Integrability and nonlinearity in field theory XVII International conference on Geometry, Integrability and Quantization 5 10 June 2015, Varna, Bulgaria Systems of MKdV equations related to the affine Lie algebras V. S. Gerdjikov Institute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria with D.M. Mladenov, A.A. Stefanov, S.K. Varbev Sofia University St. Kliment Ohridski, Bulgaria and A. B. Yanovsky Cape Town University
PLAN The inverse scattering method Hierarchies of integrable nonlinear evolution equations NLEE) Reductions of polynomial bundles mkdv equations related to simple Lie algebras The ISM as a GFT Conclusions and open questions
Based on: Mon1 V. S. Gerdjikov, G. Vilasi, A. B. Yanovski. Integrable Hamiltonian Hierarchies. Spectral and Geometric Methods Lecture Notes in Physics 748, Springer Verlag, Berlin, Heidelberg, New York 2008). ISBN: 978-3-540-77054-1. V S Gerdjikov, A B Yanovski. CBC systems with Mikhailov reductions by Coxeter Automorphism. I. Spectral Theory of the Recursion Operators. Studies in Applied Mathematics 134 2), 145 180 2015). DOI: 10.1111/sapm.12065. V. S. Gerdjikov. On new types of integrable 4-wave interactions. AIP Conf. proc. 1487 pp. 272-279; 2012). V. S. Gerdjikov. Riemann-Hilbert Problems with canonical normalization and families of commuting operators. Pliska Stud. Math. Bulgar. 21, 201 216 2012). V. S. Gerdjikov. Derivative Nonlinear Schrödinger Equations with
Z N and D N Reductions. Romanian Journal of Physics, 58, Nos. 5-6, 573-582 2013). V. S. Gerdjikov, A. B. Yanovski On soliton equations with Z h and D h reductions: conservation laws and generating operators. J. Geom. Symmetry Phys. 31, 57 92 2013). V. S. Gerdjikov, A B Yanovski. Riemann-Hilbert Problems, families of commuting operators and soliton equations Journal of Physics: Conference Series 482 2014) 012017 doi:10.1088/1742-6596/482/1/012017 V. S. Gerdjikov, D. M. Mladenov, A. A. Stefanov, S. K. Varbev. Integrable equations and recursion operators related to the affine Lie algebras A 1) r. ArXiv: 1411.0273v1 [nlin-si] Submitted to JMP. V.S. Gerdjikov, D.M. Mladenov, A. A. Stefanov, S.K. Varbev. MKdV-type of equations related to sln, C) algebra In Mathematics in Industry, Ed. A. Slavova. Cambridge Scholar Publishing, pp. 335 344 2014).
The inverse scattering method The inverse scattering method for the N-wave equations Zakharov, Shabat, Manakov 1973). Lax representation: [L, M] 0, Lψ i ψ x + U 1x, t) λj)ψx, t, λ) = 0, Mψ i ψ t + V 1x, t) λk)ψx, t, λ) = 0, where J, K constant diagonal matrices. Eq. a) is satisfied identically. λ 2 a) [J, K] = 0, λ b) [U 1, K] + [J, V 1 ] = 0, λ 0 c) iv 1,x iu 1,t + [U 1, V 1 ] = 0.
Eq. b) is satisfied identically if: U 1 x, t) = [J, Q 1 x, t)], V 1 x, t) = [K, Q 1 x, t)], Then eq. c) becomes the N-wave equation: [ i J, Q ] [ 1 i K, Q ] 1 + [[K, Q 1 ], [J, Q 1 ]] = 0. t x Simplest non-trivial case: N = 3, g sl3), Q 1 x, t) = Then the 3-wave equations take the form: 0 u 1 u 3 u 1 0 u 2 u 3 u 2 0 u 1 t a 1 a 2 u 1 b 1 b 2 x + κϵ 1ϵ 2 u 2u 3 = 0, u 2 t a 2 a 3 u 2 b 2 b 3 x + κϵ 1u 1u 3 = 0, u 3 t a 1 a 3 u 3 b 1 b 3 x + κϵ 2u 1u 2 = 0,.
where κ = a 1 b 2 b 3 ) a 2 b 1 b 3 ) + a 3 b 1 b 2 ). Solving Nonlinear Cauchy problems by the Inverse scattering method Find solution to the N-wave eqs. such that Q 1 x, t = 0) = q 0 x). q 0 L 0 L t>0 qx, t) I III T 0, λ) II T t, λ) Step I: Given Q 1 x, t = 0) = q 0 x) construct the scattering matrix T λ, 0).
