JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 613065 10.1155/2013/613065 613065 Research Article The Bäcklund Transformations, Exact Solutions, and Conservation Laws for the Compound Modified Korteweg-de Vries-Sine-Gordon Equations which Describe Pseudospherical Surfaces Sayed S. M. 1, 2 Gosse Laurent 1 Mathematics Department Faculty of Science Tabouk University Tabouk Saudi Arabia ut.edu.sa 2 Mathematics Department Faculty of Science Beni-Suef University Beni-Suef Egypt bsu.edu.eg 2013 20 3 2013 2013 14 11 2012 21 01 2013 24 01 2013 2013 Copyright © 2013 S. M. Sayed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

A differential equation (DE) for a real-valued function u(x,t) is said to describe pseudospherical surfaces (pss) if it is the necessary and sufficient condition for the existence of smooth functions fij,  1i3,  1j2, depending on u and its derivatives, such that the 1-forms ωi=fi1dx+fi2dt,  1i3 satisfy the structure equations of a surface of a constant Gaussian curvature equal to -1, that is, (1)dω1=ω3ω2,dω2=ω1ω3,dω3=ω2ω1. It is equivalent to say that the DE for u(x,t) is necessary and sufficient for the integrability of the linear system  (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-Ωdϕ=(dΩ-ΩΩ)ϕ, which requires the vanishing of the two forms: (8)ΘdΩ-ΩΩ=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 .

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 .

The results of  were obtained by inverse spectral method. The results of  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 .

The Fokas method has a much broader domain of applicability than it is possible to present in . 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 Section 2, 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.  and Sayed et al. . 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 , who observed that most of the nonlinear evolution equations (NLEEs) solvable by the method of inverse scattering , 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 , 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:

when α=0, (11) becomes the mKdV equation in ur which is retrieved,

while the neglect of the terms in h4 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 . 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-2ω-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 .

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.

Let ut=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 .

Eliminating ψ(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)Γx=ηΓ+12f11(1-Γ2)-12f31(1+Γ2),(24)Γt=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)Γx=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 solution u(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)).

Substitute u=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:

when α=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)=4xtan-1(eηρ),ρ=x-η2t,

when α=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)=4rtan-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 solutions u(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  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)],when  β1>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 constant β1 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,

we obtain the solution of mKdV equation (38), (53)u(x,t)=4x[tan-1(eηρ)+tan-1Γ],ρ=x-η2t,ξ=t+79η2ρ-112η3e-2ηρ-49η3ln(e2ηρ-3),

we obtain the solution of mKdV equation (40), (54)u(r,θ)=4x[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 , 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 , Chern and Tenenblat , and Cavalcante and Tenenblat  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. .

Theorem 3.

Suppose that ut=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=0uk+1uk,Dt=t+k=0Dxk(f)uk, are integrable on solutions of the equation ut=F(u,ux,,uxk) or generally F(x,t,u,ux,,uxntm)=0 .

This theorem provides one with at least one η-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=1ϕ2(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),n1, 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. . 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),n1, 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=1ϕ2(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 solution u(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.  and Sayed et al. .

Acknowledgment

The author would like to thank the anonymous referees for these helpful comments. The author thinks that revising the paper according to its suggestions will be quite straightforward.

