The elliptic sinh-gordon equation in the half plane

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1 Available online at J. Nonlinear Sci. Appl. 8 25), Research Article The elliptic sinh-gordon equation in the half plane Guenbo Hwang Department of Mathematics, Daegu University, Gyeongsan Gyeongbuk 72-74, Korea Communicated by K. Q. Lan Abstract Boundary value problems for the elliptic sinh-gordon equation formulated in the half plane are studied by applying the so-called Fokas method. The method is a significant extension of the inverse scattering transform, based on the analysis of the Lax pair formulation and the global relation that involves all known and unknown boundary values. In this paper, we derive the formal representation of the solution in terms of the solution of the matrix Riemann-Hilbert problem uniquely defined by the spectral functions. We also present the global relation associated with the elliptic sinh-gordon equation in the half plane. We in turn show that given appropriate initial and boundary conditions, the unique solution exists provided that the boundary values satisfy the global relation. Furthermore, we verify that the linear limit of the solution coincides with that of the linearized equation known as the modified Helmhotz equation. c 25 All rights reserved. Keywords: Boundary value problems, elliptic PDEs, sinh-gordon equation, integrable equation. 2 MSC: 47K5, 35Q55.. Introduction A unified method introduced by A. S. Fokas [7, 8] see also the monograph [2]), which can be considered a significant extension of the inverse scattering transform, is used for analyzing boundary value problems. This so-called Fokas method has been extensively applied to solve a large class of boundary value problems; for example, integrable nonlinear evolution equations such as the nonlinear Schrödinger, the Korteweg-deVries and the sine-gordon equations [] and linear and nonlinear elliptic partial differential equations [9, 9] as well as difference-differential equations [2, 3]. The rigorous result in the implementation of the Fokas method is that the solution can be expressed in terms of the solution for a Riemann-Hilbert RH) problem uniquely defined by spectral functions that involve initial and boundary values. Importantly, the global relation that couples known and unknown boundary values plays a crucial role of analyzing the boundary value problems. address: ghwang@daegu.ac.kr Guenbo Hwang) Received

2 G. Hwang, J. Nonlinear Sci. Appl. 8 25), Indeed, the global relation is not only able to allow the existence of the unique solution for the boundary value problem, but it is also able to to characterize the unknown boundary values that enter in the spectral functions. For example, for the typical Dirichlet boundary value problem, the Neumann boundary value is unknown. In this case, it is necessary to characterize the unknown Neumann boundary value. The implementation of the Fokas method to the boundary value problems has remarkable advantages. In particular, we note that i) it is possible to eliminate the unknown boundary values that appear in the representation for the solution. In general, this can be done by solving nonlinear Volterra integral equations for the unknown boundary values. However, there is a certain class of boundary conditions BCs), called linearizable, which makes it possible to bypass the solution of the nonlinear Volterra integral equation. For such BCs, one can eliminate the unknown boundary values simply by using algebraic manipulation of the global relation []. Thus, the Fokas method is as effective as the classical inverse scattering transform. ii) For evolution equations, the jump matrix of the RH problem defined by the spectral functions has an explicit x, t)-dependence of exponential form. Thus, it is possible to study the appropriate asymptotics of the solution by using the Deift-Zhou method [5] for the long time behavior, or to study the small dispersion limit by using the Deift-Venakides-Zhou method [6]. It should be noted that the classical inverse scattering transform has been used in most cases of hyperbolic evolution equations, with some exceptions. For example, in [4, 8] the inverse scattering transform was applied to solve the initial value problem for the elliptic sinh-gordon equation in the plane {x, y) < x <, < y < }. Recently, regarding boundary value problems of the nonlinear elliptic type, the Fokas method was applied to solve the elliptic sine-gordon equation in the half plan y > ). In [4, 9], it has been derived the formal representation of the solution and discussed the existence of the solution for the elliptic sine-gordon equation by using the global relation. Hence, of particular interest appears in the study of the boundary value problem for the elliptic sinh-gordon equation formulated in the half plane via the Fokas method. In this paper we study the initial-boundary problem for the elliptic sinh-gordon equation posed in the half plane q xx + q yy = sinh q, < x <, y >..) The elliptic sinh-gordon equation arises in models of interacting charged particles in plasma physics [8]. Besides applications in physics, from a mathematical point of view, this equation is also interesting because it is completely integrable and is connected to the Toda lattice equations []. The purpose of this paper is to implement the Fokas method to the elliptic sinh-gordon equation posed in the half plane.). We derive the global relation based on the spectral analysis of the Lax pair formulation. We then show that given appropriate initial and boundary conditions, the unique solution for.) exists provided that the boundary values satisfy this global relation. Moreover, we address the formal representation for the solution in terms of the solution of the RH problem defined by the spectral functions. The outline of the paper is following. In section 2, we introduce the relevant notations, formulas and the regularity assumptions for the initial and boundary data. In section 3, we apply the Fokas method to solve the elliptic sinh-gordon equation posed in the half plane. In particular, the existence of the solution is discussed by analyzing the the matrix RH problem as an inverse problem. In section 4, we verify that the linear limit of the solution for.) coincides with that of the corresponding linearized equation known as the modified Helmhotz equation. We end with concluding remarks in section Preliminaries It was shown in [4, 8] that the elliptic sinh-gordon equation admits the following Lax pair formulation ϕ x + ω k)σ 3 ϕ = Qx, y, k)ϕ, ϕ y + ω 2 k)σ 3 ϕ = iqx, y, k)ϕ, 2.a) 2.b)

