# The Bäcklund Transformations, Exact Solutions, and Conservation Laws for the Compound Modified Korteweg-de Vries-Sine-Gordon Equations which Describe Pseudospherical Surfaces

Authors: S. M. Sayed
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

I show that the compound modified Korteweg-de Vries-Sine-Gordon equations describe pseudospherical surfaces, that is, these equations are the integrability conditions for the structural equations of such surfaces. I obtain the self-Bäcklund transformations for these equations by a geometrical method and apply the Bäcklund transformations to these solutions and generate new traveling wave solutions. Conservation laws for the latter ones are obtained using a geometrical property of these pseudospherical surfaces.


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## Body

## 1. Introduction

A differential equation (DE) for a real-valued functionu(x,t) is said to describe pseudospherical surfaces (pss) if it is the necessary and sufficient condition for the existence of smooth functions fij,  1≤i≤3,  1≤j≤2, depending on u and its derivatives, such that the 1-forms ωi=fi1dx+fi2dt,  1≤i≤3 satisfy the structure equations of a surface of a constant Gaussian curvature equal to -1, that is,
(1)dω1=ω3∧ω2,dω2=ω1∧ω3,dω3=ω2∧ω1.
It is equivalent to say that the DE for u(x,t) is necessary and sufficient for the integrability of the linear system [1–6]
(2)dϕ=Ωϕ,ϕ=(ϕ1ϕ2),
where d denotes exterior differentiation, ϕ is a column vector, and the 2×2 matrix Ω (Ωij, i,j=1,2) is traceless
(3)Ω=12(ω2ω1-ω3ω1+ω3-ω2).
Take
(4)Ω=(η2dx+Adtqdx+Bdtrdx+Cdt-η2dx-Adt)=Pdx+Qdt,
from (2) and (4), we obtain
(5)ϕx=Pϕ,ϕt=Qϕ,
where P and Q are two 2×2 null-trace matrices
(6)P=(η2qr-η2),Q=(ABC-A).
Here, η is a parameter, independent of x and t, while q and r are functions of x and t. Now,
(7)0=d2ϕ=dΩϕ-Ω∧dϕ=(dΩ-Ω∧Ω)ϕ,
which requires the vanishing of the two forms:
(8)Θ≡dΩ-Ω∧Ω=0,
or in the component form:
(9)Ax=qC-rB,qt-2Aq-Bx+ηB=0,Cx=rt+2Ar-ηC.
Many partial differential equations (PDEs) which are of interest to study and investigate due to the role they play in various areas of mathematics and physics are included in this category [7–25].The formulation of classical theory of surfaces in a form is familiar to the soliton theory, which makes possible an application of the analytical methods of this theory to integrable cases [26].The results of [27] were obtained by inverse spectral method. The results of [28] were obtained using algorithm for constructing certain exact solutions, such as solutions describing the interaction of two traveling waves.The Fokas transform method for solving boundary value problems for linear and integrable nonlinear PDEs can be viewed as an extension of the Fourier transform method, and, indeed, how in simple cases it reduces to the Fourier transform. The unifying character of the steps involved in the Fokas method makes it attractive from the theoretical and formal point of view. For nonlinear integrable problems, this approach is, at present, the only existing method yielding results in a general context [29–32].The Fokas method has a much broader domain of applicability than it is possible to present in [30–33]. For example, elliptic problems can also be treated by this general approach. In this case, the analysis of the global relation, which is the crucial step in the methodology, may involve the solution of additional Reimann-Hilbert problems. As regards numerical approximations, preliminary results indicate that, using this approach, the Dirichlet-to-Neumann map for linear elliptic problems can also be evaluated, at least for some important examples, with exponential accuracy [34, 35].The current paper directions include the implementation of the geometrical properties and the Bäcklund transformations (BTs) to generate a new soliton solution and conservation laws for the compound modified Korteweg-de Vries-sine-Gordon (cmKdV-SG) equations.The paper is organized as follows. In Section2, I show that the cmKdV-SG equations describe pss. In Section 3, we find the BTs for the cmKdV-SG equations. Exact soliton solution class from a known constant solution is obtained for the cmKdV-SG equations. On the other hand, a new exact traveling wave solutions for the cmKdV-SG equations are obtained by using the BTs to generate a new soliton solution class in Section 4. In Section 5, I obtain an infinite number of conserved densities for the cmKdV-SG equations which describe pss using a theorem of Khater et al. [18] and Sayed et al. [23–25]. Finally, I give some conclusions in Section 6.