Jost solutions: Lϕx, λ) = 0, lim ϕx, x λ)eiλjx = 11, i.e. Lψx, λ) = 0, lim ψx, x λ)eiλjx = 11, T λ, 0) = ψ 1 x, λ)ϕx, λ). Step II: From the Lax representation there follows: i T t λ[k, T λ, t)] = 0, T λ, t) = e iλkt T λ, 0)e iλkt. Step III: Given T λ, t) construct the potential Q 1 x, t) for t > 0. For g sl2) GLM eq. Volterra type integral equations For higher rank simple Lie algebras GLM eq. become very complicated. But it can be reduced to Riemann-Hilbert problem. Important: Thus the nonlinear Cauchy problem reduces to a sequence of three linear Cauchy problems; each has unique solution!
Hierarchies of integrable nonlinear evolution equations We can choose more complicated M-operators: for the NLS type eqs: Then V x, t, λ) = V 2 x, t) + λv 1 x, t) λ 2 K. i T t λ2 [K, T λ, t)] = 0, for the MKdV type eqs: V x, t, λ) = V 3 x, t) + λv 2 x, t) + λ 2 V 1 x, t) λ 3 K. i T t λ3 [K, T λ, t)] = 0, With each Lax operator L one can relate a hierarchy of integrable NLEE.
Reductions of Lax pairs a) AU x, t, ϵλ )Â = Ux, t, λ), AV x, t, ϵλ )Â = V x, t, λ), b) BU x, t, ϵλ ) ˆB = Ux, t, λ), BV x, t, ϵλ ) ˆB = V x, t, λ), c) CUT x, t, λ)ĉ = Ux, t, λ), CV x, t, λ)ĉ = V x, t, λ), where ϵ 2 = 1 and A, B and C are elements of the group G such that A 2 = B 2 = C 2 = 11. As for the fundamental analytic solutions we have a) Aξ +, x, t, ϵλ )Â = ˆξ x, t, λ), b) Bξ +, x, t, ϵλ ) ˆB = ξ x, t, λ), c) Cξ +,T x, t, λ)ĉ = ˆξ x, t, λ), For the Z N -reductions we may have: Dξ ± x, t, ωλ) ˆD = ξ ± x, t, λ), DUx, t, ωλ) ˆD = Ux, t, λ), DV x, t, ωλ) ˆD where ω N = 1 and D N = 11. = V x, t, λ),
NLS and MKdV eqs with sln)-series DNLS type equations Special examples of DNLS systems of equations can be found in VSG - 1988. We will give some particular examples when M operator is from second and third degree in λ. Those equations admit the following Hamiltonian formulation q i t = x δh δq r+1 i The first interesting nontrivial case is when M is quadratic polynomial in λ and g A 1) 2 algebra. The potential of L is given by Ux, t, λ) = 0 q 1 q 2 q 2 0 q 1 q 1 q 2 0 λ ). 1 0 0 0 ω 0 0 0 ω 2 where ω = e 2πi/3. This gives us the system of integrable nonlinear partial,