3 G. Hwang, J. Nonlinear Sci. Appl. 8 25), where k C is a spectral parameter, ϕ is a 2 2 matrix-valued eigenfunction and ω k) = k ), ω 2 k) = k + 2i 4k 2 4k with Qx, y, k) = 4 i cosh q ) r sinh q i rx, y) = iq x x, y) + q y x, y), σ 3 = ) ) r + sinh q cosh q ) ), 2.2) 2.3) ). 2.4) The Lax pair given in 2.) is a slightly different formulation from that considered in [8], but they are equivalent. Using this Lax pair, we compare the linear limit of the elliptic sinh-gordon equation to the solution of the linearized equation known as the modified Helmholtz equation, which will be discussed in Section 4. It should be remarked that the elliptic sinh-gordon equation.) is the compatible condition of the Lax pair 2.) in the sense that φ xy = φ yx from 2.) implies that q solves.) if the spectral parameter k is independent of x and y. Note that Re ω k) = 8 Im k 4 + ) k 2, Re ω 2 k) = 8 Re k 4 + ) k 2, 2.5) which implies Re ω k) < for Im k >, Re ω 2 k) < for Re k >. 2.6) Also note that due to the symmetry of Qx, y, k), the eigenfunction has the same symmetry ϕ x, y, k) = ϕ 22 x, y, k), ϕ 2 x, y, k) = ϕ 2 x, y, k), 2.7) where the subscripts denote the i, j)-component of the matrix. Throughout the paper, we denote the matrix commutator by ˆσA, that is, ˆσA = [σ 3, A] = σ 3 A Aσ 3, 2.8) where A is a 2 2 matrix. This yields the notation eˆσ 3ξ A = e σ3ξ Ae σ3ξ a e = 2ξ ) a 2 e 2ξ. a 2 a 22 We also denote where the functions g and g are assumed to be qx, ) = g x), q y x, ) = g x), 2.9) g + 2mπ H R) for some m Z, g H R). 2.) Making use of the integrating factor µx, y, k) = ϕx, y, k)e ω k)x+ω 2 k)y)σ 3, it is convenient to derive the following modified Lax pair of the form µ x + ω k) [σ 3, µ] = Qx, y, k)µ, 2.a) µ y + ω 2 k) [σ 3, µ] = iqx, y, k)µ. 2.b) 3. The elliptic sinh-gordon equation in the half plane In this section, we demonstrate the Fokas method to solve the elliptic sinh-gordon equation in the half plane.

