \(\Phi \Phi '\), \(\Phi ^{(3)}\) and \(\Phi '\Psi '\) for (33), we have \(n=m=2\).
Therefore, based on Eq. (24), the solution of Eq. (15) can be assumed to be
Next, substituting Eq. (23) and Eq. (38) into Eq. (37), we make all coefficients of \(Y^{i}\) vanish and obtain the algebraic equations including unknown numbers \(a_{i}\), \(b_{i}\) (\(i=0,1,2\)), l and k. Solving these equations, we have the following solutions:
So, the exact solution of Eq. (15) is
where \(l\neq 0\), k are arbitrary constants.
When we take \(l=1\), \(k=-1\) at \(t=0\), the values of u, v are illustrated in Figs. 7 and 8.
Case 4.5. For Eq. (16), likewise, substituting Eq. (21) into (16), we get the following ordinary differential equations:
Then, balancing \(\Phi ^{(3)}\) and \(\Phi \Phi '\), \(\Phi ^{(3)}\) and \(\Phi '\Psi '\) for (41), we have \(n=m=2\).
Therefore, based on Eq. (24), the solution of Eq. (16) can be assumed to be
Next, substituting Eq. (23) and Eq. (42) into Eq. (41), we make all coefficients of \(Y^{i}\) vanish and obtain the algebraic equations including unknown numbers \(a_{i}\), \(b_{i}\) (\(i=0,1,2\)), l and c. Solving these equations, we have the following solutions:
So, the exact solution of Eq. (16) is
where \(l\neq 0\), c are arbitrary constants.
Figures 9 and 10 depict the exact solution of Eq. (16), which is obtained by taking \(l=1\), \(c=1\) at \(t=0\).
5 Construction of conservation laws
In this section, we chiefly construct conservation laws of Eq. (1) using Ibragimov’s method [27, 30]. First, we prove that Eq. (1) is nonlinear self-adjoint.
5.1 Proof of nonlinear self-adjointness
With regard to Eq. (1), conservation laws multipliers have the following form:
Moreover,
where the Euler operators \(E_{u}\), \(E_{v}\) are expressed as
Substituting (46) into (45), we obtain the following system which only has the unknown variables \(\Lambda _{1}\), \(\Lambda _{2}\):
Solving this system, we have \(\Lambda _{1}=6tuF_{1y}+(6tu+x)F_{2}(y)+uF_{3}(y)+F_{4}(y)\), \(\Lambda _{2}=F_{1}(y)\), where \(F_{1}(y)\), \(F_{2}(y)\), \(F_{3}(y)\) and \(F_{4}(y)\) are arbitrary functions.
Consider a PDE system of order m,
where \(x=(x^{1},x^{2},\ldots ,x^{n})\), \(u=(u^{1},u^{2},\ldots ,u^{m})\) and \(u_{(1)},u_{(2)},\ldots , u_{(k)}\) represent the set of all first, second,…, kth-order derivatives of u in regard to x.
The adjoint equations of Eq. (47) are written as
Besides,
where \(\mathcal{L}\) is a formal Lagrangian of the following form:
and the Euler–Lagrange operator is expressed as
Definition 5.1
([31])
The system (47) is said to be nonlinearly self-adjoint if the adjoint system is satisfied for all the solutions u of system (47) upon a substitution \(v=\varphi (x,u)\) such that \(\varphi (x,u)\neq 0\). Particularly, the system
is identical to the system
that is,
where \(\lambda ^{\beta }_{\alpha }\) is a certain function.
Theorem 5.1
([32])
The determining system of the multiplier \(\Lambda (x,u)\) of system (47) is identical to the system of nonlinearly self-adjoint substitution.
If the formal Lagrangian of Eq. (1) is given as
based on Theorem 5.1, we can get
Therefore, Eq. (1) is nonlinearly self-adjoint with substitution (48).
5.2 Construction of conservation laws
Theorem 5.2
([31])
The system of differential Eq. (47) is nonlinearly self-adjoint. Then every Lie point, the Lie–Bäcklund, nonlocal symmetry
admitted by the system of Eq. (47), gives rise to a conservation law, where the components \(\mathcal{C}^{i}\) of the conserved vector \(\mathcal{C}=(\mathcal{C}^{1},\ldots ,\mathcal{C}^{n})\) are determined by
and \(W^{\alpha }=\eta ^{\alpha }-\xi ^{j}u^{\alpha }_{j}\). The formal Lagrangian \(\mathcal{L}\) should be written in the symmetric form concerning all mixed derivatives \(u^{\alpha }_{ij},u^{\alpha }_{ijk},\ldots \) .