differential equations i q 1 t + iγ x q2 2) + γ 3 3 2 q 1 x 2 = 0, i q 2 t + iγ 3 2 q 2 x q2 1) γ 3 x 2 = 0. The corresponding Hamiltonian is H = iγ ) 3 q 1 q 2 6 x q q 2 1 γ x 3 q3 1 + q2). 3 In the case of A 1) 3 algebra using the potential 0 q 1 q 2 q 3 Ux, t, λ) = q 3 0 q 1 q 2 q 2 q 3 0 q 1 λ q 1 q 2 q 3 0 1 0 0 0 0 i 0 0 0 0 1 0 0 0 0 i we obtain the system of integrable nonlinear partial differential equations,
i q 1 t + 2iγ x q 2q 3 ) + γ 2 q 1 x 2 = 0, i q 2 t + iγ x q2 1) + iγ x q2 3) = 0, i q 3 t + 2iγ x q 1q 2 ) γ 2 q 3 x 2 = 0. The corresponding Hamiltonian is H = iγ q 1 q 3 2 x q q 3 1 x + 1 ) 2 x q2 2) γq 2 q1 2 + q3). 2 Systems of equations of mkdv type These are equations with cubic dispersion laws, therefore the M-operators are also cubic polynomials in λ. In the case of A 1) 1 algebra, with the following potential ) ) Ux, t, λ) = 0 q1 q 1 0 λ 1 0 0 1,
we obtain the well-known focusing mkdv equation where α = a3 b α q 1 t = 1 4 3 q 1 x 3 1 2 x q3 1),. In this case the Hamiltonian is H = 1 q1 ) ) 2 q1 4. 8α x In the case of A 1) 2 algebra we obtain a trivial system of equations t q 1 = 0 and t q 2 = 0 and the corresponding Hamiltonian is bilinear with respect to q 1 and q 2. In the case of A 1) 3 algebra the potential of the Lax operator is parameterized by Ux, t, λ) = 0 q 1 q 2 q 3 q 3 0 q 1 q 2 q 2 q 3 0 q 1 q 1 q 2 q 3 0 λ 1 0 0 0 0 i 0 0 0 0 1 0 0 0 0 i,
which is related to the following system of mkdv type equations α q 1 t = 1 2 ) q 1 2 x x 2 + 3 q 2 x q 3 + 3q 1 q2 2 + q3 3, α q 2 t = 1 4 α q 3 t = 1 2 x x The corresponding Hamiltonian is 2 q 2 x 2 + 3 x 2 q 3 x 2 3 q 2 x q 1 + 3q 3 q2 2 + q1 3 q 2 1 q3 2 ) + 12q1 q 2 q 3 2q2 3 ). ), H = 1 α + 1 2 q2 3 + 1 6 q 3 dx q 2 x 1 6 1 4 q4 1 1 8 q4 2 + 1 4 q4 3 + 3 2 q 1q2q 2 3 + 1 2 q q 1 1q 2 x 1 2 q2 1 ) ) q1 q3 + 1 ) 2 q2 1 x x 24 x 2 q q 3 2q 3 x ). 2 q 1 x 2 1 12 q 2 q 2 2 x 2 + 1 6 q 2 q 3 1 x 2 q 2 x
The next example is related to A 1) 4. The potential of the Lax operator now is Ux, t, λ) = 0 q 1 q 2 q 3 q 4 q 4 0 q 1 q 2 q 3 q 3 q 4 0 q 1 q 2 q 2 q 3 q 4 0 q 1 q 1 q 2 q 3 q 4 0 The set of equations is α q 1 t = x α q 2 t = x α q 3 t = x α q 4 t = x c1 2s 2 1 c 2 2s 2 2 c 2 c1 2s 2 1 2s 2 2 λ 1 0 0 0 0 0 ω 0 0 0 0 0 ω 2 0 0 0 0 0 ω 3 0 0 0 0 0 ω 4, ω = e2πi/5 2 q 1 x 2 + 3 2s 1 q 4 q 2 x + 3 2s 2 q 3 q 3 x + 3q 1q 2 q 3 + q 3 2 + 3q 3 q 2 4 2 q 2 x 2 3 q 4 q 3 2s 2 x + 3 q 1 q 1 2s 1 x + 3q 1q 2 q 4 + q4 3 + 3q 1 q3 2 2 q 3 x 2 + 3 q 1 q 2 2s 2 x 3 q 4 q 4 2s 1 x + 3q 1q 3 q 4 + q1 3 + 3q 4 q2 2 2 q 4 x 2 3 2s 1 q 1 q 3 x 3 2s 2 q 2 q 2 x + 3q 2q 3 q 4 + q 3 3 + 3q 2 q 2 1 ), ), ), ),