4 G. Hwang, J. Nonlinear Sci. Appl. 8 25), Eigenfunctions. In order to analyze the eigenfunctions associated with the Lax pair formulation 2.), we introduce a differential -form W given by W x, t, k) = [Qx, y, k)dx + iqx, y, k)dy] µx, y, k), 3.) and hence, equations 2.) can be written in the form [ ] d e ω k)x+ω 2 k)y)ˆσ 3 µx, y, k) = e ω k)x+ω 2 k)y)ˆσ 3 W x, y, k). 3.2) We now define eigenfunctions that satisfy both parts of the Lax pair 2.) as x,y) µ j x, y, k) = I + e ω k)x ξ)+ω 2 k)y η))ˆσ 3 W j ξ, η, k), 3.3) x j,y j ) where x, y), x j, y j ) D = { < x <, < y < }, W j is the differential form defined in 3.) with µ j and x j, y j ) is a fixed point in D. If D is the interior of the convex polygon, it has been shown in [2] that the all vertices of the polygon can be taken for x j, y j ). We choose three distinct points x j, y j ) in D, j =, 2, 3 see fig. ), x, y ) =, y), x 2, y 2 ) =, y), x 3, y 3 ) = x, ) and the corresponding eigenfunctions satisfy the following integral equations: µ x, y, k) = I + µ 2 x, y, k) = I µ 3 x, y, k) = I i x x y e ω k)x ξ)ˆσ 3 Qµ ) ξ, y, k)dξ, e ω k)x ξ)ˆσ 3 Qµ 2 ) ξ, y, k)dξ, e ω 2k)y η)ˆσ 3 Qx, η, k)µ 3 x, η, k)) dη. 3.4a) 3.4b) 3.4c) Note that the eigenfunctions µ j, j =, 2, 3, also enjoy the same symmetry as in the form of 2.7). Since the off-diagonal components of the matrix-valued eigenfunctions µ involve the explicit exponential terms, the regions of analyticity of the eigenfunctions can be determined: [µ ] and [µ 2 ] 2 are analytic for Im k > [µ ] 2 and [µ 2 ] are analytic for Im k < [µ 3 ] is analytic for Re k <, while [µ 3 ] 2 is analytic for Re k >, where [ ] j denotes the j-th column of the matrix. Hereafter, we then write each column of µ j as the following notations: µ = µ +, ) µ, k C +, C ), µ 2 = µ 2, ) µ+ 2, k C, C +), 3.5a) µ 3 = µ 3, ) µ+ 3, k Re k, Re k ), 3.5b) where C ± denote the upper/lower half plane of the complex k-plane, respectively. Spectral functions. The matrix eigenfunctions µ and µ 2 are both fundamental solutions of the Lax pair 2.). Hence, they are related by the so-called spectral function Sk): µ 2 x, y, k) = µ x, y, k)e ω k)x+ω 2 k)y)ˆσ 3 Sk), k R, x R, y <. 3.6)

5 G. Hwang, J. Nonlinear Sci. Appl. 8 25), Μ 3,y Μ Μ 2 Figure : The eigenfunctions µ, µ 2 and µ 3. Letting y = and x in 3.6), we find the spectral function given by Sk) = I e ω k)ξˆσ 3 Qµ 2 ) ξ,, k)dξ. 3.7) From the symmetry of the eigenfunctions µ 2, we write the spectral function Sk) in the form ) ak) b k) Sk) =. 3.8) bk) a k) We define Φx, k) = µ 2 x,, k). More specifically, the function Φ satisfies Φx, k) = I where Q x, k) = Qx,, k), that is, x Q x, k) = 4 e ω k)x ξ)ˆσ 3 Q Φ)ξ, k)dξ, k C, C +), x R, 3.9) i cosh qx, ) ) rx, ) + rx, ) sinh qx,) i sinh qx,) cosh qx, ) ) )). 3.) From 3.7), it follows that the spectral function Sk) can be determined by the function Φ [ ] Sk) = e ω k)xˆσ 3 Φx, k). 3.) lim x This justifies that ak) has an analytic continuation for Im k < and bk) is defined only for k R. Moreover, due to the symmetry of the eigenfunction µ 2, Φ can be written as ) Φ x, k) Φ Φx, k) = 2 x, k). 3.2) Φ 2 x, k) Φ x, k) Recalling the domain of analyticity of the eigenfunction µ 2, we also denote Φ = Φ, Φ +), k C, C +). 3.3) Thus, we obtain the integral representations for the spectral functions ak) and bk) and we summarize these representations below.

