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Selfadaptive moving mesh schemes for short pulse type equations and their Lax pairs
Pacific Journal of Mathematics for Industry volume 6, Article number: 8 (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.
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.
The short pulse (SP) equation [8,32,33,37–39]
is linked with the socalled coupled dispersionless (CD) system [21,26,28,29]
through the reciprocal transformation (this is often called the hodograph transformation in many literatures)
where X_{0} is a constant. The reciprocal (hodograph) transformation (4) yields
Note that the reciprocal (hodograph) transformation (4) originates from the conservation law (2). Applying the reciprocal (hodograph) transformation (4) to (3) yields
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].
The CD system (2) and (3) can be derived from the compatibility condition of the following linear 2×2 system (Lax pair) [28]:
where
and Ψ is a twocomponent vector. By applying the reciprocal (hodograph) transformation (4) into the above linear 2×2 system (Lax pair) (8) and (9), we obtain the linear 2×2 system (Lax pair) for the short pulse equation [38]:
where
which can be rewritten as
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
The SP equation (1) can be discretized by means of the following two methods: Method 1:The discretization method using bilinear equations

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.
Method 2:The discretization method using a Lax pair

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.
Here we show the details of procedures to construct the selfadaptive moving mesh scheme for the SP equation by means of the above two methods. Method 1:

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.
Method 2:

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.
Multisoliton solutions of the SP equation are given by
where
and I_{ N } is the N×N identity matrix, a^{⊤} is the transpose of a,
For example, the τfunctions f and g of the 2soliton solution are written as
where
There are two types of 2loop soliton solutions and a type of breather solutions:

(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.
Here we show three examples of numerical simulations of the SP equation. We use the number of mesh grid points N=200, the width of the computational domain D=80, and the time interval dt=0.0001. Figure 1 shows the numerical simulation of the 2loop soliton solution of the SP equation. Figure 2 shows the numerical simulation of the solution describing the interaction of a loop soliton and antiloop soliton of the SP equation. In Figure 3, we show the numerical simulation of the breather solution of the SP equation. In these three examples, numerical results have good agreement with exact solutions of the SP equation. It is possible to have more accurate numerical results if we increase the number of mesh grid points and use smaller time steps.
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.
Consider the following linear 2×2 system:
where
where Ψ is a twocomponent vector. The compatibility condition yields the coupled CD system [24,25]
Note the coupled CD system (45), (46) and (47) can be transformed into the bilinear equations
via the dependent variable transformation
Applying the reciprocal (hodograph) transformation (4) into the above linear problem (43) and (44), we obtain the linear 2×2 system (Lax pair) for the coupled SP equation [31]:
where
which can be rewritten as
by replacing λ by iλ. The compatibility condition yields the coupled SP equation [9,31]
Letting u be a complex function and v=u^{∗} where u^{∗} is a complex conjugate of u, the compatibility condition of the above linear 2×2 systems yields the complex CD system [26,30]
and the complex SP equation [10]
Note the complex CD system (59), (60) and (61) can be transformed into the bilinear equations
via the dependent variable transformation
Using the dependent variables u^{(R)} and u^{(I)} such that u^{(R)}=Reu, u^{(I)}=Imu, the complex SP equation can be written as
Consider the following linear 2×2 system (32) with
where Ψ_{ k } is a twocomponent vector. The compatibility condition (32) with (70) and (71) yields the semidiscrete coupled CD system
We can rewrite U_{ k } and V_{ k } by using lattice intervals δ_{ k }(=aρ_{ k }=x_{k+1}−x_{ k }) and replacing λ by iλ:
The compatibility condition of (32) with (75) and (76) provides the selfadaptive moving mesh scheme for the coupled SP equation
where \(x_{k}=X_{k}+\sum _{j=0}^{k1}\delta _{j}\) and δ_{ k }=x_{k+1}−x_{ k }, x_{0}=X_{0}.
The discrete reciprocal (hodograph) transformation \(x_{k}=X_{k}+\sum _{j=0}^{k1}\delta _{j}\) yields
Applying this to eqs. (78) and (79), we obtain
In the continuous limit δ_{ k }→0, this leads to the coupled SP equation (57) and (58).
Note that the semidiscrete coupled CD system and the selfadaptive moving mesh scheme for the coupled SP equation can be transformed into the bilinear equations
via the dependent variable transformation
By letting u_{ k } be a complex function and adding a constraint \(v_{k}=u_{k}^{*}\), we obtain the semidiscrete complex CD system
and the selfadaptive moving mesh scheme for the complex SP equation
Note that the semidiscrete complex CD system and the selfadaptive moving mesh scheme for the complex SP equation can be transformed into the bilinear equations
via the dependent variable transformation
Using the dependent variables \(u_{k}^{({R})}\) and \(u_{k}^{(\mathrm {I})}\) such that \(u_{k}^{({R})}=\text {Re} u_{k}\), \(u_{k}^{(\mathrm {I})}=\text {Im} u_{k}\), the complex SP equation can be written as
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.
The multisoliton solutions of the complex SP equation (62) and (63) are given by the following formula:
where
and I_{ N } is the N×N identity matrix, a^{⊤} is the transpose of a,
Note that this formula is obtained from the gram type determinant solution of the coupled SP equation in Appendix.
For example, the τfunctions f, g and g^{∗} of the 2soliton solution of the complex SP equation are written as
where
Figure 4 shows the numerical simulation of the 2 soliton solution of the complex SP equation (62) and (63) by means of the selfadaptive moving mesh scheme (91), (92), and (93). Figures 5 and 6 show the graphs of the real part and the imaginary part of u of the complex SP equation (62) and (63), respectively. Thus these graphs are the solution of equations (68) and (69). We use the number of mesh grid points N=300, the width of the computational domain D=100, and the time interval dt=0.00005. Again, the numerical result has good agreement with the exact solution of the complex SP equation.
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
The multisoliton solutions of the coupled SP equation (57) and (58) are given by the following formula:
where
and I_{ N } is the N×N identity matrix, a^{⊤} is the transpose of a,
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Feng, BF., Maruno, K. & Ohta, Y. Selfadaptive moving mesh schemes for short pulse type equations and their Lax pairs. Pac. J. Math. Ind. 6, 8 (2014). https://doi.org/10.1186/s4073601400087
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DOI: https://doi.org/10.1186/s4073601400087