where ) ) kπ kπ s k = sin, c k = cos, s 1 = 1 5 5 4 c 1 = 1 4 1 + 5), s 2 = 1 4 The Hamiltonian is H = 2b 3a 3 dx + 3 8s 1 + 3 8s 2 10 2 5, 10 2 5, c 2 = 1 4 5 1). c 1 q 1 q 4 2s 2 1 x x + c 2 q 2 q 3 2s 2 2 x x + q 1q3 3 + q2q 3 4 + q 3 q4 3 ) q4 2 q 2 x 2q q 4 2q 4 x + 2q q 1 1q 3 x q 3 q2 1 + 3q 1 q 2 q 3 q 4 + q 3 x 1q 2 ) ) q2 2 q 1 x 2q q 2 1q 2 x + 2q q 3 3q 4 x q 4 q2 3. x
Additional Involutions. Real Hamiltonian forms Along with the Z r+1 -reduction we can introduce one of the following involutions Z 2 -reductions) on the Lax pair: a) K0 1 x, t, κ 1 λ))k 0 = Ux, t, λ), κ 1 λ) = ω 1 λ ; b) K0 1 x, t, κ 1 λ))k 0 = Ux, t, λ), κ 1 λ) = ω 1 λ ; c) U T x, t, λ) = Ux, t, λ), where K 2 0 = 11. If we choose K 0 = r+1 k=1 E k,r k+2 then the action of K 0 on the basis is as follows K 0 J s k) ) K0 = ω ks 1) J k) s, K 0 J s k) ) K0 = ω k J k) s.
An immediate consequences are the constraints on the potentials a) K0 1 Q x, t)k 0 = Qx, t), K0 1 1) J 0 ) K 0 = ω 1 J 1) 0, b) K0 1 Q x, t)k 0 = Qx, t), K0 1 1) J 0 ) K 0 = ω 1 J 1) 0, c) Q T x, t) = Qx, t), J 1) 0 )T = J 1) 0. Thus we obtain the algebraic relations below a) q j x, t) = q j x, t), α = α ; b) q j x, t) = q r j+1 x, t), α = α ; c) q j x, t) = q r j+1 x, t), where j = 1,..., r, are compatible with the evolution of the mkdv equations. If we apply case a) we get the same set of mkdv equations with q 1, q 2 and q 3 being purely real functions. In the case b) we put q 1 = q 3 = u
and q 2 = q 2 = iv and we get α v t = 1 3 v 4 x 3 + 3 u 2 4i x 2 u ) 2) 3 x u 2 v) + 1 2 x v3, α u t = 1 3 u 2 x 3 i3 u v ) 3 2 x x 2 x uv2 ) x u ) 3, 2 where u is a complex function but v is a purely real function. corresponding Hamiltonian is The H = 1 α 1 4 u4 1 8 v4 + 1 4 u ) 4 + 3 2 u 2 v 2 + i u uv 2 + 1 u 6 x 2 1 24 v x x i v u2 2 x + i 2 u ) 2 v x ) 2 i 2 u v u x 1 6 u 2 u x 2 + 1 12 v 2 v x 2 1 u 6 u 2 x 2 The case c) leads to the well known defocusing mkdv equation α u t = 1 2 3 u x 3 x u3 ), ).