6 G. Hwang, J. Nonlinear Sci. Appl. 8 25), Proposition 3.. Given qx, ) = g x), the spectral functions {ak), bk)} are defined by ak) = { i 4 cosh g ξ) ) Φ ξ, k) iġ ξ) + q y ξ, ) + sinh g ) } ξ) Φ 2 ξ, k) dξ, Im k <, 3.4a) bk) = { e 2ω k)ξ iġ ξ) + q y ξ, ) sinh g ) ξ) Φ ξ, k) 4 i } cosh g ξ) ) Φ 2 ξ, k) dξ, k R, 3.4b) where the functions Φ and Φ 2 are solutions of the x-part of the Lax pair 2.a) with y =, that is, Φ and Φ 2 solve the following system of ordinary differential equations: Φ x = [ i 4 cosh g x) ) Φ iġ x) + q y x, ) + sinh g ) ] x) Φ 2, 3.5a) Φ 2x 2ω k)φ 2 = [ iġ x) + q y x, ) sinh g ) x) Φ i ] 4 cosh g x) ) Φ 2 3.5b) with lim x Φ, Φ 2 ) =, ). Global relation. Note that the eigenfunctions µ and µ 3 solve the same differential equations 2.) with the same boundary condition at infinity. Similarly, µ 2 and µ 3 solve the same differential equations with the same boundary condition at infinity. Thus, we know µ 3 x, y, k) = µ 2 x, y, k), k C III, 3.6a) µ 3 x, y, k) = µ+ x, y, k), k C II, 3.6b) where C II and C III denote the second and third quadrants of the complex k-plane, respectively. Φ x, k) = µ 2 x,, k), we find the following global relation Since Φ x, k) = µ 3 x,, k), x R, k C III. 3.7) In particular, letting x in 3.7) yields ak) = for k C III and bk) = for k. From the analytic continuation, we find the global relation in terms of the spectral functions ak) and bk) ak) =, Im k, bk) =, k. 3.8) We also obtain an alternative global relation by analyzing eigenfunctions µ and µ 3. Indeed, let Ψx, k) = µ x,, k), that is, Ψx, k) = I + According to the domain of analyticity, we denote Then, we also find the following global relation e ω k)x ξ)ˆσ 3 Q ξ, k)ψξ, k)dξ, x R, k C +, C ). 3.9) Ψ = Ψ +, Ψ ), k C +, C ). 3.2) Ψ + x, k) = µ 3 x,, k), x R, k C II. 3.2)

7 G. Hwang, J. Nonlinear Sci. Appl. 8 25), Riemann-Hilbert problem. As an inverse problem, we formulate a matrix Riemann-Hilbert RH) problem. First, note that from the global relation 3.8), for k R the spectral function Sk) simply becomes ) b k) Sk) =. 3.22) bk) Recalling equation 3.6), we then formulate the following matrix RH problem where the sectionally analytic functions M ± are defined by M x, y, k) = M + x, y, k)jx, y, k), k R, 3.23) M + = µ + 2, ) µ+, k C +, M = µ, ) µ 2, k C 3.24) and the jump matrix is given by b k)e J = 2θx,y,k) ) bk)e 2θx,y,k), 3.25) where Note that θx, y, k) = ω k)x + ω 2 k)y. 3.26) det M ± =, M ± = I + O/k), k. 3.27) The solution of the elliptic sinh-gordon equation can be obtained from the unique solution of the RH problem. In this respect, we expand the solution of the RH problem M as Mx, y, k) = I + M ) x, y) k + M 2) x, y) k 2 + O/k 2 ), k. 3.28) Substituting this expansion into the x-part of the Lax pair 2.a), from the 2, )-component at O), we find iq x x, y) + q y x, y) = 4iM ) 2 x, y) 3.29) and the, )-component at O/k) yields M ) x x, y) = 8i cosh qx, y) ) 4 iq xx, y) + q y x, y)) M ) x, y). 3.3) Simplifying the above equation with 3.29), we obtain the reconstruction formula for the solution of.) given by ) cosh qx, y) = 8iM ) x x, y) 8 M ) ) Note that the RH problem also can be solved via the Cauchy projector, which results in the integral representation for the solution of.) in terms of the solution M. Indeed, let J = I J. Then, the matrix RH problem can be written as M + M = M + J, k R, 3.32) where the jump matrix J is given by b k)e Jx, y, k) = ) bk)e 2θx,y,k), k R. 3.33) Employing the Cauchy projector, the solution of the RH problem is found in terms of the Cauchy type of the integral equation: M + x, y, k) = I + M + x, y, l) 2iπ Jx, y, l) dl l k. 3.34) 2