The Lagrangian \(\mathcal{L}\) of Eq. (1) is given as follows:
For the generator \(X=\xi ^{1}\partial _{x}+\xi ^{2}\partial _{y}+\xi ^{3}\partial _{t}+ \phi \partial _{u}+\varphi \partial _{v}\), in line with the Theorem 5.2, we obtain \(W^{1}=\phi -\xi ^{1}u_{x}-\xi ^{2}u_{y}-\xi ^{3}u_{t}\), \(W^{2}= \varphi -\xi ^{1}v_{x}-\xi ^{2}v_{y}-\xi ^{3}v_{t}\), so the components of the conservation vector have the following form:
By substituting the Lagrangian \(\mathcal{L}\) into above components of the conservation vector, \(\mathcal{C}^{x}\), \(\mathcal{C}^{y}\), \(\mathcal{C}^{t}\) are simplified as
For the generator \(X_{1}=\frac{1}{3}x\partial _{x}+t\partial _{t}-\frac{2}{3}u\partial _{u}- \frac{1}{3}v\partial _{v}\), we have \(W^{1}=-\frac{2}{3}u-\frac{1}{3}xu_{x}-tu_{t}\), \(W^{2}=-\frac{1}{3}v- \frac{1}{3}xv_{x}-tv_{t}\). According to Eqs. (49)–(51), the components of the conserved vector of generator \(X_{1}\) have the following form:
For the generator \(X_{2}=\partial _{t}\), we have \(W^{1}=-u_{t}\), \(W^{2}=-v_{t}\). According to Eqs. (49)–(51), the components of the conserved vector of generator \(X_{2}\) can be expressed as follows:
For the generator \(X_{3}=\partial _{x}\), we have \(W^{1}=-u_{x}\), \(W^{2}=-v_{x}\). According to Eqs. (49)–(51), the components of the conserved vector of generator \(X_{3}\) can be written the following form:
For the generator \(X_{4}=y\partial _{y}-v\partial _{v}\), we have \(W^{1}=-yu_{y}\), \(W^{2}=-v-yv_{y}\). According to Eqs. (49)–(51), the components of the conserved vector of generator \(X_{4}\) are given as
For the generator \(X_{5}=\partial _{y}\), we have \(W^{1}=-u_{y} W^{2}=-v_{y}\). According to Eqs. (49)–(51), the components of the conserved vector of the generator \(X_{5}\) can be expressed as
6 Conclusions
In this paper, Lie group analysis is applied to the \((2+1)\)-dimensional integrable coupling of the KdV equation. The optimal system of the obtained symmetries and reduced equations are obtained based on symmetry method. Moreover, explicit solutions of the reduced equations are constructed by using the tanh method. Through the figures related to solutions, we can show the rules of the wave propagation corresponding to Eq. (1). Finally, nonlinearly self-adjointness of Eq. (1) is manifested and its conservation laws are derived by using Ibragimov’s method.
Availability of data and materials
Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.
References
Ablowitz, M.J., Solitons, C.P.A.: Nonlinear Evolution Equations and Inverse Scattering. Cambridge University Press, Cambridge (1991)
Gu, C.H.: Soliton Theory and Its Applications. Springer, Berlin (1995)
Mancas, S.C., Adams, R.: Dissipative periodic and chaotic patterns to the KdV–Burgers and Gardner equations. Chaos Solitons Fractals 126, 385–393 (2019)
Carvajal, X., Panthee, M.: On sharp global well-posedness and ill-posedness for a fifth-order KdV-BBM type equation. J. Math. Anal. Appl. 479, 688–702 (2019)
Wang, X.F., Dai, W.Z.: A conservative fourth-order stable finite difference scheme for the generalized Rosenau–KdV equation in both 1D and 2D. J. Comput. Appl. Math. 355, 310–331 (2019)
Jackaman, J., Papamikos, G., Pryer, T.: The design of conservative finite element discretisations for the vectorial modified KdV equation. Appl. Numer. Math. 137, 230–251 (2019)
Ayano, T., Buchstaber, V.M.: Construction of two parametric deformation of KdV-hierarchy and solution in terms of meromorphic functions on the sigma divisor of a hyperelliptic curve of genus 3. SIGMA 15, 032 (2019)
Wang, C.J., Fang, H., Tang, X.X.: State transition of lump-type waves for the \((2+1)\)-dimensional generalized KdV equation. Nonlinear Dyn. 95, 2943–2961 (2019)
Ma, W.X., Fuchssteiner, B.: The bi-Hamiltonian structure of the perturbation equations of the KdV hierarchy. Phys. Lett. A 213(1–2), 49–55 (1996)
