 ORIGINAL ARTICLE
 Open Access
Selfadaptive moving mesh schemes for short pulse type equations and their Lax pairs
 BaoFeng Feng^{1},
 Kenichi Maruno^{1, 2}Email author and
 Yasuhiro Ohta^{3}
https://doi.org/10.1186/s4073601400087
© Feng et al.; Licensee Springer 2014
 Received: 10 March 2014
 Accepted: 14 March 2014
 Published: 12 September 2014
Abstract
Integrable selfadaptive moving mesh schemes for short pulse type equations (the short pulse equation, the coupled short pulse equation, and the complex short pulse equation) are investigated. Two systematic methods, one is based on bilinear equations and another is based on Lax pairs, are shown. Selfadaptive moving mesh schemes consist of two semidiscrete equations in which the time is continuous and the space is discrete. In selfadaptive moving mesh schemes, one of two equations is an evolution equation of mesh intervals which is deeply related to a discrete analogue of a reciprocal (hodograph) transformation. An evolution equations of mesh intervals is a discrete analogue of a conservation law of an original equation, and a set of mesh intervals corresponds to a conserved density which play an important role in generation of adaptive moving mesh. Lax pairs of selfadaptive moving mesh schemes for short pulse type equations are obtained by discretization of Lax pairs of short pulse type equations, thus the existence of Lax pairs guarantees the integrability of selfadaptive moving mesh schemes for short pulse type equations. It is also shown that selfadaptive moving mesh schemes for short pulse type equations provide good numerical results by using standard timemarching methods such as the improved Euler’s method.
Keywords
 Solitons
 Selfadaptive moving mesh schemes
 Short pulse equation
Background
The studies of discrete integrable systems were initiated in the middle of 1970s. Hirota discretized various soliton equations such as the KdV, the mKdV, and the sineGordon equations based on the bilinear equations [16–20], Ablowitz and Ladik proposed a method of integrable discretizations of soliton equations, including the nonlinear Schrödinger equation and the modified KdV (mKdV) equation, based on the AblowitzKaupNewellSegur (AKNS) form [1–5]. Following the pioneering works of Hirota and AblowitzLadik, the studies of discrete integrable systems have been expanded in diverse areas (see, for example, [6,7,15,40]).
It is known that there is a class of soliton equations which are derived from the WadatiKonnoIchikawa (WKI) type 2×2 linear system [5,27,43]. Soliton equations in the WKI class are transformed to certain soliton equations which are derived from the AKNS type 2×2 linear system through reciprocal (hodograph) transformations [5,22,23,36,44].
Integrable discretization of soliton equations in the WKI class had been regarded as a difficult problem until recently. A systematic treatment of reciprocal (hodograph) transformations in integrable discretizations had been unknown for three decades. Recently, the present authors proposed integrable discretizations of some soliton equations in the WKI class by using the bilinear method, and it was confirmed that those integrable discrete equations work effectively on numerical computations of the above class of soliton equations as selfadaptive moving mesh schemes [11–14,34,35]. However, the method employed in our previous papers was rather technical, thus it is not easy to extract a fundamental structure of discretizations to apply this method to a broader class of nonlinear wave equations including nonintegrable systems.
The aim of this article is to present two systematic methods (in sophisticated forms), one is based on bilinear equations (this method is regarded as an extension of Hirota’s discretization method) and another is based on Lax pairs (this method is regarded as an extension of AblowitzLadik’s discretization method), to construct selfadaptive moving mesh schemes for soliton equations in the WKI class. We demonstrate how to construct selfadaptive moving mesh schemes for short pulse type equations whose Lax pairs are written in the WKI type form which is transformed into the AblowitzKaupNewellSegur (AKNS) type form by reciprocal (hodograph) transformations. We clarify that moving mesh is generated by following discrete conservation law and mesh intervals are nothing but discrete conserved densities which is a key of selfadaptive moving mesh schemes. Lax pairs of selfadaptive moving mesh schemes for short pulse type equations are constructed by discretization of Lax pairs of short pulse type equations. It is also shown that selfadaptive moving mesh schemes for short pulse type equations provide good numerical results by using standard timemarching methods such as the improved Euler’s method.
This can be rewritten as eq. (1). Thus the SP equation is equivalent to the CD system with the reciprocal (hodograph) transformation. As we mentioned in our previous paper, the reciprocal (hodograph) transformation between the CD system and the SP equation is nothing but the transformation between the Lagrangian coordinate and the Eulerian coordinate [11].
by replacing λ by iλ. This is nothing but the Lax pair of the SP equation. Note that the Lax pair of the SP equation is of the WKI type [38]. In general, soliton equations derived from WKItype eigenvalue problems are transformed into soliton equations derived from AKNStype eigenvalue problems by reciprocal (hodograph) transformations [22,23,36,44].
A selfadaptive moving mesh scheme for the SP equation and its Lax pair

Step 1: Transform the SP equation (1) into the CD system (2) and (3) by the reciprocal (hodograph) transformation (4).

