Project supported by the National Natural Science Foundation of China (Grant Nos. 11371293 and 11505090), the Natural Science Foundation of Shaanxi Province, China (Grant No. 2014JM2-1009), the Research Award Foundation for Outstanding Young Scientists of Shandong Province, China (Grant No. BS2015SF009), and the Science and Technology Innovation Foundation of Xi’an, China (Grant No. CYX1531WL41).
Abstract
We explore the (2+1)-dimensional dispersive long-wave (DLW) system. From the standard truncated Painlevé expansion, the Bäcklund transformation (BT) and residual symmetries of this system are derived. The introduction to an appropriate auxiliary dependent variable successfully localizes the residual symmetries to Lie point symmetries. In particular, it is verified that the (2+1)-dimensional DLW system is consistent Riccati expansion (CRE) solvable. If the special form of (CRE)-consistent tanh-function expansion (CTE) is taken, the soliton-cnoidal wave solutions and corresponding images can be explicitly given. Furthermore, the conservation laws of the DLW system are investigated with symmetries and Ibragimov theorem.
It is well known that soliton theory, an important topic in nonlinear science, has been studied thoroughly.[1–5] Many measures can be taken to describe nonlinear phenomena, of which to construct exact solution including the soliton and different wave solutions to an integrable system is an effective one. Symmetry and Painlevé analyses play important roles in deriving the exact solutions.[6–8] Obtained from the classical or nonclassical Lie group method, Lie point symmetries of a differential system can reduce the dimensions of the partial differential equations (PDEs) and help construct group invariant solutions by similarity reductions. Nevertheless, the nonlocal symmetries of the integrable system can be acquired through inverse recursion operators,[9,10] Darboux transformation (DT),[11,12] Bäcklund transformation (BT),[13,14] conformal invariance,[15] negative hierarchies,[16] and so on.
Recently, Lou and his colleagues[17,18] found that Painlevé analysis can also be applicable in acquiring nonlocal symmetries, also known as residual symmetries, since the symmetries correspond to the residues with respect to the singular manifold of the truncated Painlevé expansion. However, the nonlocal symmetries cannot be directly used to establish explicit solutions to differential equations, so the transformation of the nonlocal symmetries into local ones is needed with the help of suitable prolonged systems.[18,19] In Ref. [18], the residual symmetries are localized and the related finite transformation was found by Lou. Furthermore, advancing the truncated Painlevé expansion, Lou introduced the definition of consistent Riccati expansion (CRE) solvable,[20] which is of great efficiency in constructing interaction solutions and possible new integrable systems. In Ref. [21], a consistent tanh expansion (CTE) was proposed to identify CTE solvable, a special simplified form of the CRE.
On the other hand, conservation laws, essential in the study of differential equations mathematically and physically, propose one of the primary principles to formulate and investigate models, especially in existence, uniqueness, and stability of the solutions. In addition, the integrability of the system is quite possible if conservation laws exist.[22] For conservation laws, different methods have been mobilized. The celebrated Noether’s theorem[23] proves to be a systematic and efficient approach in finding conservation laws of PDEs unless there exists a Lagrangian. However, there exist some equations not having a Lagrangian. Hence the Noether theorem cannot be used to obtain conservation laws directly because of the equation symmetries. This, however, can be solved with the general concept of nonlinear self-adjointness proposed by Ibragimov[24,25] and Gandarias[26] to construct the conservation laws for any differential equation. This procedure can be true of classes of single differential equations of any order but of the systems where the number of equations is equal to that of dependent variables.[27]
This paper concentrates on investigating the residual symmetries, CRE solvable, interaction solution, and conservation laws of the (2+1)-dimensional dispersive long-wave (DLW) system
(1)
System (1), modeling nonlinear and dispersive long gravity waves in two horizontal directions on shallow waters of uniform depth, was initially proposed by Boiti et al.[28] By now, system (1) has been extensively studied by many professionals. Reference [29] has presented some new soliton-like solutions to system (1). Reference [30] provided the rational series solutions and multi solitary wave solutions to system (1), while in Refs. [31] and [32], with the extended mapping method and variable separation approach, respectively, new variable separation excitations with arbitrary function and numerous localized coherent structures such as multi-ring solitons, dromion, breathers, and instantons of system (1) have been obtained. In Ref. [33], the generalized singular manifold approach yields the Darboux transformations and Lax pair of systems (1). In Ref. [34], two kinds of new multiple soliton solutions have been constructed through the Bäcklund transformation and Hirota bilinear method, and the fusion and fission interaction phenomena among the different localized structures have been used to investigate into system (1). However, we can see that the residual symmetry, soliton-cnodial wave solution, and conservation laws of system (1) have not been studied yet in the above literature.