where u is a complex function. The corresponding Hamiltonian is u H = 1 ) ) 2 + u 4. 4α x find And finally, considering A 1) 5 algebra with D 6 -reduction, case c) we α u t = 2 3 u x 3 2 3 x u v ) x α v t = 3 2 x 2 u 2 ) 6 x u2 v), 6 x uv2 ), where u and v are complex functions. The Hamiltonian is given by u H = 1 ) 2 + ) ) v 3u 2 + 3u 2 v 2. α x x
Figure 1: The contour for the RHP of L with Z -symmetry. l 3 λ l 4 Ω 3 Ω 2 l 2 l 5 Ω 4 Ω 1 l 1 Ω 5 Ω 0 l 6 l 0 Ω 6 Ω 11 l 7 Ω 7 Ω 8 Ω 9 Ω 10 l 11 l 8 l 10 l 9
ISP and RHP Fundamental analytic solutions of L χ n ux, t, λ) and solutions to the RHP: m ν x, t, λ) = χx, t, λ)e ijλx. The rays l ν are defined by: Im λαj) = 0, α δ ν g ν g. The RHP is: m + ν x, λ) = m ν x, λ)e ijλx g ν λ)e ijλx g ν λ) = Ŝ ν λ)s + ν λ) = ˆD ν λ) ˆT + ν λ)t ν λ)d + ν λ). Here S ν ± λ), T ν ± λ), D ν ± λ) are defined by the asymptotic of m ± ν x, λ) when x ± : S ν ± λ) = lim e iλjx m ± ν x, λ)e iλjx) = lim x x eijλx χ ± ν x, λ) Tν λ)d ν ± λ) = lim e iλjx m ± ν x, λ)e iλjx) = lim x x + eijλx χ ± ν x, λ). 2) 1)
One could write S ν ±, T ν ±, D ν ± also into the form S ν ± λ) = exp s ± ν,αλ)e ±α, T ν ± λ) = exp t ± ν,αλ)e ±α 3) α δ + ν D ± ν,αλ) = exp± α δ + ν α π ν d ± ν,αλ)h α ). 4) In other words S ν ±, T ν ±, D ν ± belong to the subgroup G ν with Lie algebra g ν. The fact that the factors S ν ±, T ν ±, D ν ± have the above form is a consequence of the following relations that hold for λ l ν lim x ± E α, m ± ν E β ˆm ± ν = 0, α, β, Im λα β)j)) 0 lim x ± H, m± ν E β ˆm ± ν = 0, β, H h, Im λβj)) 0 lim E β, m ± ν H ˆm ± ν = 0, β, H h, Im λβj)) 0. x ± 5) The minimal sets of scattering data that determine uniquely T λ) and
Qx, t) are T S = T T = 2 {s ± ν,αλ) : α δ ν +, λ l ν } 6) ν=0 2 {t ± ν,αλ) : α δ ν +, λ l ν }. 7) ν=0 Completeness of squared solutions and generalized Fourier transforms. Theorem The sets of squared solutions e ν,il x, λ) form complete sets of functions in M J. The completeness relation has the form: = 1 π δx y)π 0 = 2h 1 ν=0 1) ν l ν dλg ν+1 x, y, λ) G ν x, y, λ)) 2i N Res λ=λj G ν x, y, λ) j=1
Π 0 = α>0e α E α E α E α ) G ν+1 x, y, λ) = e ν+1,α x, λ) e ν+1, α y, λ), α + ν G ν x, y, λ) = e ν, α x, λ) e ν,α y, λ) + α ν 2 h ν,s x, λ) h ν,s y, λ). s=1
l 1 λ + l 2 + l 0 + + l 3 + + l 5 l 4 Figure 2: The contours γ ν = l ν γ ν, l ν+1.
Expansions over the squared solutions : Qx, t) = i 5 1) ν α ν J) dλ s + α 2π ν,νλ)e ν+1;α x, λ) s α ν,νλ)e ν; α x, λ) ) l ν ν=0 + DS 8) ad 1 J i 2π δqx, t) = 5 1) ν ν=0 dλ δs + α ν,νλ)e ν+1;α x, λ) + δ s α ν,νλ)e ν; α x, λ) ) + l ν DS 9) e αν ;νx, λ) are generalizations of e iλx. We need the analogs of id/dx for which id/dx)e iλx = λe iλx Λ + λ)e αν ;νx, λ) = 0, Λ λ)e αν ;νx, λ) = 0, Λ ± Xx) ad 1 J i dx [ x ]) dx + i [J, Qx)], dy [[J, Qy)], Xy)]. ±
