Rogue periodic waves for mkdv and NLS equations

Transcription

1 Rogue periodic waves for mkdv and NLS equations Jinbing Chen and Dmitry Pelinovsky Department of Mathematics, McMaster University, Hamilton, Ontario, Canada AMS Sectional meeting, Buffalo NY, September D.Pelinovsky (McMaster University) Rogue periodic waves 1 / 4

2 The rogue wave of the cubic NLS equation The focusing nonlinear Schrödinger (NLS) equation admits the exact solution iψ t +ψ xx + ( ψ 1)ψ = ψ(x, t) = 1 4(1+4it) 1+4x + 16t. It was discovered by H. Peregrine (1983) and was labeled as the rogue wave. Properties of the rogue wave: It is developed due to modulational instability of the constant wave background ψ (x, t) = 1. It comes from nowhere: ψ(x, t) 1 as x + t. It is magnified at the center: M := ψ(, ) = 3. D.Pelinovsky (McMaster University) Rogue periodic waves / 4

3 Periodic waves of the modified KdV equation The modified Korteweg de Vries (mkdv) equation u t + 6u u x + u xxx = appears in many physical applications, e.g., in models for internal waves. The mkdv equation admits two families of the travelling periodic waves: positive-definite periodic waves u dn (x, t) = dn(x ct; k), c = c dn (k) := k, sign-indefinite periodic waves u cn (x, t) = kcn(x ct; k), c = c cn (k) := k 1, where k (, 1) is elliptic modulus. As k 1, the periodic waves converge to the soliton u(x, t) = sech(x t). As k, the periodic waves converge to the small-amplitude waves. D.Pelinovsky (McMaster University) Rogue periodic waves 3 / 4

4 Modulation theory for the Gardner equation References: E. Parkes, J. Phys. A, 5-36 (1987); R. Grimshaw et al., Physica D 159, (1) Start with the following Gardner equation with the parameter α: u t +αuu x + 6u u x + u xxx = and use the small-amplitude slowly-varying approximation u(x, t) = ǫ 1/[ ψ( ] ǫ(x + c t),ǫt)e i(k x+ω t) + c.c +O(ǫ), where ω = ω(k ) = k 3, c = ω (k ) = 3k, and ψ in scaled variables X and T satisfies the cubic NLS equation iψ T + 1 ω (k )ψ XX +β ψ ψ =, where ω (k ) = 6k and β = 6k α 6k. D.Pelinovsky (McMaster University) Rogue periodic waves 4 / 4

5 Application of the modulation theory The cubic NLS equation with ω (k ) = 6k and β = 6k α 6k : iψ T + 1 ω (k )ψ XX +β ψ ψ =. sign-indefinite periodic waves u cn (x, t) = kcn(x ct; k), c = c cn (k) := k 1, As k, u cn (x, t) kcos(x + t), hence k = 1, α = and β >. cn periodic waves are modulationally unstable. positive-definite periodic waves u dn (x, t) = dn(x ct; k), c = c dn (k) := k, As k, u dn (x, t) 1+k cos(x t), hence k =, α = 1, β =. dn periodic waves are modulationally stable. (See also Bronski Johnson Kapitula, 11 and Deconinck Nivala, 11) D.Pelinovsky (McMaster University) Rogue periodic waves 5 / 4

6 Main questions 1. Can one construct rogue waves for periodic waves of mkdv?. Can one compute the magnification factor for such rogue waves? u t + 6u u x + u xxx = Background for these questions: Numerically constructed rogue periodic waves for NLS (Kedziora Ankiewicz Akhmediev, 14) Numerically constructed rogue waves for two-phase solutions of NLS (Calini Schober, 17) Rogue waves from a superposition of nearly identical solitons for mkdv (Shurgalina E.Pelinovsky, 16) - two solitons (Slunyaev E.Pelinovsky, 16) - N solitons The magnification factor for such N-soliton rogue waves = N. D.Pelinovsky (McMaster University) Rogue periodic waves 6 / 4

7 Main results MKdV equation for u(x, t) R u t + 6u u x + u xxx = is a compatibility condition of the Lax pair ϕ(x, t) C : ϕ x = U(λ, u)ϕ, ϕ t = V(λ, u)ϕ. 1 For periodic waves u, we compute explicitly the periodic eigenfunctions ϕ for four particular eigenvalues λ. For each periodic eigenfunction ϕ, we construct the second linearly independent non-periodic solution ψ for the same values of λ. 3 By using Darboux transformations of mkdv with non-periodic function ψ, we define the rogue periodic waves in the closed (implicit) form. 4 From the implicit solutions, we compute the magnification factor explicitly. D.Pelinovsky (McMaster University) Rogue periodic waves 7 / 4