Step 2: Transform the CD system into the bilinear equations.

Step 3: Discretize the bilinear equations of the CD system.

Step 4: Transform the (semi)discrete bilinear equations into the (semi)discrete CD system.

Step 5: Discretize the reciprocal (hodograph) transformation and transform the (semi)discrete CD system via the discrete reciprocal (hodograph) transformation.

Step 1: Transform the Lax pair of the SP equation (1) by the reciprocal (hodograph) transformation (4). The Lax pair obtained by the reciprocal (hodograph) transformation is the one of the CD system (2) and (3).

Step 2: Discretize the Lax pair of the CD system. The compatibility condition of the discretized Lax pair yields the (semi)discrete CD system.

Step 3: Discretize the reciprocal (hodograph) transformation and transform the discretized Lax pair of the (semi)discrete CD system via the discrete reciprocal (hodograph) transformation.

Step 4: The compatibility condition of the discretized Lax pair obtained in Step 3 yields the (semi)discrete SP equation.
Since the SP equation (1) is equivalent to the CD system (2) and (3) with the reciprocal (hodograph) transformation (4), the semidiscrete CD system with the discrete reciprocal (hodograph) transformation is equivalent to the semidiscrete SP equation.
 Step 1:
The SP equation (1) is transformed into the CD system (2) and (3) via the reciprocal (hodograph) transformation (4).
 Step 2:The CD system (2) and (3) can be transformed into the bilinear equations$$ {D_{T}^{2}} f \cdot f =\frac{1}{2} g^{2}\,, $$(15)$$ D_{X}D_{T} f \cdot g =fg\,, $$(16)via the dependent variable transformation$$ u=\frac{g}{f}\,, \quad \rho=12 (\ln f)_{XT} \,. $$(17)Here D_{ X } and D_{ T } are Hirota’s Doperators defined as$${D_{X}^{m}}f\cdot g=(\partial_{X}\partial_{X^{\prime}})^{m}f(X)g(X^{\prime})_{X^{\prime}=X}\,. $$
 Step 3:Discretize the space variable X in the bilinear equations (15) and (16).$$ {D_{T}^{2}} f_{k} \cdot f_{k} =\frac{1}{2} {g_{k}^{2}}\,, $$(18)$$ \begin{aligned} &\frac{1}{a}D_{T} \left(\,f_{k+1} \cdot g_{k}f_{k}\cdot g_{k+1}\right)\\ &\qquad\quad=\frac{1}{2}\left(\,f_{k+1}g_{k}+f_{k}g_{k+1}\right)\,. \end{aligned} $$(19)
 Step 4:Consider the dependent variable transformation$$ u_{k}=\frac{g_{k}}{f_{k}}\,, \quad \rho_{k}=1\frac{2}{a} \left(\ln \frac{f_{k+1}}{f_{k}}\right)_{T} \,, $$(20)which is a discrete analogue of (17). Then the bilinear equations (18) and (19) are transformed into$$ \partial_{T} \rho_{k}\frac{\left(\frac{u_{k+1}^{2}}{2}\right) \left(\frac{{u_{k}^{2}}}{2}\right)}{a}=0\,, $$(21)$$ \partial_{T} \left(\frac{u_{k+1}u_{k}}{a}\right)=\rho_{k} \frac{u_{k+1}+u_{k}}{2}\,, $$(22)
which is a semidiscrete analogue of the CD system.
 Step 5:Consider a discrete analogue of the reciprocal (hodograph) transformation$$ x_{k}={X}_{0}+\sum\limits_{j=0}^{k1}a\rho_{j}\,, $$(23)where x_{0}=X_{0}. Now we introduce the mesh interval$$ \delta_{k}=x_{k+1}x_{k}\,. $$(24)Note that the mesh interval satisfies the relation$$ \delta_{k}=a\rho_{k}\,, $$(25)so we can rewrite equations (21) and (22) with the discrete reciprocal (hodograph) transformation (23) into the selfadaptive moving mesh scheme for the SP equation$$ \partial_{T} \delta_{k}=\frac{u_{k+1}^{2}+{u_{k}^{2}}}{2}\,, $$(26)$$ \partial_{T}(u_{k+1}u_{k})=\delta_{k} \frac{u_{k+1}+u_{k}}{2}\,, $$(27)where δ_{ k } is related to x_{ k } by δ_{ k }=x_{k+1}−x_{ k } which originates from the discrete reciprocal (hodograph) transformation$$ x_{k}={X_{0}}+\sum\limits_{j=0}^{k1}\delta_{j}\,. $$(28)
The set of points {(x_{ k },u_{ k })}_{k=0,1,⋯} provides a solution of the semidiscrete SP equation. Note that the above discrete reciprocal (hodograph) transformation can be interpreted as the transformation between Eulerian description and Lagrangian description in a discretized space [11].