Beals R. Rabelo M. Tenenblat K. Bäcklund transformations and inverse scattering solutions for some pseudospherical surface equations Studies in Applied Mathematics 1989 81 2 125 151 MR1016586 ZBL0697.58059 Chern S. S. Tenenblat K. Pseudospherical surfaces and evolution equations Studies in Applied Mathematics 1986 74 1 55 83 MR827492 ZBL0605.35080 Ablowitz M. J. Kaup D. J. Newell A. C. Segur H. The inverse scattering transform-Fourier analysis for nonlinear problems Studies in Applied Mathematics 1974 53 4 249 315 MR0450815 ZBL0408.35068 Chadan K. Sabatier P. C. Inverse Problems in Quantum Scattering Theory 1977 New York, NY, USA Springer xxii+344 MR0522847 Zakharov V. E. Shabat A. B. Integration of nonlinear equations of mathematical physics by the method of inverse scattering. II Functional Analysis and Its Applications 1979 13 3 166 174 10.1007/BF01077483 Crampin M. Solitons and SL(2,R) Physics Letters A 1978 66 3 170 172 10.1016/0375-9601(78)90646-1 MR598750 Konno K. Wadati M. Simple derivation of Bäcklund transformation from Riccati form of inverse method Progress of Theoretical Physics 1975 53 6 1652 1656 MR0375992 10.1143/PTP.53.1652 ZBL1079.35505 Miura R. M. Bäcklund Transformations, the Inverse Scattering Method, Solitons, and Their Applications 1976 515 Berlin, Germany Springer viii+295 Lecture Notes in Mathematics MR0497409 Scott A. C. Chu F. Y. F. McLaughlin D. W. The soliton: a new concept in applied science 1973 61 1443 1483 MR0358045 Reyes E. G. On geometrically integrable equations and hierarchies of pseudo-spherical type The Geometrical Study of Differential Equations 2001 285 Providence, RI, USA American Mathematical Society 145 155 Contemporary Mathematics 10.1090/conm/285/04740 MR1874296 ZBL1198.37099 Tenenblat K. Transformations of Manifolds and Applications to Differential Equations 1998 93 Harlow, UK Longman x+209 Pitman Monographs and Surveys in Pure and Applied Mathematics MR1771214 Khater A. H. El-Kalaawy O. H. Callebaut D. K. Bäcklund transformations and exact solutions for Alfvén solitons in a relativistic electron-positron plasma Physica Scripta 1998 58 6 545 548 2-s2.0-0038416362 Khater A. H. Callebaut D. K. Sayed S. M. Exact solutions for some nonlinear evolution equations which describe pseudo-spherical surfaces Journal of Computational and Applied Mathematics 2006 189 1-2 387 411 10.1016/j.cam.2005.10.007 MR2202986 ZBL1093.35005 Zakharov V. E. Shabat A. B. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media Journal of Experimental and Theoretical Physics 1972 34 62 69 Sasaki R. Soliton equations and pseudospherical surfaces Nuclear Physics B 1979 154 2 343 357 10.1016/0550-3213(79)90517-0 MR537509 Cavalcante J. A. Tenenblat K. Conservation laws for nonlinear evolution equations Journal of Mathematical Physics 1988 29 4 1044 1049 10.1063/1.528020 MR940372 Wadati M. Sanuki H. Konno K. Relationships among inverse method, Bäcklund transformation and an infinite number of conservation laws Progress of Theoretical Physics 1975 53 419 436 MR0371297 10.1143/PTP.53.419 ZBL1079.35506 Khater A. H. Callebaut D. K. Sayed S. M. Conservation laws for some nonlinear evolution equations which describe pseudo-spherical surfaces Journal of Geometry and Physics 2004 51 3 332 352 10.1016/j.geomphys.2003.11.009 MR2079415 ZBL1069.37058 Reyes E. G. Pseudo-spherical surfaces and integrability of evolution equations Journal of Differential Equations 1998 147 1 195 230 10.1006/jdeq.1998.3430 MR1632681 ZBL0916.35047 Reyes E. G. Conservation laws and Calapso-Guichard deformations of equations describing pseudo-spherical surfaces Journal of Mathematical Physics 2000 41 5 2968 2989 10.1063/1.533284 MR1755482 ZBL0992.53005 Antonova M. Biswas A. Adiabatic parameter dynamics of perturbed solitary waves Communications in Nonlinear Science and Numerical Simulation 2009 14 3 734 748 10.1016/j.cnsns.2007.12.004 MR2449750 ZBL1221.35321 Johnson S. Biswas A. Perturbation of dispersive topological solitons Physica Scripta 2011 84 1 2-s2.0-79960491777 10.1088/0031-8949/84/01/015002 015002 Sayed S. M. Elkholy A. M. Gharib G. M. Exact solutions and conservation laws for Ibragimov-Shabat equation which describe pseudo-spherical surface Journal of Computational and Applied Mathematics 2008 27 3 305 318 10.1590/S0101-82052008000300005 MR2457187 ZBL1157.35474 Sayed S. M. Elhamahmy O. O. Gharib G. M. Travelling wave solutions for the KdV-Burgers-Kuramoto and nonlinear Schrödinger equations which describe pseudospherical surfaces Journal of Applied Mathematics 2008 10 576783 10.1155/2008/576783 MR2447470 ZBL1162.35449 Sayed S. M. Gharib G. M. Canonical reduction of self-dual Yang-Mills equations to Fitzhugh-Nagumo equation and exact solutions Chaos, Solitons & Fractals 2009 39 2 492 498 10.1016/j.chaos.2007.01.076 MR2518907 ZBL1197.35139 Bobenko A. I. Surfaces in terms of 2 by 2 matrices. Old and new integrable cases Harmonic Maps and Integrable Systems 1994 Braunschweig, Germany Vieweg 83 127 Aspects of Mathematics E23 MR1264183 ZBL0841.53003 Fokas A. S. Gelfand I. M. Finkel F. Liu Q. M. A formula for constructing infinitely many surfaces on Lie algebras and integrable equations Selecta Mathematica. New Series 2000 6 4 347 375 10.1007/PL00001392 MR1847380 ZBL0976.35064 Fokas A. S. Liu Q. M. Generalized conditional symmetries and exact solutions of non-integrable equations Theoretical and Mathematical Physics 1994 99 2 571 582 10.1007/BF01016141 MR1308788 ZBL0850.35097 Fokas A. S. Gelfand I. M. Surfaces on Lie groups, on Lie algebras, and their integrability Communications in Mathematical Physics 1996 177 1 203 220 MR1382226 10.1007/BF02102436 ZBL0864.53003 Fokas A. S. Gelfand I. M. Integrability of linear and nonlinear evolution equations and the associated nonlinear Fourier transforms Letters in Mathematical Physics 1994 32 3 189 210 10.1007/BF00750662 MR1299036 ZBL0807.35138 Fokas A. S. A unified transform method for solving linear and certain nonlinear PDEs Proceedings of the Royal Society of London Series A 1997 453 1962 1411 1443 10.1098/rspa.1997.0077 MR1469927 ZBL0876.35102 Fokas A. S. Two-dimensional linear partial differential equations in a convex polygon Proceedings of the Royal Society of London Series A 2001 457 2006 371 393 10.1098/rspa.2000.0671 MR1848093 ZBL0988.35129 Fokas A. S. Integrable nonlinear evolution equations on the half-line Communications in Mathematical Physics 2002 230 1 1 39 10.1007/s00220-002-0681-8 MR1930570 ZBL1010.35089 Fokas A. S. Pelloni B. Integral transforms, spectral representation and the d-bar problem Proceedings of the Royal Society of London Series A 2000 456 1996 805 833 10.1098/rspa.2000.0538 MR1805082 Fokas A. S. Sung L.-Y. Generalized Fourier transforms, their nonlinearization and the imaging of the brain Notices of the American Mathematical Society 2005 52 10 1178 1192 MR2186901 ZBL1076.42004