Some Elliptic Traveling Wave Solutions to the Novikov-Veselov Equation

Progress In Electromagnetics Research Symposium 006, Cambridge, USA, March 6-9 59 Some Elliptic Traveling Wave Solutions to the Novikov-Veselov Equation J. Nickel, V. S. Serov, and H. W. Schürmann University of Osnabrück, Germany University of Oulu, Finland Abstract An approach is proposed to obtain some exact explicit solutions in terms of elliptic functions to the Novikov-Veselov equation NVE[V x,y,t = 0. An expansion ansatz V ψ = j=0 a jf j is used to reduce the NVE to the ordinary differential equation f = Rf, where Rf is a fourth degree polynomial in f. The well-known solutions of f = Rf lead to periodic and solitary wave like solutions V. Subject to certain conditions containing the parameters of the NVE and of the ansatz V ψ the periodic solutions V can be used as start solutions to apply the linear superposition principle proposed by Khare and Sukhatme.. Introduction The Novikov-Veselov NV equation [ as a natural two-dimensional generalization of the celebrated Korteweg-de Vries KV equation [ has relevance in nonlinear physics in particular in inverse scattering theory [, 4 and mathematics cf. e. g., [5, 6. As regards to physics, Tagami [ derived solitary solutions of the NV equation by means of the Hirota method. Cheng [4 investigated the NV equation associated with the spectral problem x y + uψ = 0 in the plane and presented solutions by applying the inverse scattering transform. With regards to mathematics, Taimanov [5 investigated applications of the modified NV equation to differential geometry of surfaces. Ferapontov [6 used the stationary NV equation to describe a certain class of surfaces in projective differential geometry the so-called isothermally asymptotic surfaces. Apart from these applications solutions of the NV equation are interesting in and of themselves. In the following we derive some solutions of the NV equation by combining a symmetry reduction method [7, 8 and the Khare-Sukhatme superposition principle [9.. Elliptic Solutions.. General Considerations Following Novikov and Veselov [ we consider the system V t = V + V + uv + uv, u = V, where = x i y, = x + i y are the Cauchy-Riemann operators in R. System, is equivalent to V t = 4 V xxx V xyy + V u x + u y + u V x + u V y, V x = u x u y, V y = u y + u x 4 with ux,y,t = u x,y,t+iu x,y,t, where u is defined up to an arbitrary holomorphic function ϕ = ϕ +iϕ so that ϕ x = ϕ y, ϕ y = ϕ x. 4 imply u = x y DV + V + ϕ, u = DV + ϕ. 5 The operator D := x y + y x is well-defined [, 6, so that u, u can be inserted into. Traveling wave solutions V x,y,t = ψz, z = x + ky vt 6

50 Progress In Electromagnetics Research Symposium 006, Cambridge, USA, March 6-9 imply x = k y and thus lead to ϕ const. = C 0 + ic. Hence, can be written as vψ z = k ψ zzz + 6 k 4 k + ψψ z + ψ z C 0 + C k. 7 Following an approach outlined previously [7, 8, 4 it seems useful to find elliptic traveling wave solutions of the form p = follows from balancing the linear term of highest order with the nonlinear term in 7 p= ψz = a j fz j 8 j=0 with [5 dfz = αf 4 + 4βf + 6γf + 4δf + ɛ Rf. 9 dz The coefficients a 0, a, a, α, β, γ, δ, ɛ are assumed to be real but otherwise either arbitrary or interrelated. Inserting 8 into 7 and using 9 we obtain a system of algebraic equations that can be reduced to yield the nontrivial solutions α = 0, β = a + k, γ = 4a 0 + k + F k, δ, ɛ arbitrary, subject to a = 0, k 0, 0 α = a + k, β = a + k, γ = F 6k a + 4a 0 a 4a + k, δ = a a 0 a a 8a + k + a a F k, ɛ arbitrary, subject to a 0, k 0 with F = v + C 0 + kc. Thus, the coefficients of the polynomial Rf are partly determined leading to solutions fz of 9. As is well known [5, pp. 4 6, [6, p. 454 fz can be expressed in terms of Weierstrass elliptic function z;g,g according to R f 0 fz = f 0 + [ 4 z;g,g, 4 R f 0 where the primes denote differentiation with respect to f and f 0 is a simple root of Rf. The invariants g,g of z;g,g and the discriminant = g 7g are related to the coefficients of Rf [7, p. 44. They are suitable to classify the behaviour of fz and to discriminate between periodic and solitary wave like solutions [8. Solitary wave like solutions are determined by cf. and Ref. [8, pp. 65 65 R f 0 fz = f 0 + [ 4 e R f 0 + e csch, = 0, g < 0, e z 4 where e = g in. In general, fz according to is neither real nor bounded. Conditions for real and bounded solutions fz can be obtained by considering the phase diagram of Rf [9, p. 5. They are denoted as phase diagram conditions PDC in the following. An example of a phase diagram analysis is given in [4.