8 G. Hwang, J. Nonlinear Sci. Appl. 8 25), In particular, the first column of 3.34) yields ) ) M + x, y, k) M 2 + = x, y, k) 2iπ M + 2 x, y, l)bl)e 2θx,y,l) ) dl M 22 + x, y, l)bl)e2θx,y,l) l k. 3.35) Substituting equation 3.35) into the Lax pair and collecting the 2, )-component of the resulting expression, finally, we obtain the integral representation of the solution for the elliptic sinh-gordon equation in the half plane given by iq x x, y) + q y x, y) = 2 ) µ + π 22 x, y, k)bk)eω k)x+ω 2 k)y dk. 3.36) In summary, we now state the existence theorem of the solution for the elliptic sinh-gordon equation in the half plane. Theorem 3.2. Assume the functions g + 2mπ H R) for some m Z, and g H R) with the sufficiently small H norms of g and g. Let the functions ak) and bk) be given by 3.4) in Proposition 3.. Suppose that given g x), there exists a function g x) such that the global relation is satisfied ak) =, Im k, bk) =, k. 3.37) Let Mx, y, k) be the solution of the following matrix Riemann-Hilbert RH) problem M x, y, k) = M + x, y, k)jx, y, k), k R, 3.38) where det M ± ) =, M ± = I + O/k) as k and the jump matrix is given in 3.25). Then the Reimann-Hilbert problem is uniquely solvable and the function qx, y) defined by iq x + q y = 4i lim k km 2, solves the elliptic sinh-gordon equation.) in the half plane satisfying cosh qx, y) = 8i lim k km x 8 lim k km 2) ) qx, ) = g x), q y x, ) = g x). 3.4) Proof. First, note that the vanishing lemma [2] verifies the unique solvability of the RH problem 3.42), that is, if M = O/k) as k, the RH problem has only the trivial solution. The proof that qx, y) defined in 3.39) solves the elliptic sinh-gordon equation.) is based on the so-called dressing method [2] which was also discussed before the theorem. To prove 3.4), define the map {ak), bk)} {g x), g x)} given by x) cosh g x) = 8i lim km x 8 lim k k km x) 2 ) 2, 3.4a) x) iġ x) + g x) = 4i lim km 2, 3.4b) k where M x) is the solution of the following matrix Riemann-Hilbert problem M x) x) x, k) = M + x, k)j x) x, k), k R, 3.42) with M x) = I + O/k) as k and the jump matrix J x) defined by J x) b k) ) a k) x, k) = e 2ω k)x, k R. 3.43) bk) ak) e2ω k)x Note that if the global relation 3.37) is satisfied, then J x) x, k) = Jx,, k), 3.44) which implies that M x) x, k) = Mx,, k). Hence, letting y = in 3.39), equation 3.4) follows.

9 G. Hwang, J. Nonlinear Sci. Appl. 8 25), Linear limit We next show that the linear limit of the solution of.) coincides with the solution of the linearized equation for the elliptic sinh-gordon equation known as the modified Helmholtz equation in the half plane q xx + q yy = q, < x <, y >, 4.) qx, ) = g x), < x <. 4.2) It has been shown in [9] that the Fokas method was applied to solve 4.) and the solution of 4.) was found in the form qx, y) = e ω k)x ω 2 k)y ω 2 k)ĝ k) dk 2π k, 4.3) where the linear dispersion relations are defined by ω k) = k ), ω 2 k) = k + ), 4.4) 2i k 2 k and the spectral function ĝ is given by ĝ k) = e ω k)x qx, )dx. 4.5) Regarding the linear limit, we consider equation 3.36). First, we note that in the linear limit, µ + ) 22, Φ, Φ ) Hence, the spectral function bk) becomes bk) = 4 e 2ω k)ξ rξ) g ) ξ) dξ, 4.7) where rx) = iġ x) + q y x, ) to suppress the notation for brevity, and then equation 3.36) leads to iq x x, y) + q y x, y) = 2 π e 2ω k)ξ bk)e 2θx,y,k) dk. 4.8) Noting iω k) + ω 2 k) = k, we find the linear limit of the solution for the elliptic sinh-gordon equation in the half plane given by qx, y) = 2θx,y,k) dk bk)e π k. 4.9) It should be now required to characterize the spectral function bk) appearing in 4.9). Note that b k) = for k thanks to the global relation 3.8) and hence, we find rξ) + g ) ξ) dξ =, k. 4.) 4 Employing the change of variable k 4 4k in 4.), we then obtain e 2ω k)ξ rξ) + g ξ)) dξ =. 4.) Adding the above equation with 4.7), the function bk) can be written as bk) = ω 2 k) Thus, substituting 4.2) into 4.9) yields the linear limit given by qx, y) = π e 2θx,y,k) ω 2 k) e 2ω k)ξ g ξ)dξ. 4.2) e 2ω k)ξ g ξ)dξ dk k. 4.3) Finally, performing the change of variable k, we find the linear limit of the solution for the elliptic sinh-gordon equation in the half plane, which coincides with 4.3).

