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Some symmetries, similarity solutions and various conservation laws of a type of dispersive water waves
Advances in Difference Equations volume 2019, Article number: 435 (2019)
Abstract
We investigate the point symmetries, Lie–Bäcklund symmetries for a type of dispersive water waves. We obtain some Lie transformation groups, various group-invariant solutions, and some similarity solutions. Besides, we produce different formats of conservation laws of the dispersive water waves by using different schemes. Finally, we consider some special solutions of the stationary dispersive water-wave equations.
1 Introduction
The classical dispersiveless long wave equations
have a number of dispersive generalizations [1]. Kupershmidt [2] investigated the commuting hierarchy and the Hamiltonian structures of the following generation of (1):
and further turned (2) into the following system:
by using the invertible change of variables: \(u=\bar{u}\), \(h=\bar{h}+ \gamma \bar{u}_{x}\). For \(\alpha =\frac{1}{3}\), \(\beta =0\), system (2) was given by Broer [1]. For \(\beta =0\), in terms of the potential \(\varphi :u=\varphi _{x}\), system (2) was derived by Kaup [3], who found its multisoliton solutions. Matveev and Yavor [4] algebro-geometrically found a large class of almost periodic solutions of system (3). In the paper, we want to apply the Lie group analysis method [5] to study the point symmetries and Lie–Bäcklund transformation symmetries of system (2). In fact, many symmetries, similarity reductions, and conservation laws were obtained by Lie group analysis [6,7,8,9,10]. Lou et al. [11, 12] applied the symmetry group method to study some coherent solutions of nonlocal KdV systems and primary branch solutions of a first-order autonomous system. In addition, Ma [13] obtained some new conservation laws of some discrete evolution equations by symmetries and adjoint symmetries. Qu and Ji [14] studied inhomogeneous nonlinear diffusion equations by invariant subspace and conditional Lie–Backlund symmetry methods. It was shown that the equations admit a class of invariant subspaces governed by the nonlinear ordinary differential equations, which is equivalent to a kind of higher-order conditional Lie–Backlund symmetries of the equations. Ji and Qu [15] used the conditional Lie–Backlund symmetry method to study the invariant subspaces of nonlinear diffusion equations with convection and source terms and obtained a complete list of canonical forms for such equations, which admit higher-order conditional Lie–Backlund symmetries and multidimensional invariant subspaces. Ma [16, 17] discussed the conservation laws of differential and discrete equations, respectively. Recent studies by Ma et al. [18,19,20,21,22] also show a remarkable richness of rational function solutions, called lumps, as well as interaction solutions and solutions of other kinds. In addition, the invariant solutions can be formulated from the invariant submanifold method in [23].
Ibragimov and Avdonina [24] applied the Lie group method to propose a new approach for looking for conservation laws and exact solutions to nonlinear self-adjoint differential equations. For a system of m differential equations
where \(u_{(1)}=\{u_{i}^{\alpha }\}, \ldots , u_{(s)}=\{u^{\alpha }_{i _{1}\cdots i_{s}}\}\), the steps are as follows.
Step 1: Introducing the adjoint equations of (4) by using the variational derivative:
where \(\varphi =\sum_{\beta =1}^{m}v^{\beta }F_{\beta }(x,u,u _{(1)},\ldots , u_{(s)})\).
Step 2: Let
and require the following relations to be satisfied:
We say that system (4) is nonlinearly self-adjoint if (7) holds for the solutions of (4), where \(\lambda _{\alpha }^{\beta }\) are functions dependent on \(x,u,u_{(1)},\ldots \) .
Step 3: Assume that the infinitesimal symmetry of the nonlinear self-adjoint system (4) is given by
Then a conservation law for the system is presented by
where the components of the conserved vector are the following:
where \(W^{\alpha }=\eta ^{\alpha }-\xi ^{j}u_{j}^{\alpha }\).
By applying (9) and (10) some conservation laws of some nonlinear self-adjoint differential equations were obtained in [5,6,7,8,9]. In addition, Göktas and Hereman [25] proposed a new method for looking for conservation laws of nonlinear differential equations without using the symmetries of differential equations. The explicit steps are as follows:
(1) Consider the form of conservation laws
for system (4). Assuming the uniformity in rank in the ith equation, form the linear system
and then gather the \(A_{i}\) to form the global linear system \(\mathcal{A}=\bigcup_{i=1}^{\alpha }A_{i}\).
(2) Solving for the unknown weights \(w(u_{i})\), \(w(\partial _{t})\).
(3) Set \(\mathcal{V}=\{v_{1},\ldots , v_{Q}\}\) to be the sorted list of all the variables with positive weights, excluding \(\partial _{t}\). Form all monomials of rank R or less by taking combinations of the variables in \(\mathcal{V}\) and form sets consisting of ordered pairs.