In order to treat NLEE consider variations of the form: δq Q t δt + Oδt)2 ), 10) and keep only first order of δt. Then we have the expansion: iad 1 J i 2π Qx, t) = t 5 1) l ν dλ ν ν=0 i s+ α ν,ν t e ν+1;α x, λ) + i s α ν,ν t e ν; α x, λ) ) + DS Λad 1 J [J 2, Qx, t)] = = i 5 1) ν α ν J) dλ λ 3 s + α 2π ν,νλ)e ν+1;α x, λ) s α ν,νλ)e ν; α x, λ) ) + l ν ν=0
iad 1 Qx, t) J t i 2π + Λad 1 J [J 2, Qx, t)] mkdv = 5 ) 1) ν dλ i ν=0 l s+ α ν,ν + λ 3 s + α ν t ν,ν e ν+1;α x, λ)+ ) ) 11) + i s α ν,ν λ 3 s α t ν,νλ) e ν; α x, λ) + = 0 DS i.e. these mkdv equations are equivalent to the following linear equations i s+ α ν,ν t + λ 3 s + α ν,ν = 0, i s α ν,ν λ 3 s α t ν,ν = 0. These GFT linearize the NLEE of mkdv type!. 12) Solving the RHP and soliton solutions The dressing Zakharov-Shabat method - 1974), Mikhailov - 1981)
Assume we have a regular solution of the RHP ξ 0 ν+1x, t, λ) = ξ 0 νx, t, λ)g 0 νx, t, λ) Regular: det m 0 ν 0 for λ Ω ν Construct a new, singular solution of the RHP ξ 1 νx, t, λ) = ux, t, λ)ξ 0 ν+1x, t, λ), ux, t, λ) is the dressing factor, which may have poles and zeroes in λ. The regular solution corresponds to potential Q 0 of L; we may even choose Q 0 = 0. The new singular solution of RHP corresponds to new potential Q which will depend on additional parameters. One soliton solution of first type: ux, t, λ) = 11 + 1 A1 + J 1 A 1 J 3 λ λ 1 λω 2 + J 2 A 1 J 2 ) 13) λ 1 λω λ 1 where A 1 ξ, η) is a 3 3 degenerate matrix of the form A 1 x, t) = nx, t) m T x, t) A 1 ) ij x, t) = n i x, t)m j x, t). 14)
By construction ux, t, λ) satisfies the Z 3 -symmetry. The Z 2 -symmetry on ux, t, λ) can be put in the form uξ, η, λ)a 1 0 u ξ, η, λ )A 0 = 11. 15) and leads to algebraic equations which allow us to express the components of n j x, t) in terms of m k x, t): n 1 = 2λ 3 1m 3 λ 2 1 m 3 m 1 + λ 1 2 m 2 2 + λ 2, 1 m 1 m 3 2λ 3 1m 2 = 2iρ 1 m 3 2m 1 m 3 m 2 2 n 2 = λ 2, 1 m 3 m 1 + λ 2 1 m 2 2 + λ 1 2 m 1 m 3 = 2iρ 1 m 2 16) n 3 = 2λ 3 1m 1 λ 1 2 m 3 m 1 + λ 2, 1 m 2 2 + λ 2 1 m 1 m 3 = 2iρ 1m 1 m 2 2. After putting λ 1 = iρ 1 we obtain the 1-soliton solution of the first
type for Tzitzeica eq.: ) ϕ 1s x, t) = 1 2 ln µ 01 2 e 3X 1 4 cos 2 Ω 1 ) 6 8 µ 01 µ 02 cos Ω 1 ) + µ 2 02e 3X 1 4 µ 01 2 e 3X 1 cos2 Ω 1 ) + 4 µ 01 µ 02 cos Ω 1 ) + µ 2 02 e3x 1 17) where X 1 = 1 2 ρ 1 x t ), Ω 1 = ρ 1 3 2 ρ 1 x + t ). 18) ρ 1 Note: it is not traveling wave solution; it may have singularities! In the limit µ 02 0 we obtain a traveling wave solution of the form [ ) ] ϕx, t) = 1 2 ln 3 3 2 tanh2 2 ρ 1ξ + ρ 1 1 η) α 01 + 1. 19) 2 One Soliton Solutions of Second Type