This paper is organized as follows. Section 2 focuses on the non-auto Bäcklund transformation theorem and residual symmetry of the (2+1)-dimensional DLW system by using the truncated Painlevé expansion approach. Then extending the original system makes the residual symmetry localized. Consequently, the corresponding finite transformation group can be obtained with Lie’s first theorem. Section 3 verifies that the (2+1)-dimensional DLW system is CRE solvable and leads to new interactions between solitons and cnoidal periodic waves. In Section 4, the conservation laws of the (2+1)-dimensional DLW system are derived from the Ibragimov theorem. In the last section, some conclusions and a discussion are presented.
2. The residual symmetry of DLW system and its localization
For the (2+1)-dimensional DLW system, its truncated Painlevé expansion can be expressed as
(2)
with being the functions of x, y, and t. Substituting Eq. (2) into system (1) and eliminating all the coefficients of different powers of , we have
(3)
and the field ϕ satisfies the following Schwarzian form:
(4)
where k is an arbitrary integral parameter,
The Schwarzian form (4) is invariant under the Möbious transformation
That is to say, equation (4) bears three symmetries , , and with arbitrary constants , and .
After substituting expression (2) into system (1), we have the following theorem.
3. CRE solvability and interaction solutions to DLW system
3.1. CRE solvability
The leading order analysis lays the basis for the acquisition of the following truncated Painlevé expansion solution to the DLW system:
(12)
in which is a solution to the Riccati equation
(13)
with and being arbitrary constants. Substituting Eqs. (12) and (13) into Eq. (1) and then vanishing all the coefficients of the like powers of , we obtain
(14)
where the function w only needs to satisfy
(15)
with and .
From the above, it can be concluded that the (2+1)-dimensional DLW system is CRE solvable since it has the truncated Painlevé expansion solution related to the Riccati equation (13), and the following theorem can be established.
3.2. CTE solvable and interaction solutions to the DLW system
It is known that the Riccati equation (13) has a special solution when , which substantiates that the CRE solvable system is CTE solvable, and vice versa. Then the following CTE solvable theorem can be acquired.
Theorem 5 shows that solving the w equation (15) can result in various interaction solutions between solitons and other nonlinear excitations, which posess a form[36]
(18)
where , and satisfies
(19)
with being constants. Substituting Eqs. (18) and (19) into Eq. (15), with the help of software Maple, we obtain
(20)
while all the other constants , and remain free.
Obviously, the general solution to Eq. (19) can be expressed in terms of Jacobi elliptic functions. Here, just take w in the following special Jacobi elliptic function
(21)
as the solution to Eq. (15), which characterizes the interactions between a soliton and a cnoidal wave. Here, sn is the usual Jacobi elliptic sine function and
(22)
is the third type of incomplete elliptic integral. Substituting Eq. (21) into Eq. (15) and solving the over-determined equations with the help of maple will yield
(23)
(24)
(25)
where
(26)
where , and m in Eq. (23) are arbitrary constants, and .