Set \(\mathcal{B}_{0}=\{(1;0)\}\). For \(q=1, 2,\ldots , Q\), \(m=0,1,\ldots , M-1\), where M is the number of pairs in \(\mathcal{B}_{q-1}\), form \(\mathcal{B}_{q,m}=\bigcup_{i=1}^{p_{q,m}}\{(T_{q,s};W_{q,s})\}\), \(T _{q,s}=T_{q-1,m}v_{q}^{s}\), \(W_{q,s}=W_{q-1,m}+sw(v_{q})\), \(p_{q,m}=[\!\![\frac{R-W _{q-1,m}}{w(v_{q})}]\!\!]\).
Denote \(\mathcal{B}_{q}=\bigcup_{m=0}^{M-1}\mathcal{B}_{q,m}\).
(4) Let \(\mathcal{G}=\mathcal{B}_{Q}\), which consists of all possible combinations of powers of the variables that produce rank R or less. For each pair \((T_{Q,s};W_{q,s})\) in \(\mathcal{G}\), apply \(\frac{ \partial ^{l}}{\partial x^{l}}\) to the term \(T_{Q,s}\), where \(l=R-W_{Q,s}\). Set \(\mathcal{H}\) to contain the terms that result from computing the various \(\frac{\partial ^{l}}{\partial x^{l}}(T_{Q,s})\).
(5) Removing those terms in \(\mathcal{H}\) that can be written as a total derivative with respect to x,or as a derivative up to terms kept previously in the set, we denote such a set by I.
(6) For all terms from I with desired rank R, let
where \(I(i)\) is the ith element in I, σ is the number of the terms in I, and \(c_{i}\) are constants to be determined later.
(7) Computing \(D_{t}(\rho )\) and replacing the terms with derivatives with respect to \(\partial _{t}\) by using the original system (4), we denote the result by E. Then act on E by the variational derivatives \(\frac{\delta E}{\delta u^{\alpha }}\) (\(\alpha =1,\ldots ,m\)). Next, solving the system \(\frac{\delta E}{\delta u^{\alpha }}=0\) (\(\alpha =1, \ldots ,m\)), we obtain the relations among the coefficients \(c_{i}\). Thus we get the density ρ in (12). Substituting \(c_{i}\) into the expression E, we can obtain the resulting fluxes of differential equations.
In the paper, we apply the Lie group analysis method and the above approach to investigate the conservation laws of system (2).
2 Point symmetries, Lie–Bäcklund symmetries, and conservation laws
By using the Lie-group analysis method we easily get the point symmetry of the system (2):
Set
Then the adjoint equations of system (2) are given by
where u, h are solutions to system (2). System (14) has the solutions
Hence system (2) is strictly self-adjoint. Besides, system (14) has some special solutions for given u and h. For example, when \(u=h\), (14) has the solution
where \(c_{1}\), \(c_{2}\), \(c_{3}\) are constants. In particular, for \(\beta =0\), system (2) reduces to
Obviously, (15) is solvable. Since system (2) is nonlinearly self-adjoint, we can look for the conservation laws by using the Lie group method. For system (2), the conservation laws are of the following form:
For the vector \(V_{3}\), the conservation law is
Note that \(W^{u}=-1-tu_{t}\) and \(W^{h}=-th_{t}\). Then we have
To cancel the trivial operation in computing conservation laws, we may assume that
Then the conserved vector \(C=(X^{1},C^{2},\ldots , c^{m})=0\) can be written as
with the components
Based on versions (18)–(21), we get the reduced forms of the components of the conserved density:
In particular, when \(u=h\), \(p=-c_{1}x+c_{2}\), and \(q=c_{1}(x-t)+c_{3}\), we can obtain the special components of the conserved density:
For the symmetry vector \(V_{4}\), we have that
Similarly as before, we can obtain the components of the density of system (2):
In what follows, we investigate the Lie–Bäcklund symmetries of system (2) and the resulting conservation laws.
Set
Substituting (22) into system (2), we infer the following Lie-Bäcklund symmetries by using the software Maple:
Applying (10), we can deduce the components of the conserved density for system (2). For \(X_{1}\), we have
For \(X_{2}\), \(W^{u}=\eta ^{u}=1+tu_{x}\) and \(W^{h}=\eta ^{h}=h_{x}\); substituting into (16) and (17), we have that
Similarly, for \(X_{3}\), we get that
For \(X_{4}\), we have
For \(X_{5}\), we have
For \(X_{6}\), we infer that
where \(p=h\) and \(q=u\), which can substituted into the above conserved densities so that we obtain more explicit formulas for \(c^{1}\) and \(C^{2}\). Here we omit the computations.