The discrete reciprocal (hodograph) transformation (28) yields$$ \begin{aligned} \frac{\Delta}{\Delta X_{k}}&=\frac{\Delta}{a}= \frac{\Delta x_{k}}{a}\frac{\Delta}{\Delta x_{k}}=\rho_{k}\frac{\Delta}{\Delta x_{k}} =\rho_{k}\frac{\Delta}{\delta_{k}}\,,\\ \frac{\partial}{\partial T}&=\frac{\partial}{\partial t} +\frac{\partial x_{k}}{\partial T}\frac{\partial}{\partial x_{k}} =\frac{\partial}{\partial t} +\sum\limits_{j=0}^{k1}\frac{\partial \delta_{j}}{\partial T}\frac{\partial}{\partial x_{k}}\\ &=\frac{\partial}{\partial t} +\left(\sum\limits_{j=0}^{k1}\frac{u_{j+1}^{2}+{u_{j}^{2}}}{2}\right) \frac{\partial}{\partial x_{k}}\\ &=\frac{\partial}{\partial t} +\left(\frac{u_{k+1}^{2}+{u_{0}^{2}}}{2}\right) \frac{\partial}{\partial x_{k}}\\[2pt] \end{aligned} $$(29)$$\hspace*{24.5pt} =\frac{\partial}{\partial t} +\left(\frac{u_{k+1}^{2}}{2}\right) \frac{\partial}{\partial x_{k}}\,, \quad \text{if}\,\, u_{0}=0\,, $$(30)where Δ is a difference operator defined as Δf_{ k }≡f_{k+1}−f_{ k }. Applying this to eq. (27), we obtain$$ \begin{aligned} &\frac{1}{\delta_{k}}\frac{\partial (u_{k+1}u_{k})}{\partial t} \frac{u_{k+1}^{2}}{2} \frac{1}{\delta_{k}}\frac{\partial (u_{k+1}u_{k})}{\partial x_{k}}\\ &\qquad\qquad =\frac{u_{k+1}+u_{k}}{2}\,.\\[15pt] \end{aligned} $$(31)In the continuous limit δ_{ k }→0, this leads to the SP equation (1).
We remark that eq. (21) describes the evolution of the mesh interval δ_{ k }, and this equation is nothing but a discrete analogue of the conservation law (2). This means that the mesh interval δ_{ k } is a conserved density of the selfadaptive moving mesh scheme. Thus the mesh interval δ_{ k } is determined by the semidiscrete conservation law. From the semidiscrete conservation law, one can find the following property: If \(\frac {u_{k+1}^{2}+{u_{k}^{2}}}{2}<0\), i.e., the slope between \({u_{k}^{2}}\) and \(u_{k+1}^{2}\) is positive, then the mesh interval δ_{ k } becomes smaller. If \(\frac {u_{k+1}^{2}+{u_{k}^{2}}}{2}>0\), i.e., the slope between \({u_{k}^{2}}\) and \(u_{k+1}^{2}\) is negative, then the mesh interval δ_{ k } becomes larger. Thus this scheme creates refined mesh grid for given data {x_{ k },u_{ k }} for k=0,1,2,⋯,N, i.e., mesh grid is refined in which slopes are steep.
 Step 1:
The Lax pair of the SP equation is given by (10) with (13) and (14). This is transformed into (8) with (9) via the reciprocal (hodograph) transformation (4).
 Step 2:By discretizing the Lax pair (8) with (9), we obtain the following linear 2×2 system (Lax pair):$$ \Psi_{k+1}=U_{k}\Psi_{k}\,,\qquad \frac{\partial \Psi_{k}}{\partial T}=V_{k}\Psi_{k} \,, $$(32)where$$ \begin{aligned} U_{k}= \left(\begin{array}{cc} 1\mathrm{i} \lambda a \rho_{k} & \mathrm{i} \lambda \frac{u_{k+1}u_{k}}{a}\\ \mathrm{i} \lambda \frac{u_{k+1}u_{k}}{a} & 1+\mathrm{i} \lambda a \rho_{k} \end{array} \right)\,,\quad \\[5pt] \end{aligned} $$(33)$$ \begin{aligned} V_{k}= \left(\begin{array}{cc} \frac{\mathrm{i}}{4\lambda} & \frac{u_{k}}{2} \\ \frac{u_{k}}{2} &  \frac{\mathrm{i}}{4\lambda} \end{array} \right)\,, \end{aligned} $$(34)
where Ψ_{ k } is a twocomponent vector. The compatibility condition yields the semidiscrete CD system (21) and (22).
 Step 3:Consider a discrete analogue of the reciprocal (hodograph) transformation (23). Now we introduce the mesh interval δ_{ k }=x_{k+1}−x_{ k } which satisfies the relation δ_{ k }=aρ_{ k }, so one can rewrite U_{ k } and V_{ k } by using lattice intervals δ_{ k } and replacing λ by iλ:$$ \begin{aligned} &U_{k}= \left(\begin{array}{cc} 1+\lambda \delta_{k} & \lambda \frac{u_{k+1}u_{k}}{a}\\ \lambda \frac{u_{k+1}u_{k}}{a} & 1\lambda \delta_{k} \end{array} \right)\,,\quad \\ \end{aligned} $$(35)$$ \begin{aligned} &V_{k}= \left(\begin{array}{cc} \frac{1}{4\lambda} & \frac{u_{k}}{2} \\ \frac{u_{k}}{2} &  \frac{1}{4\lambda} \end{array} \right)\,. \end{aligned} $$(36)
 Step 4:
This Lax pair provides (26) and (27) which is nothing but the selfadaptive moving mesh scheme for the SP equation.