When equation (23) is substituted into expression (21), then equation (21) will become
(27)
where
is the first type of incomplete elliptic integral. Substituting Eqs. (14) and (27) into Eq. (17), we can obtain the interaction solution between soliton and cnoidal periodic waves. The result is omitted here because of its prolixity. The corresponding images are shown in Figs. 1 and 2, and the parameters used in the figure are selected as .
Fig. 1. (color online) The soliton-cnodial periodic wave solutions to u: (a) the profile of the special structure with t = 0 and y = 0, (b) the profile of the special structure with t = 0 and x = 0, (c) perspective view of the wave.
Fig. 2. (color online) The soliton-cnodial periodic wave solutions to v: (a) the profile of the special structure with t = 0 and y = 0, (b) the profile of the special structure at t = 0 and x = 0, (c) perspective view of the wave.
When equation (24) or (25) with and is substituted into expression (21), then equation (21) becomes
(28)
Substituting Eqs. (14) and (28) into Eq. (17) will lead to the following two-soliton solutions of system (1):
(29)
(30)
where
Since the solutions are complex numbers, the figures drawn here are modules to them (Fig. 3), and the parameters used in the figure are selected as .
Fig. 3. (color online) The two soliton-solution structure of (a) u and (b) v determined by Eqs. (29) and (30) for the DLW system at t = 0.
4. Conservation laws for DLW system
In this section, we consider the conservation laws of the DLW system (1) by Ibragimov’s theorem with the introduction of some notations and theorems.
We utilize Theorem 6 to construct the conservation law for system (1). For system (1), we introduce the Lagrangian in the following symmetrized form
(36)
where are new dependent variables, and the adjoint equations of system (1) from definition 1 are
(37)
Based on Theorem 6, for a general vector field
(38)
the conservation law is under the control of
(39)
Here the conserved vector of Eq. (35) will change into the following concrete forms
(40)
By substituting Eq. (36) into Eq. (40), it will change into
with
The following symmetries of system (1) are obtained from the classical Lie group theory[35]
(41)
(42)
(43)
where is an arbitrary function of of t.
To proceed, we consider the following cases.
Conservation laws are mathematical expressions of physical laws, such as conservation of energy, mass, and momentum. Not all of the conservation laws of a PDE have physical interpretations but the existence of a great number of conservation laws is a strong indication of its integrability. The celebrated Noether theorem provides us with a corresponding relationship between symmetry and conservation law; she pointed out that each kind of symmetry corresponds to a conservation law, such as the invariance of the spatial transformation ensures the conservation of momentum and the invariance of the temporal transformation guarantees the energy conservation, and vice versa. Therefore, it is of great significance to find infinite conservation laws for a soliton system.
5. Discussion and summary
With the standard truncated Painlevé expansion, we explore the non-auto Bäcklund transformation, residual symmetry, and interaction solution between soliton-cnoidal periodic waves of the (2+1)-dimensional DLW system (1). Nonlocal in the original nonlinear system, the residual symmetries are localized with a properly auxiliary dependent variable introduced. Besides, solving the standard Lie’s initial value problem leads to the finite transformation of the residual symmetry. Moreover, the DLW system is proved to be CRE solvable, and also CTE solvable in a special case. In the CTE, abundant soliton and cnoidal periodic wave solutions can be given by the Jacobi elliptic functions. Furthermore, countless conservation laws of system (1) have been obtained from Ibragimov’s new conservation laws. The basic conserved quantity can be applied in obtaining various estimates for smooth solutions and defining suitable norms for weak solutions. So it is worth being further investigated.
A good understanding of the solutions to system (1) is very helpful for coastal and civil engineers by applying the nonlinear water model to coastal harbor design. One can also consider the relationship between residual symmetry and other nonlocal symmetries. The research into the CTE method and more types of interaction solutions to different kinds of nonlinear excitations should be pushed forward.
Reference
[1]
HasegawaA1989Optical Solitons in FibersBerlinSpringer-Verlag