## 2. The cmKdV-SG Equations which Describe pss

The notion of a DE describing pss was first introduced by Chern and Tenenblat [2], who observed that most of the nonlinear evolution equations (NLEEs) solvable by the method of inverse scattering [3–5], such as the KdV and mKdV equations, have the property of describing pss. They also showed that if f21=η and the functions f11 and f31 do not depend on η, then the linear system (2) reduces to the inverse scattering problem (ISP) considered by Ablowitz et al. in [3], with η corresponding to the spectral parameter. Let M2 be a two-dimensional differentiable manifold parameterized by coordinates x,  t. We consider a metric on M2 defined by ω1,  ω2. The first two equations in (1) are the structure equations which determine the connection from ω3, and the last equation in (1), the Gauss equation, determines that the Gaussian curvature of M2 is -1,  that is,  M2 is a pss. Moreover, the one-forms
(10)ω1=f11dx+f12dt,ω2=f21dx+f22dt,ω3=f31dx+f32dt
satisfy the structure equations (1) of a pss. It has been known, for a long time, that the SG equation describes a pss. In this paper, we extend the same analysis to include the cmKdV-SG equations:
(11)βurθ+116h4(ur)2urr+124h4urrrr-αsinu=0,
where α,  β,  h are constants. This equation can be thought of as a generalization of the mKdV and SG equations. As particular cases:(i)
whenα=0, (11) becomes the mKdV equation in ur which is retrieved,(ii)
while the neglect of the terms inh4 leads to the SG equation. Moreover, the introduction of the variables
(12)x=(24)1/4rh,t=(24)-1/4hθβ,reduces (11) to the form
(13)uxt+uxxxx+32ux2uxx-αsinu=0.
Let M2 be a differentiable surface, parameterized by coordinates x,  t. Consider that
(14)ω1=(ηuxx+αηsinu)dt,ω2=ηdx+(αηcosu-η3-η2ux2)dt,ω3=uxdx+(-uxxx-η2ux-12ux3)dt,
then M2 is a pss if and only if u satisfies the cmKdV-SG equations (13).