Now the anzatz for the dressing factor is ux, t, λ) = 11 + 1 A1 + J 1 A 1 J 3 λ λ 1 λω 2 + J 2 A 1 J 2 ) λ 1 λω λ 1 1 A 1 3 λ + λ + J 1 A 1J 1 λω 2 + λ + J 2 A 1J 2 ) 20) 1 λω + λ 1 which obviously satisfies the Z 3 -reduction and the first Z 2 -reduction. Again we obtain an algebraic relations between n j x, t) in terms of m k x, t) which are more complicated: µ = m 3 m 2 m 1 m 3 m 2 m 1, ν = n 1 n 2 n 3 n 1 n 2 n 3, µ = M ν 21)
where M = c 1 P 1 0 0 ζ 1 K 1 0 0 0 c 1 P 2 0 0 ζ 1 K 2 0 0 0 c 1 P 3 0 0 ζ 1 K 3 ζ 1 K1 0 0 c 1 P1 0 0 0 ζ 1 K2 0 0 c 1 P2 0 0 0 ζ 1 K3 0 0 c 1 P3 22) The result is ν = M 1 ν c P 1 1 0 0 ζ 1 K1 0 0 0 c P 1 2 0 0 ζ 1 K2 0 M 1 0 0 c = P 1 3 0 0 ζ 1 K3 ζ 1 K 1 0 0 c 1 P1 0 0 0 ζ1 K 2 0 0 c 1 P2 0 0 0 ζ1 K 3 0 0 c 1 P3 23)
where P s = P s d s, Ps = P s d s, Ks = K s d s, K s = K s d 1 d 1 = ζ 1 ζ 1 K 1 K 1 c 1 c 1P 1 P 1 d 2 = ζ 1 ζ 1 K 2 K 2 c 1 c 1P 2 P 2 d 3 = ζ 1 ζ 1 K 3 K 3 c 1 c 1P 3 P 3. 24) From the above equations we obtain n in terms of m T n 1 = 1 d 1 c 1P 1 m 3 + ζ 1 K 1 m 3) n 2 = 1 d 2 c 1P 2 m 2 + ζ 1 K 2 m 2) n 3 = 1 d 3 c 1P 3 m 1 + ζ 1 K 3 m 1). The explicit x, t dependence of m j x, t) is 25) m 1 = ω 2 µ 01 e ix 1 Y 1 + µ 02 e ix 2 Y 2 + ωµ 03 e ix 3 Y 3 m 2 = µ 01 e ix 1 Y 1 + µ 02 e ix 2 Y 2 + µ 03 e ix 3 Y 3 26) m 3 = ωµ 01 e ix 1 Y 1 + µ 02 e ix 2 Y 2 + ω 2 µ 03 e ix 3 Y 3
where X 1 = xρ 1 + t ) cos β 1 2π ), Y 1 = xρ 1 t ) sin β 1 2π ρ 1 3 ρ 1 3 X 2 = xρ 1 + t ) cos β 1 ), Y 2 = xρ 1 t ) sin β 1 ) ρ 1 ρ 1 X 3 = xρ 1 + t ) cos β 1 + 2π ), Y 3 = xρ 1 t ) sin β 1 + 2π ρ 1 3 ρ 1 3 27) We determine the 1-soliton solution for the second kind of solitons using exactly the same technique Φ = 1 2 ln 1 1 n 1 m 1 1 λ 1 λ n 1m 1. 28) 1 ) ). Multisoliton solutions N = N 1 +N 2 with N 1 solitons of first type and N 2 solitons of second type can also be derived: They would correspond to 6N 1 + 12N 2 singularities of the RHP.
Reconstructing the potential Qx, t) from ux, t, λ) After constructing the dressing factor we use the fact that it satisfies the equation: i u x + Qx, t) λj)ux, t, λ) ux, t, λ)q 0x, t) λj) = 0, 29) Take the limit λ and use that and choose also Q 0 x, t) = 0. Then lim ux, t, λ) = 11, λ Qx, t) = lim λ λj ux, t, λ)jûx, t, λ)) 30) which allows you to express Qx, t) in terms of the residue A 1 x, t) = n 1 m 1.
l 1 λ Ω 1 Ω 0 l 2 + + l 0 + Ω 2 Ω 5 + l 3 + + l 5 Ω 3 Ω 4 l 4 Figure 3: The discrete eigenvalues of L with Z 3 -symmetry and Z 2 - symmetries. Two types of discrete eigenvalues, two types of soliton solutions.
Conclusions and some open questions The mkdv eqs. are Hamiltonian. View the jets Ux, t, λ) and V x, t, λ) as elements of co-adjoint orbits of some Kac-Moody algebra. Each of these eqs. has two types of soliton solutions. Find constraints on the soliton parameters that render them regular. One can derive their soliton interactions by evaluating the limits of the dressing factors for x ±. Thank you for your attention!