Numerical simulations Here we show some examples of numerical simulations using the selfadaptive moving mesh scheme (26) and (27). As a time marching method, we use the improved Euler’s method.
 (1)
Interactions of two loop solitons if both a_{1} and a_{2} are positive, or if both a_{1} and a_{2} are negative. The wave numbers p_{1} and p_{2} are chosen real.
 (2)
Interactions of a loop soliton and an antiloop soliton if a_{1} and a_{2} have opposite signs. The wave numbers p_{1} and p_{2} are chosen real.
 (3)
Breather solutions if the wave numbers p_{1} and p_{2} are chosen complex and satisfy \(p_{2}=p_{1}^{*}\) and \(a_{2}=a_{1}^{*}\) in the above τfunctions.
Selfadaptive moving mesh schemes for the coupled short pulse equation and the complex short pulse equation
By means of the above methods (Method 1 or Method 2) for constructing selfadaptive moving mesh schemes, we can also construct selfadaptive moving mesh schemes for the coupld SP equation and the complex SP equation. Here we show only the results obtained by using Method 1 and Method 2. Note that both methods give the same results.
where \(x_{k}=X_{k}+\sum _{j=0}^{k1}\delta _{j}\) and δ_{ k }=x_{k+1}−x_{ k }, x_{0}=X_{0}.
In the continuous limit δ_{ k }→0, this leads to the coupled SP equation (57) and (58).
We remark that the discretization of the generalized CD systems were proposed by Vinet and Yu recently [41,42]. Our results are consistent with their results.
Numerical simulations Here we show some examples of numerical simulations of the complex SP equation using the selfadaptive moving mesh scheme (91), (92) and (93). As a time marching method, we use the improved Euler’s method.
Concluding remarks
We have proposed two systematic methods for constructing selfadaptive moving mesh schemes for a class of nonlinear wave equations which are transformed into a different class of nonlinear wave equations by reciprocal (hodograph) transformations. We have demonstrated how to create selfadaptive moving mesh schemes for short pulse type equations which are transformed into coupled dispersionless type systems by a reciprocal (hodograph) transformation. Selfadaptive moving mesh schemes have exact solutions such as multisoliton solutions and Lax pairs, thus those schemes are integrable. Selfadaptive moving mesh schemes consist of two semidiscrete equations in which the time is continuous and the space is discrete. In selfadaptive moving mesh schemes, one of two equations is an evolution equation of mesh intervals which is deeply related to a discrete analogue of a reciprocal (hodograph) transformation. An evolution equations of mesh intervals is a discrete analogue of a conservation law of an original equation, and a set of mesh intervals corresponds to a conserved density which play a key role in generation of adaptive moving mesh. We have shown several examples of numerical computations of the short pulse type equations by using selfadaptive moving mesh schemes.
In our previous papers, we have investigated how to discretize the CamassaHolm [13,34], the HunterSaxton [35], the short pulse [11,14], the WKI elastic beam [11], the Dym equation [12,14] by using bilinear methods or by using a geometric approach. Based on our previous studies, we have proposed two systematic methods in sophisticated forms, one uses bilinear equations and another uses Lax pairs, for producing selfadaptive moving mesh schemes. Although we have discussed only short pulse type equations in this paper, our methods can be used to construct selfadaptive moving mesh schemes for other nonlinear wave equations in the WKI class.
More details about exact solutions, fully discretizations, and numerical computations of selfadaptive moving mesh schemes for the coupled SP equation, the complex SP equation, and their generalized equations will be discussed in our forthcoming papers.
Appendix
Authors’ Affiliations
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