Progress In Electromagnetics Research Symposium 006, Cambridge, USA, March 6-9 5.. Periodic Solutions At first the coefficients according to 0 are considered. For simplicity we assume ɛ = 0, so that f 0 = 0 is a simple root of 9. The solution can be evaluated to yield V x,y,t = a 0 + a + k k δ + k F + 6a 0 k + + k k x + ky vt;g,g with g, g according to 0 and [8. Evaluating with coefficients according to with ɛ = 0 for simplicity in the same manner we obtain periodic solutions depending on a 0, a and a... Solitary Wave like Solutions To find the subset of solitary wave like solutions of the NV equation according to 0, the discriminant must vanish. This is given if δ = 0 or δ = 6a 0 k + + k F 8a k + k. For g < 0 we obtain solitary wave like solutions and here the PDC is fulfilled automatically for g < 0. If δ = 0, ɛ = 0, f 0 = 6a 0 k + + k F a k, we obtain cf. 8, V x,y,t = a 0 + 6a 0 k + + k F k sech [ 6a 0 + k + F k x + ky vt 4. 5 If δ = 6a 0 k + + k F 8a k + k, ɛ = 0, f 0 = 0, 8 reads [ V x,y,t = a 0 + 6a 0 k + + k F 4k tanh F k + a 0 x + ky vt. 6 + k Subject to 0 5, 6 represent general physical traveling solitary wave solutions of the NV equation for ɛ = 0. While periodic solutions depend on a 0 and a, solitary solutions only depend on a 0. Solitary wave like solutions according to can be obtained by an analogous procedure.. Superposition Solutions Khare and Sukhatme proposed a superposition principle for nonlinear wave and evolution equations NL- WEEs [9. They have shown that suitable linear combinations of periodic traveling-wave solutions expressed by Jacobian elliptic functions lead to new solutions of the nonlinear equation in question. Combining the approach above with this superposition principle we have evaluated the following start solutions for superposition [0 fz = γ+ 9γ 6βδ 4β dn 4δ γ+ 9γ 6βδ sn γ + 9γ 6βδz, 9γ 6βδ γ+ 9γ 6βδ, βδ > 0, γ > 0, γ + 9γ 6βδz, γ+ 9γ 6βδ, βδ > 0, γ < 0, γ 9γ 6βδ γ+ 9γ 6βδ 4β cn 9γ 6βδ 4 z, γ+ 9γ 6βδ, βδ < 0. 9γ 6βδ In 0 we choose ɛ = 0 for simplicity and thus, we obtain start solutions for superposition according to 7. As an example we consider solutions of the form dn for p =, further results for cn, sn and according to can be obtained in the same manner. According to 8, 0 the start solution for superposition reads 7 V x,y,t = a 0 + a A dn µx + ky vt,m, 8 with A, µ, m according to 7, so that the superposition ansatz can be written as Ṽ x,y,t = a 0 + a A i= dn [ µx + k y v t + i Km,m. 9

5 Progress In Electromagnetics Research Symposium 006, Cambridge, USA, March 6-9 Inserting Ṽ x,y,t denoting d i = dn µx + ky v t + i Km,m into 7 v v and using well known relations for c i and s i [, p. 6 leads to 6Aa µm k µ Aa + k c i d is i A a mµ k + k d i c j d j s j 0 i= i= j i 6a0 k Aa µm + k + C 0 + C k + m k µ + v c i d i s i = 0. Remarkably, µ Aa vanishes automatically [0,. By use of [, reads + k Aa µm Aa µm 6a0 k + k + C 0 + C k + m k µ + v c i d i s i i= Aa k m + q + k q c i d i s i = 0. Thus, the speed v in the superposition solution 9 is given by v = 6a 0k + k C 0 + C k + mk µ + Aa k m + q + k q. The start solution V and the superposition solution Ṽ are shown in Fig.. V, V i= i= 6 5 V 4 V -4-4 z Figure : V and Ṽ cf. 8, 9 for c =, k =, a 0 =, a =, C 0 =, C =, δ = 4 therefore: v = 8.66008. 4. Conclusion For the NV equation we have shown that a rather broad set of traveling wave solutions according to 6, 8 and subject to the nonlinear ordinary differential equation 9 can be obtained. Periodic and solitary wave solutions can be presented in compact form in terms of Weierstrass elliptic function and its limiting cases = 0, g 0, respectively. The phase diagram conditions PDC yield constraints for real and bounded solutions. Finally, it is shown that application of the Khare-Sukhatme superposition principle yields new periodic real, bounded solutions of the NV equation. Acknowledgment One of us J. N. gratefully acknowledges support by the German Science Foundation DFG Graduate College 695 Nonlinearities of optical materials.

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