3 Lie symmetry groups and similarity solutions
In this section, we apply the point symmetries (13) to consider the Lie symmetry groups and some similarity solutions to system (2). Denote the Lie symmetry groups generated by \(V_{1}\), \(V_{2}\), \(V_{3}\), \(V_{4}\) by \(g_{1}\), \(g_{2}\), \(g_{3}\), \(g_{4}\), respectively. It is easy to see that
If \(u=f(x,t)\), \(h=g(x,t)\) are solutions to e system (2), then we can get the following new solutions based on these symmetry groups:
Of course, we can go on getting some iteration solutions following the work [8]:
where \(n\in N^{+}\).
For the transformation groups \(g_{1}\) and \(g_{2}\), the invariant solutions are traveling wave solutions. Indeed, set
Substituting (23) into system (2), we have
Integrating gives rise to
From (24) we find that
Let \(U^{\prime }=y(\xi )\). Then (25) turns to
Suppose \(U^{\prime }=aU+U^{2}\). Then inserting this into (26), we have that
Thus we get
Solving (27) yields
and hence
where c̄ is an integral constant, which does not vanish. Again applying (23) and (28)–(29), we have
The characteristic equation of the vector field \(V_{3}\) presents
which gives
The resulting group-invariant solution reads
Substituting (30) into system (2), we infer
which has the following set of solutions:s
where c̃, \(c_{0}\) are constants. Substituting (31) into (30), we obtain the similarity solutions of system (2):
Similarly, for \(g_{4}\), the characteristic equation reads
from which we get the invariants
where \(f(\xi )\), \(g(\xi )\) are arbitrary invariant functions with respect to the variable ξ. Inserting (32) into the system, we get the following ordinary differential equations with variable coefficients:
Now we look for the formal series solutions to (33), so we assume that
Substituting (34) into system (2) and comparing the coefficients of both sides in the system (33), one infers that
When \(a_{1}\neq 0\), there exists a constraint between α and β:
In terms of the above relations among \(a_{i}\), \(b_{i}\) (\(i=0,1,2,\ldots ,n\)), we can write out the formal series solutions where \(a_{1}\) is a free parameter. As for the convergence of the series, we skip the discussion. However, we can follow the way presented in [8] to proceed.
4 A new scheme for seeking conservation laws of system (2)
In this section, we adopt the approach given by Göktas and Hereman to investigate the conserved densities and the fluxes of the system for the given rank of ever term in the equations. We rewrite the system (2) as follows
where \(u_{i,nx}\) denotes the nth derivative with respect to x for the variables \(u_{i}\) (\(i=1,2\)). We denote the weight of the variables \(u_{i}\) and derivative \(\partial _{t}\) by \(w(u_{i})\) and \(w(\partial _{t})\), respectively. We easily find that
Solving
we get
List of the variables in system (2):
In what follows, we consider the case where \(\operatorname{rank}=4\) and
For \(q=1\) and \(m=0\), a direct calculation gives
Hence \(s=0,1,2\), and
For \(q=2\) and \(m=0\), we have
Hence \(s=0,1,2,3,4\), and
For \(q=2\) and \(m=1\), we have that
Thus \(s=0,1,2\). It follows that
For \(q=2\) and \(m=2\), we find that
Hence \(s=0\). Thus we get
Computation of l for each pair of \(\mathcal{B}_{2}\) reads
Gathering the terms by applying the number l of partial derivatives with respect to x, we get that
Removing from \(\mathcal{H}\) the constant terms and the terms that can be written as an x-derivative, or an x-derivative up to terms retained earlier in the set \(\mathcal{H}\), we have that
Let
where \(c_{i}\) (\(i=1,2,3,4\)) are constants to be determined. Then we have
Substituting (35) into (36) to cancel the terms with t-derivatives, we get
Acting on the variational derivative to \(D_{t}(\rho )|_{\text{(35)}}\) and comparing the coefficients of the same terms, we get the following relations:
Solving S gives that
where \(c_{4}\) is a free parameter, and \(c_{2}\) can be an arbitrary constant.
Suppose \(c_{4}=1\) and \(c_{2}=0\). Then \(c_{1}=-\alpha \), \(c_{3}=-2\beta \), \(c_{5}=1\). Thus, we have the conserved density:
The resulting flux of system (2) when \(\operatorname{rank}=4\) is given by
If \(c_{1}=c_{3}=c_{4}=c_{5}=0\) and \(c_{2}\neq 0\), then the conserved density is of the form
whereas
Hence the flux has nonlocal differential form
When \(\operatorname{rank}=5,6,\ldots \) , we can obtain the resulting conserved densities and fluxes of system (2), and we omit the complicated computations.
