Rogue periodic waves for mkdv and NLS equations

Rogue periodic waves for mkdv and NLS equations Jinbing Chen and Dmitry Pelinovsky Department of Mathematics, McMaster University, Hamilton, Ontario, Canada http://dmpeli.math.mcmaster.ca AMS Sectional meeting, Buffalo NY, September 16-17 17 D.Pelinovsky (McMaster University) Rogue periodic waves 1 / 4

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

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

Modulation theory for the Gardner equation References: E. Parkes, J. Phys. A, 5-36 (1987); R. Grimshaw et al., Physica D 159, 35-57 (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

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

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

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

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 -1 1 5 t -5-1 -15-1 -5 x 5 1 15 Figure: The rogue dn-periodic wave of the mkdv for k =.99. D.Pelinovsky (McMaster University) Rogue periodic waves 8 / 4

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 -1 3 1 t -1 - -3-3 - -1 x 1 3 Figure: The rogue cn-periodic wave of the mkdv for k =.99. D.Pelinovsky (McMaster University) Rogue periodic waves 9 / 4

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

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

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 4. 1 8 6 4 - - λ + - λ - λ - λ + -4-6 -8-1 -1 -.8 -.6 -.4 -...4.6.8 1 D.Pelinovsky (McMaster University) Rogue periodic waves 1 / 4

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 4 1.8.6.4 - λ * I λ I. -. -.4 -.6 - λ I λ * I -.8-1 -.5 -.4 -.3 -. -.1.1..3.4.5 D.Pelinovsky (McMaster University) Rogue periodic waves 13 / 4

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

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

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

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

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

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 -1 3 1 t -1 - -3-3 - -1 x 1 3 Figure: The rogue cn-periodic wave of the mkdv for k =.99. D.Pelinovsky (McMaster University) Rogue periodic waves / 4

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

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. 3.5.5 1.5 1 1.5 1.5.5 4 1 - -4-1 -5 5 1 5-5 -1 - -1 1 Figure: The rogue dn-periodic wave of the NLS for k =.5 and k =.99. D.Pelinovsky (McMaster University) Rogue periodic waves / 4

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). 1.8 1.5.6.4 1..5-15 -1-5 5 1 15 - -1 1-3 - -1 1 3-4 - 4 Figure: The rogue cn-periodic wave of the NLS for k =.5 and k =.99. D.Pelinovsky (McMaster University) Rogue periodic waves 3 / 4

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