## 3. The Self-Bäcklund Transformation for the cmKdV-SG Equations

In this section, we show how the geometrical properties of a pss may be applied to obtain analytical results for the cmKdV-SG equations which describe pss.The classical Bäcklund theorem originated in the study of pss, relating solutions of the SG equation. Other transformations have been found relating solutions of specific equations in [6–9]. Such transformations are called BTs after the classical one. A BT which relates solutions of the same equation is called a self-Bäcklund transformation (sBT). An interesting fact which has been observed is that DEs which have sBT also admit a superposition formula. The importance of such formulas is due to the following: if u0 is a solution of the NLEE and u1,  u2 are solutions of the same equation obtained by the sBT, then the superposition formula provides a new solution u′ algebraically. By this procedure, one obtains the soliton solutions of an NLEE. In what follows, we show that geometrical properties of pss provide a systematic method to obtain the BTs for some NLEEs which describe pss.Proposition 1.
Given a coframe{ω-1,ω-2} and corresponding connection one-form ω-3 on a smooth Riemannian surfaces M2, there exists a new coframe {ω-1′,ω-2′} and new connection one-form ω-3′ satisfying the equations
(15)dω-1′=0,dω-2′=ω-2′∧ω-1′,ω-3′+ω-2′=0,
if and only if the surface M2 is pss. For the sake of clarity, one gives a revised proof of [10].Proof.
Assume that the orthonormal dual to the coframes{ω-1,ω-2} and {ω-1′,ω-2′} possess the same orientation. The one-forms ω-i and ω-i′   (i=1,2,3) are connected by means of
(16)ω-1′=ω-1cosψ-ω-2sinψ,ω-2′=ω-1sinψ+ω-2cosψ,ω-3′=ω-3-dψ.
It follows that ω-1′,  ω-2′,  ω-3′  satisfying (15) exist if and only if the Pfaffian system
(17)ω-3-dψ+ω-1sinψ+ω-2cosψ=0,
on the space of coordinates (x,t,ψ) is completely integrable for ψ(x,t), and this happens if and only if M2 is pss.
Geometrically, (15) and (17) determine geodesic coordinates on M2. Now, if ut=F(u,ux,…,uxk)(uxk=∂ku/∂xk) describes pss with associated one-forms ωi=fi1dx+fi2dt, (15) and (17) imply that the Pfaffian system,
(18)ω3-dψ+ω1sinψ+ω2cosψ=0,
is completely integrable for ψ(x,t) whenever u(x,t) is a local solution of ut=F(u,ux,…,uxk) [2, 11].Proposition 2.
Letut=F(u,ux,…,uxk) be an NLEE which describes a pss with associated one-forms (10). Then, for each solution u(x,t) of ut=F(u,ux,…,uxk), the system of equations for ψ(x,t),
(19)ψx-f31+f11sinψ+ηcosψ=0,ψt-f32+f12sinψ+f22cosψ=0,
is completely integrable. Moreover, for each solution of u(x,t) of ut=F(u,ux,…,uxk) and corresponding solution ψ,
(20)(f11cosψ-ηsinψ)dx+(f12cosψ-f22sinψ)dt,
is a closed one-form [2].Eliminatingψ(x,t) from (19), by using the substitution
(21)cosψ=2Γ1+Γ2,
where
(22)Γ=ϕ1ϕ2,
then (19) is reduced to the Riccati equations:
(23)∂Γ∂x=ηΓ+12f11(1-Γ2)-12f31(1+Γ2),(24)∂Γ∂t=f22Γ+12f12(1-Γ2)-12f32(1+Γ2).
The procedure in the following is that one constructs a transformation Γ′ satisfying the same equation as (24) with a potential u′(x), where
(25)u′(x)=u(x)+f(Γ,η).
Thus eliminating Γ in (23), (24), and (25), we have a BT to a desired NLEE. We consider the following example (BT for the cmKdV-SG equations).For (13), we consider the functions defined by
(26)f11=0,f12=ηuxx+αηsinu,f21=η,f22=αηcosu-η3-η2ux2,f31=ux,f32=-u3x-η2ux-12ux3,
for any solution u(x,t) of (13), the above functions satisfy (8). Then, (23) becomes
(27)∂Γ∂x=ηΓ-ux2(1+Γ2).
If we choose Γ′ and u′ as
(28)Γ′=1Γ,u′=u+4tan-1Γ,
then Γ′ and u′ satisfy  (27). If we eliminate Γ in (27) and (24) with (28), we get the BT
(29)(u′+u)x=-2ηsin12(u-u′),(u-u′)t=2f32-2f12cos12(u-u′)+2f22sin12(u-u′).
Equation (29) is the BT for the cmKdV-SG equations (13) with f12,  f22, and f32 given in (26).