8 Rogue dn-periodic waves For dn-periodic waves the magnification factor is u dn (x, t) = dn(x ct; k), c = c dn (k) := k, M dn (k) = + 1 k, k [, 1]. The rogue dn-periodic wave is a superposition of the (modulationally stable) dn-periodic wave and a travelling algebraic soliton. 1 u t x Figure: The rogue dn-periodic wave of the mkdv for k =.99. D.Pelinovsky (McMaster University) Rogue periodic waves 8 / 4

9 Rogue cn-periodic waves For cn-periodic waves the magnification factor is u cn (x, t) = kcn(x ct; k), c = c cn (k) := k 1, M dn (k) = 3, k [, 1]. The rogue cn-periodic wave is a result of the modulational instability of the cn-periodic wave. 1 u t x 1 3 Figure: The rogue cn-periodic wave of the mkdv for k =.99. D.Pelinovsky (McMaster University) Rogue periodic waves 9 / 4

10 How did we complete the job? 1. For periodic waves u, we compute explicitly the periodic eigenfunctions ϕ for four particular eigenvalues λ. The AKNS spectral problem for ϕ(x, t) C : ( λ u ϕ x = U(λ, u)ϕ, U(λ, u) := u λ ), where u R is any solution of the mkdv. An algebraic technique based on the nonlinearization of Lax pair. Cao Geng, 199; Cao Wu Geng, 1999; Zhou, 9; Chen, 1; Fix λ = λ 1 C with an eigenfunction ϕ = (ϕ 1,ϕ ) C. Set u = ϕ 1 +ϕ R and consider the Hamiltonian system { dϕ1 dx dϕ dx = λ 1 ϕ 1 +(ϕ 1 +ϕ )ϕ = H ϕ, = λ 1 ϕ (ϕ 1 +ϕ )ϕ 1,= H ϕ 1 related to the Hamiltonian function H(ϕ 1,ϕ ) = 1 4 (ϕ 1 +ϕ ) +λ 1 ϕ 1 ϕ. D.Pelinovsky (McMaster University) Rogue periodic waves 1 / 4

11 How did we complete the job? 1. For periodic waves u, we compute explicitly the periodic eigenfunctions ϕ for four particular eigenvalues λ. The AKNS spectral problem for ϕ(x, t) C : ( λ u ϕ x = U(λ, u)ϕ, U(λ, u) := u λ ), where u R is any solution of the mkdv. An algebraic technique based on the nonlinearization of Lax pair. Cao Geng, 199; Cao Wu Geng, 1999; Zhou, 9; Chen, 1; Fix λ = λ 1 C with an eigenfunction ϕ = (ϕ 1,ϕ ) C. Set u = ϕ 1 +ϕ R and consider the Hamiltonian system { dϕ1 dx dϕ dx = λ 1 ϕ 1 +(ϕ 1 +ϕ )ϕ = H ϕ, = λ 1 ϕ (ϕ 1 +ϕ )ϕ 1,= H ϕ 1 related to the Hamiltonian function H(ϕ 1,ϕ ) = 1 4 (ϕ 1 +ϕ ) +λ 1 ϕ 1 ϕ. D.Pelinovsky (McMaster University) Rogue periodic waves 1 / 4

12 Periodic eigenfunctions Besides u = ϕ 1 +ϕ, we also have constraints u x = λ 1 (ϕ 1 ϕ ) and E u = 4λ 1 ϕ 1 ϕ, where E = 4H(ϕ 1,ϕ ) is conserved. Moreover, the nonlinear Hamiltonian system satisfies the compatibility condition of the Lax pair if and only if µ := ux u satisfies the following (Dubrovin?) equation ( ) 1 dµ = (µ λ 4 dx 1)(µ λ 1 E ). This ODE is satisfied if u satisfies the travelling wave reduction of the mkdv: d u dx + u3 = cu, where with real constants c and d given by ( ) du + u 4 = cu + d, dx c = 4λ 1 + E, d = E. Moreover, if u(x ct), then ϕ(x ct) is compatible with the time evolution. D.Pelinovsky (McMaster University) Rogue periodic waves 11 / 4

13 dn-periodic waves The connection formulas: For dn-periodic waves c = 4λ 1 + E, d = E. u dn (x, t) = dn(x ct; k), c = c dn (k) := k, we have d = k 1. Hence E = ± 1 k and λ 1 = 1 [ ] k 1 k λ + - λ - λ - λ D.Pelinovsky (McMaster University) Rogue periodic waves 1 / 4