5 Special solutions of the stationary system (2)
An important application of the conservation laws of evolution equations is the study of nonvariant solutions of symmetry groups. In this section, we want to apply the conservation laws of system (2) to investigating some noninvariant solutions of the symmetries. For simplicity, we only choose simpler conservation laws and only consider the solutions of the stationary system (2). System (2) can be written as
where the conserved densities are u and h, and the corresponding fluxes are \(\frac{1}{2}u^{2}+h+\beta u_{x}\) and \(uh+\alpha u_{xx}- \beta h_{x}\), which are the simplest conservation laws. We further want to use them to deduce some stationary solutions of system (2). Taking
and substituting into the second conserved equation in (38), we have that
Assume that \(f(t)=0\) and then integrate (39):
Case 1: When \(\alpha >0\), \(\beta =0\), Eq. (39) reduces to
which has a special solution
Thus we obtain a set of special solutions to system (2):
Case 2: When \(\beta \neq 0 \), assume a formal solution of (40)
Then we get
Inserting (42) and (43) into (40) and comparing the coefficients at the powers of x, we infer that
Taking \(B=1\), we have the special solution to system (2)
where \(C\neq 0\) is a parameter.
In the case \(f(t)=0\), we find that \(u_{t}=h_{t}=0\). Hence the system becomes the following ordinary differential equations:
which can be written as
where \(c_{1}\), \(c_{2}\) are integral constants. It is easy to rewrite (46) as the following differential equation with respect to the variable u:
When \(c_{1}=c_{2}=0\), solving (47) yields
Substituting (48) into the first equation in (46), we get
When \(c_{1}\neq 0\) and \(c_{2}\neq 0\), (47) becomes
Let \(u_{x}=y\). Then (50) can be written as
Integrating (5) with respect to u, we get
where \(c_{0}\) is an integral constant. Thus (52) can be written as
Fan [26] studied the solutions to the formal ODE (53) taking different parameters. Now we follow his method to give some exact solutions to (53).
(i) When \(c_{0}=0\), (53) has the following solutions:
(ii) Equation (53) still has the following three Jacobi elliptic function solutions: when \(\frac{c_{1}}{\alpha +\beta ^{2}}>0\), \(c_{0}=\frac{2c_{1}^{2}m ^{2}(m^{2}-1)}{(\alpha +\beta ^{2})(2m^{2}-1)^{2}} \),
when \(\frac{c_{1}}{\alpha +\beta ^{2}}<0\), \(c_{0}=\frac{2c_{1}^{2}m ^{2}}{(\alpha +\beta ^{2})(m^{2}+1)} \),
when \(\frac{c_{1}}{\alpha +\beta ^{2}}>0\), \(c_{0}=\frac{2c_{1}^{2}(1-m ^{2})}{(\alpha +\beta ^{2})(2-m^{2})^{2}} \),
where m denotes the module of Jacobi elliptic functions. Substituting all these u into (46), we can obtain the resulting h.
From the above discussion we find that we indeed obtain some new special solutions of system (2). However, there are two questions to further consider. One is that when \(f(t)\neq 0\), can we obtain solutions dependent on the variable t? Of course, we can. Because we may investigate the traveling-wave solutions of system (2) by setting \(\xi =x-ct\), which can transform system (2) to the ODEs with respect to the variable ξ. As for this, we do not further discuss them. Another one is that when \(f(t)=0\), can we obtain solutions only dependent on the variable x and not on the variable t? We do not think so. In fact, if we replace system (2) with the system
then we can get a set of solutions dependent on the variable t. In fact, taking \(u_{t}=g(x)\), \(\frac{1}{2}u^{2}+h+\beta h_{x}=0\), we have
Substituting this into the second equation in (54) gives rise to
where \(G(x,t)=hg(x)+\alpha g_{xx}-\beta h_{x}\), which has the solution
from which we have
where \(\sigma (t)\) is a function to be determined. A direct verification indicates that
is a set of solutions of (54). We see that the function h is dependent on the variable t.
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This work is supported by the Fundamental Research Funds for the Central University (No. 2017XKZD11).
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The idea to deduce point symmetry and Lie–Bäcklund transformations of the system similarity solutions was finished by YZ, and conservation laws were done by NB and YZ. Some exact solutions of the stationary system (2) belong to HG. All authors read and approved the final manuscript.
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Zhang, Y., Bai, N. & Guan, H. Some symmetries, similarity solutions and various conservation laws of a type of dispersive water waves. Adv Differ Equ 2019, 435 (2019). https://doi.org/10.1186/s13662-019-2373-0
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DOI: https://doi.org/10.1186/s13662-019-2373-0
PACS Codes
- 05.45.Yv
- 02.30.Jr
- 02.30.Ik
Keywords
- Symmetry
- Similarity solution
- Conservation law