## 4. A New Traveling Wave Solutions for the cmKdV-SG Equations

For any solutionu(x,t) of the cmKdV-SG equations (13), the matrices P and Q are(30)P=(η2-ux2ux2-η2),Q=(12(-η3-ηux22+αηcosu)12(ηuxx+uxxx+ux32+η2ux+αηsinu)12(ηuxx-uxxx-ux32-η2ux+αηsinu)-12(-η3-ηux22+αηcosu)).Substituteu=nΠ,  n=0,±1,±2,±3,… into the matrices P and Q in (30), then by (5) we have
(31)dϕ=ϕxdx+ϕtdt=Pϕdρn,
where
(32)P=(η200-η2),ρn=x-kt,k=η2-αη2(-1)n.
The solution of (31) is
(33)ϕn=eρnPϕ0=(I+ρnP+ρn2P22!+ρn3P33!+⋯)ϕ0,
where ϕ0 is a constant column vector. The solution of (33) is
(34)ϕn=(coshη2ρn+sinhη2ρn00coshη2ρn-sinhη2ρn)ϕ0.
Now, we choose ϕ0=(1,1)T in (34), then we have
(35)ϕn=(eηρn/2e-ηρn/2).
Substitute (35) into (22), then, by (28), we obtain the new solutions of the cmKdV-SG equations (13):
(36)u′(x,t)=nΠ+4tan-1(eηρn),n=0,±1,±2,±3,….
Consequently, the solution of (11) is
(37)u′(r,θ)=nΠ+4tan-1(eηρn),ρn=(24)1/4rh-(24)-1/4hθβk,k=η2-αη2(-1)nn=0,±1,±2,±3,….
By means of the same procedures above, we obtain the solution of mKdV equation:(i)
whenα=0 in (13), we obtain the mKdV equation in ux(38)(ux)t+(ux)xxx+32(ux)2(ux)x=0,   and its solutions is
(39)u′(x,t)=4∂∂xtan-1(eηρ),ρ=x-η2t,(ii)
whenα=0 in (11), we obtain the mKdV equation in ur(40)β(ur)θ+116h4(ur)2(ur)r+124h4(ur)rrr=0,   and its solutions is
(41)u′(r,θ)=4∂∂rtan-1(eηρ),ρ=(24)1/4rh-(24)-1/4hθβη2.Now, we use a known traveling wave solutions for the cmKdV-SG equations to generate a new solution for the cmKdV-SG equations by means of the BTs.We will find a new traveling wave solutionsu′(x,t) of the cmKdV-SG equations (13) and substitute these solution into the corresponding matrices P and Q. Next we solve (22) for ϕ1 and ϕ2. Then by (22) and the corresponding BTs (28) we will obtain the new solution classes. We take
(42)u=4tan-1(eηρ),ρ=x-kt,k=η2-αη2,
as a traveling wave solution class of the cmKdV-SG equations (13). The traveling wave known solution of the cmKdV-SG equations takes the form
(43)u=u(ρ),ρ=x-kt.
In this case the AKNS system (5) and (6) has a general solution. Let us consider the more general case. Suppose that the components q and r of the matrix P are function of ρ [8, 12]:
(44)q=q(ρ),r=r(ρ);
then the components A, B and C of the matrix Q as determined by (6) are also functions of ρ:
(45)A=A(ρ),B=B(ρ),C=C(ρ).
Under these assumptions, the following result holds, which is crucial in the subsequent exact solution. The quantity
(46)β1=(A+kη2)2+(B+kq)(C+kr),
is constant with respect to ρ (or x and t). Using the result of [13] and the constant β1 defined by (46) is greater than zero and therefore the corresponding solution of the AKNS system (5) and (6) is: (47)[ϕ1ϕ2]=[c1(C+kr)-1/2[(A+kη2)sinhω(ξ+c2)+ωcoshω(ξ+c2)]c1(C+kr)1/2sinhω(ξ+c2)],whenβ1>0,ω2=β1,where c1 and c2 are constants and
(48)ξ=t+∫‍rdρC+kr.Now applying the results obtained here and the known traveling wave solutions for the cmKdV-SG equations respectively to construct new solution class of the corresponding cmKdV-SG equations by means of the BTs. The constantβ1 and ξ defined by (46), (48) can be determined by using (42)
(49)ξ=t-[η22η4+2α-8η109η12+3η8α-5α2η4+α3]ρ-(η12η4-4α)e-2ηρ-[4η99η12+3η8α-5α2η4+α3]×ln(e2ηρ+α-3η4α+η4).
Consequently, we obtain Γ from (47) for β1>0(50)Γ=(C+kr)-1[(A+kη2)+ωcothω(ξ+c2)],
then substituting this Γ into the BTs (28) and using (42), we arrive at the new solution u′ of the cmKdV-SG equations (13) corresponding to the known traveling wave solution class (42), then
(51)u′(x,t)=4[tan-1(eηρ)+tan-1Γ].Consequently, the solution of (11) is
(52)u′(r,θ)=4[tan-1(eηρ)+tan-1Γ],ρ=(24)1/4rh-(24)-1/4hθβk,k=η2-αη2.By means of the same procedures above,(i)
we obtain the solution of mKdV equation (38),
(53)u′(x,t)=4∂∂x[tan-1(eηρ)+tan-1Γ],ρ=x-η2t,ξ=t+79η2ρ-112η3e-2ηρ-49η3ln(e2ηρ-3),(ii)
we obtain the solution of mKdV equation (40),
(54)u′(r,θ)=4∂∂x[tan-1(eηρ)+tan-1Γ],ρ=(24)1/4rh-(24)-1/4hθβη2,ξ=(24)-1/4hθβ+79η2ρ-112η3e-2ηρ-49η3ln(e2ηρ-3).