14 cn-periodic waves The connection formulas: For cn-periodic waves c = 4λ 1 + E, d = E. u cn (x, t) = kcn(x ct; k), c = c cn (k) := k 1, we have d = k (1 k ). Hence E = ±ik 1 k and λ 1 = 1 [ ] k 1 ik 1 k λ * I λ I λ I λ * I D.Pelinovsky (McMaster University) Rogue periodic waves 13 / 4

15 How did we complete the job?. For each periodic eigenfunction ϕ, we construct the second linearly independent non-periodic solution ψ for the same values of λ. For λ = λ 1 C, we have one periodic solution ϕ = (ϕ 1,ϕ ) of ( ) λ u ϕ x = U(λ, u)ϕ, U(λ, u) :=, u λ where u R is any solution of the mkdv. Let us define the second solution ψ = (ψ 1,ψ ) by ψ 1 = θ 1 ϕ, ψ = θ + 1 ϕ 1, such that ϕ 1 ψ ϕ ψ 1 = (Wronskian is constant). Then, θ satisfies the first-order reduction dθ dx = uθ ϕ ϕ 1 ϕ 1 ϕ + u ϕ 1 +ϕ ϕ 1 ϕ. D.Pelinovsky (McMaster University) Rogue periodic waves 14 / 4

16 Non-periodic solutions Because u = ϕ 1 +ϕ, u x = λ 1 (ϕ 1 ϕ ), and E u = 4λ 1 ϕ 1 ϕ, we can rewrite the ODE for θ as dθ dx = θ uu u E 4λ 1u u E, where u E is assumed. Integration yields x θ(x) = 4λ 1 (u(x) E ) u(y) (u(y) E ) dy. Moreover, if u(x ct) and ϕ(x ct), then the time evolution yields [ ] x ct θ(x, t) = 4λ 1 (u(x ct) u(y) E ) (u(y) E ) dy t. up to translation in t. D.Pelinovsky (McMaster University) Rogue periodic waves 15 / 4

17 How did we complete the job? 3. By using Darboux transformations of mkdv with non-periodic function ψ, we define the rogue periodic waves in the closed form. One-fold Darboux transformation: ũ = u + 4λ 1pq p + q, where u and ũ are solutions of the mkdv and ϕ = (p, q) is a nonzero solution of the Lax pair with λ = λ 1 and u. Two-fold Darboux transformation: 4(λ 1 ũ = u+ λ )[ λ 1 p 1 q 1 (p + q ) λ p q (p1 + q 1 )] (λ 1 +λ )(p 1 + q 1 )(p + q ) λ [ 1λ 4p1 q 1 p q +(p1 q 1 )(p q )] where (p 1, q 1 ) and (p, q ) are nonzero solutions of the Lax pair with λ 1 and λ such that λ 1 ±λ. D.Pelinovsky (McMaster University) Rogue periodic waves 16 / 4

18 Rogue dn-periodic waves Using one-fold transformation with periodic eigenfunction (ϕ 1,ϕ ) yields ũ = u + 4λ 1ϕ 1 ϕ 1 k ϕ = = dn(x ct + K(k); k), 1 +ϕ dn(x ct; k) which is a translation of the dn-periodic wave. Using one-fold transformation with non-periodic (ψ 1,ψ ) yields ũ = u + 4λ 1ψ 1 ψ ψ 1 +ψ = u + 4λ 1 ϕ 1 ϕ (θ 1) (ϕ 1 +ϕ )(1+θ ) (ϕ 1 ϕ )θ, which is not a translation of the dn-periodic wave. As θ (as x + t almost everywhere): 1 k ũ(x, t) = dn(x ct + K(k); k). dn(x ct; k) At θ = (at (x, t) = (, )), the rogue wave is at the maximum point: ũ(, ) = + 1 k. D.Pelinovsky (McMaster University) Rogue periodic waves 17 / 4

19 Rogue dn-periodic waves Using one-fold transformation with periodic eigenfunction (ϕ 1,ϕ ) yields ũ = u + 4λ 1ϕ 1 ϕ 1 k ϕ = = dn(x ct + K(k); k), 1 +ϕ dn(x ct; k) which is a translation of the dn-periodic wave. Using one-fold transformation with non-periodic (ψ 1,ψ ) yields ũ = u + 4λ 1ψ 1 ψ ψ 1 +ψ = u + 4λ 1 ϕ 1 ϕ (θ 1) (ϕ 1 +ϕ )(1+θ ) (ϕ 1 ϕ )θ, which is not a translation of the dn-periodic wave. As θ (as x + t almost everywhere): 1 k ũ(x, t) = dn(x ct + K(k); k). dn(x ct; k) At θ = (at (x, t) = (, )), the rogue wave is at the maximum point: ũ(, ) = + 1 k. D.Pelinovsky (McMaster University) Rogue periodic waves 17 / 4