10 G. Hwang, J. Nonlinear Sci. Appl. 8 25), Concluding remarks In conclusion, we have studied the boundary value problems for the elliptic sinh-gordon equation posed in the half plane. We have done so by applying the Fokas method, which is a significant extension of the inverse scattering transform. We have derived the global relation for the elliptic sinh-gordon equation that involves all known and unknown boundary values. Also, we have presented the existence theorem for the unique solution provided that the boundary values satisfy the global relation. Moreover, we have derived the representation of the solution in terms of the unique solution of the Riemann-Hilbert problem whose jump matrix are uniquely defined by the spectral functions. In addition to solving the elliptic sinh-gordon equation, we have verified that the linear limit coincides with the solution of the linearized equation known as the modified Helmhotz equation. We remark that one of the most difficult steps in the implementation of the Fokas method is to characterize unknown boundary values, known as the Dirichlet-to-Neumann map []. For example, for the Dirichlet boundary value problem of the elliptic sinh-gordon equation in the half plane, the Dirichlet boundary datum qx, ) is given, whereas the Neumann boundary value q y x, ) is unknown. This unknown boundary value appears in the representation of the solution through the spectral functions and hence, it is required to characterize the unknown Neumann boundary value for the explicit solution. This can be done by analyzing the global relation as was done in [4]. It is noted that in [4] the Dirichlet-to-Neumann map for a nonlinear elliptic partial differential equation, namely the elliptic sine-gordon equation, was reported for the first time. Moreover, we expect that it could be possible to study the effective characterization of the Dirichlet-to-Neumann map by employing the perturbative scheme as in [3, 5, 6, 7]. We will discuss regarding these issues in the future. Acknowledgement The work is supported by the Daegu University Research Grant, 23. References [] M. J. Ablowitz, P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering, Cambridge: Cambridge University Press, 99). [2] G. Biondini, G. Hwang, Initial-boundary value problems for discrete evolution equations: discrete linear Schrödinger and integrable discrete nonlinear Schrödinger equations, Inverse. Problems, 24 28), 44 pages. [3] G. Biondini, D. Wang, Initial-boundary-value problems for discrete linear evolution equations, IMA J. Appl. Math., 75 2), [4] M. Boiti, J. J-P. Leon, F. Pempinelli, Integrable two-dimensional generalisation of the sine- and sinh-gordon equations, Inverse Problems, 3 987), , 2 [5] P. Deift, X. Zhou, A steepest descent method for oscillatory Riemann-Hilbert problems, Bull. Amer. Math. Soc., ), [6] P. Deift, S. Venakides, X. Zhou, New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems, Int. Math. Res. Notices, 6 997), [7] A. S. Fokas, A unified transform method for solving linear and certain nonlinear PDEs, Proc. Roy. Soc. London A, ), [8] A. S. Fokas, On the integrability of certain linear and nonlinear partial differential equations, J. Math. Phys., 4 2), [9] A. S. Fokas, Two dimensional linear PDEs in a convex polygon, Proc. Roy. Soc. London A, 457 2), [] A. S. Fokas, Integrable nonlinear evolution equations on the half-line, Comm. Math. Phys., 23 22), 39. [] A. S. Fokas, The generalized Dirichlet-to-Neumann map for certain nonlinear evolution PDEs, Comm. Pure Appl. Math., LVIII 25), [2] A. S. Fokas, A Unified Approach to Boundary Value Problems, CBMS-NSF regional conference series in applied mathematics, SIAM, 28)., 3, 3 [3] A. S. Fokas, J. Lenells, The unified method: I. Non-linearizable problems on the half-line, J. Phys. A: Math. Theor., 45 22),

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