## 5. Conservation Laws for the cmKdV-SG Equations

One of the most widely accepted definitions of integrability of PDEs requires the existence of soliton solutions, that is, of a special kind of traveling wave solutions that interact “elastically,” without changing their shapes. The analytical construction of soliton solutions is based on the general ISM. In the formulation of Zakharov and Shabat [14], all known integrable systems supporting solitons can be realized as the integrability condition of a linear problem of the form (5). Thus, an equation (5) is kinematically integrable if it is equivalent to the curvature condition
(55)Px-Qt+[P,Q]=0.
As mentioned in the previous sections, Sasaki [15], Chern and Tenenblat [2], and Cavalcante and Tenenblat [16] have given a geometrical method for constructing conservation laws of equations describing pss. The formal content of this method is contained in the following theorem, which may be seen as generalizing the classical discussion on conservation laws appearing in Wadati et al. [17].Theorem 3.
Suppose thatut=F(u,ux,…,uxk) or more generally F(x,t,u,ux,…,uxntm)=0 is an NLEE describing pss. The systems
(56)Dxϕ1=qr+(Dxqq-η)ϕ1-ϕ12,(57)Dt(η2+ϕ1)=Dx(A+Bqϕ1),(58)Dxϕ2=-qr+(Dxrr+η)ϕ2+ϕ22,(59)Dt(η2+ϕ2)=Dx(A+Crϕ2),
in which Dx and Dt are the total derivative operators defined by
(60)Dx=∂∂x+∑k=0∞‍uk+1∂∂uk,Dt=∂∂t+∑k=0∞‍Dxk(f)∂∂uk,
are integrable on solutions of the equation ut=F(u,ux,…,uxk) or generally F(x,t,u,ux,…,uxntm)=0 [18].This theorem provides one with at least oneη-dependent conservation law of the NLEE ut=F(u,ux,…,uxk) or F(x,t,u,ux,…,uxntm)=0, to wit, (56) and (57) or ((58) and (59)). One obtains a sequence of η-independent conservation laws by expanding ϕ1 or ϕ2 in inverse powers of η [19, 22]. Moreover,
(61)ϕ2=∑n=1∞‍ϕ2(n)η-n,
and the consideration of (58) yields the recursion relation
(62)ϕ2(1)=-qr,ϕ2(n+1)=Dxrrϕ2(n)+Dxϕ2(n)+∑i=1n-1‍ϕ2(i)ϕ2(n-i),n≥1,
which in turn, by replacing into (59), yields the sequence of conservation laws of equations integrable by AKNS inverse scattering found by Wadati et al. [17]. This section ends with the example.For (13), we consider the functions of u(x,t) defined by
(63)r=ux2,q=-ux2,(64)A=12(-η3-ηux22+αηcosu),B=12(ηuxx+uxxx+ux32+η2ux+αηsinu),(65)C=12(ηuxx-uxxx-ux32-η2ux+αηsinu).
Equation (58) becomes
(66)Dxϕ2=14ux2+(uxxux+η)ϕ2+ϕ22.
Assume that ϕ2 can be expanded in a series of the form (61).Equation (64) implies that ϕ2 is determined by the recursion relation
(67)ϕ2(1)=14ux2,ϕ2(n+1)=uxxuxϕ2(n)+Dxϕ2(n)+∑i=1n-1‍ϕ2(i)ϕ2(n-i),n≥1,
whenever u(x,t) is a solution of the the cmKdV-SG equations. This recursion relation yields a sequence of conserved densities given by the coefficients of the series in η(68)η2+∑n=1∞‍ϕ2(n)η-n,
which one obtains from (59).


## 6. Conclusions

In this paper, I show how the geometrical properties of a pss may be applied to obtain analytical results for the cmKdV-SG equations which describe pss. It has been shown that the implementation of certain BTs for a class of NLEE requires the solution of the underlying linear differential equation whose coefficients depend on the known solutionu(x,t) of the NLEE. I obtain a new traveling wave solutions for the cmKdV-SG equations by using BTs. Next, an infinite number of conservation laws is derived for the cmKdV-SG equations just mentioned using a theorem by Khater et al. [18] and Sayed et al. [23–25].



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