20 Rogue dn-periodic waves For dn-periodic waves the magnification factor is u dn (x, t) = dn(x ct; k), c = c dn (k) := k, M dn (k) = + 1 k, k [, 1]. The rogue dn-periodic wave is a superposition of the (modulationally stable) dn-periodic wave and a travelling algebraic soliton. 1 u t x Figure: The rogue dn-periodic wave of the mkdv for k =.99. D.Pelinovsky (McMaster University) Rogue periodic waves 18 / 4

21 Rogue cn-periodic waves Since λ 1 / R, one-fold transformation yields complex solutions of the mkdv. Using two-fold transformation with periodic (ϕ 1,ϕ ) and its conjugate yields ũ = u + 4k (1 k )u (k 1)u u 4 k (1 k ) (u ) = u, which is a translation of the cn-periodic wave. Using two-fold transformation with non-periodic (ψ 1,ψ ) and its conjugate: ] 4(λ I λ I) [λ I ψ 1 ψ (ψ 1 +ψ ) λ I ψ 1 ψ (ψ 1 +ψ ) ũ = u + (λ I +λ I) ψ1 +ψ λ I [. 4 ψ 1 ψ + ψ1 ψ ] As θ (as x + t everywhere): ũ(x, t) u(x, t). At θ = (at (x, t) = (, )), the rogue wave is at the maximum point: ũ(, ) = 3k. D.Pelinovsky (McMaster University) Rogue periodic waves 19 / 4

22 Rogue cn-periodic waves Since λ 1 / R, one-fold transformation yields complex solutions of the mkdv. Using two-fold transformation with periodic (ϕ 1,ϕ ) and its conjugate yields ũ = u + 4k (1 k )u (k 1)u u 4 k (1 k ) (u ) = u, which is a translation of the cn-periodic wave. Using two-fold transformation with non-periodic (ψ 1,ψ ) and its conjugate: ] 4(λ I λ I) [λ I ψ 1 ψ (ψ 1 +ψ ) λ I ψ 1 ψ (ψ 1 +ψ ) ũ = u + (λ I +λ I) ψ1 +ψ λ I [. 4 ψ 1 ψ + ψ1 ψ ] As θ (as x + t everywhere): ũ(x, t) u(x, t). At θ = (at (x, t) = (, )), the rogue wave is at the maximum point: ũ(, ) = 3k. D.Pelinovsky (McMaster University) Rogue periodic waves 19 / 4

23 Rogue cn-periodic waves For cn-periodic waves the magnification factor is u cn (x, t) = kcn(x ct; k), c = c cn (k) := k 1, M cn (k) = 3, k [, 1]. The rogue cn-periodic wave is a result of the modulational instability of the cn-periodic wave. 1 u t x 1 3 Figure: The rogue cn-periodic wave of the mkdv for k =.99. D.Pelinovsky (McMaster University) Rogue periodic waves / 4

24 Rogue periodic waves in NLS The NLS equation iu t + u xx + u u = has a similar Lax pair, e.g. ( λ u ϕ x = Uϕ, U = ū λ ). The NLS equation admits two families of the periodic waves: positive-definite periodic waves u dn (x, t) = dn(x; k)e ict, c = k, sign-indefinite periodic waves u cn (x, t) = kcn(x; k)e ict, c = k 1, where k (, 1) is elliptic modulus. Both periodic waves are modulationally unstable. D.Pelinovsky (McMaster University) Rogue periodic waves 1 / 4

25 Rogue dn-periodic waves For dn-periodic waves the magnification factor is still u dn (x, t) = dn(x; k)e ict, c = k, M dn (k) = + 1 k, k [, 1]. The rogue dn-periodic wave is a generalization of the Peregrine s rogue wave Figure: The rogue dn-periodic wave of the NLS for k =.5 and k =.99. D.Pelinovsky (McMaster University) Rogue periodic waves / 4

26 Rogue cn-periodic waves For cn-periodic waves u cn (x, t) = kcn(x; k)e ict, c = k 1, the magnification factor is M cn (k) = for every k (, 1) as the rogue wave is obtained from the one-fold Darboux transformation. Exact solutions are computed compared to the numerical approximation in (Kedziora Ankiewicz Akhmediev, 14) Figure: The rogue cn-periodic wave of the NLS for k =.5 and k =.99. D.Pelinovsky (McMaster University) Rogue periodic waves 3 / 4

27 Summary New method to obtain eigenfunctions of the periodic spectral (AKNS) problem associated with the periodic waves. New exact solutions to generalize the Peregrine s rogue waves to the dn-periodic and cn-periodic waves in mkdv and NLS. D.Pelinovsky (McMaster University) Rogue periodic waves 4 / 4