Miura, R.M.: The Korteweg–deVries equation: a survey of results. SIAM Rev. 18(3), 412–459 (1976)
Ablowitz, M.J., Segur, H.: Solitons and the Inverse Scattering Transform. SIAM, Philadelphia (1981)
Li, Y.S., Ma, W.X.: A nonconfocal involutive system and constrained flows associated with the MKdV-equation. J. Math. Phys. 43, 4950–4962 (2002)
Sakovich, S.Y.: On integrability of a \((2+1)\)-dimensional perturbed KdV equation. J. Nonlinear Math. Phys. 5(3), 230–233 (1998)
Fan, E.G.: A new algebraic method for finding the line soliton solutions and doubly periodic wave solution to a two-dimensional perturbed KdV equation. Chaos Solitons Fractals 15(3), 567–574 (2003)
Yan, Z.Y.: The \((2+1)\)-dimensional integrable coupling of the KdV equation: auto-Backlund transformation and new non-traveling wave profiles. Phys. Lett. A 345(4–6), 362–377 (2005)
Qiao, Z.H., Yang, X.G.: A multiple-relaxation-time lattice Boltzmann method with Beam–Warming scheme for a coupled chemotaxis-fluid model. Electron. Res. Arch. 28(3), 1207–1225 (2020)
Chorfi, N., Abdelwahed, M., Berselli, L.C.: On the analysis of a geometrically selective turbulence model. Adv. Nonlinear Anal. 9, 1402–1419 (2020)
Bathory, M., BulĂÄŤek, M., Málek, J.: Large data existence theory for three-dimensional unsteady flows of rate-type viscoelastic fluids with stress diffusion. Adv. Nonlinear Anal. 10, 501–521 (2021)
Olver, P.J.: Applications of Lie Groups to Differential Equations. Grauate Texts in Mathematics. Springer, New York (1993)
Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, Berlin (1989)
Sahoo, S., Ray, S.S.: Lie symmetry analysis and exact solutions of \((3+1)\) dimensional Yu–Toda–Sasa–Fukuyama equation in mathematical physics. Comput. Math. Appl. 73(2), 253–260 (2017)
Gao, B., Invariant, W.Y.X.: Solutions and nonlinear self-adjointness of the two-component Chaplygin gas equation. Discrete Dyn. Nat. Soc. 2019, 9609357 (2019)
Gao, B.: Symmetry analysis and explicit power series solutions of the Boussinesq–Whitham–Broer–Kaup equation. Waves Random Complex Media, 27(4), 700–710 (2017)
Wazwaz, A.M.: The tanh method: solitons and periodic solutions for the Dodd–Bullough–Mikhailov and the Tzitzeica–Dodd–Bullough equations. Chaos Solitons Fractals 25, 55–63 (2005)
Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60, 650–654 (1992)
Bertsch, M., Smarrazzo, F., Terracina, A., Tesei, A.: Radon measure-valued solutions of first order scalar conservation laws. Adv. Nonlinear Anal. 9(1), 65–107 (2018)
Ibragimov, N.H.: A new conservation theorem. J. Math. Anal. Appl. 333, 311–328 (2007)
Bluman, G.W., Anco, S.C.: Symmetry and Integration Methods for Differential Equations. Springer, New York (2004)
Grigoriev, Y.N., Kovalev, V.F., Meleshko, S.V.: Symmetries of Integro-Differential Equations: With Applications in Mechanics and Plasma Physics. Springer, New York (2010)
Ibragimov, N.H.: Integrating factors, adjoint equations and Lagrangians. J. Math. Anal. Appl. 318, 742–757 (2006)
Ibragimov, N.H.: Nonlinear self-adjointness and conservation laws. J. Phys. A 44, 432002 (2011)
Ibragimov, N.H.: Nonlinear self-adjointness in constructing conservation laws. Arch. ALGA 7, 1–99 (2011)
Acknowledgements
The authors would like to thank the editors and referees for their constructive comments and suggestions.
Funding
The authors are supported by the Natural Science Foundation of Shanxi (No. 201801D121018).
Author information
Authors and Affiliations
Contributions
The authors have equally contributed to this article and read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Gao, B., Yin, Q. Construction of invariant solutions and conservation laws to the \((2+1)\)-dimensional integrable coupling of the KdV equation. Bound Value Probl 2020, 169 (2020). https://doi.org/10.1186/s13661-020-01466-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-020-01466-6