Exponential asymptotics theory for stripe solitons in two-dimensional periodic potentials

Transcription

1 NLS Workshop: Crete 2013 Exponential asymptotics theory for stripe solitons in two-dimensional periodic potentials Jianke Yang University of Vermont Collaborators: Sean Nixon (University of Vermont) Triantaphyllos Akylas (MIT) 1

2 Talk Outline: We develop an exponential asymptotics thoery for bifurcations of line solitons in two-dimensional periodic potentials. Introduction Exponential asymptotics theory for line solitons bifurcated from band edges. We will show that for any rational slope, two line solitons exist Construction of infinite number of multi-line-soliton bound states Line solitons bifurcated from interiors of Bloch bands. We will show that such embedded line solitons exist, but only at special angles Numerical examples Summary 2

3 1. Introduction For a nonlinear wave system, the study of its solitary waves is a fundamental question. Solitary waves often bifurcate from infinitesimal continuous waves at zero-group-velocity points. continuous wave solitary-wave packet The envelope of this wave packet A(X) is generally governed by the stationary NLS equation: 3

4 1. Introduction This stationary NLS equation admits localized solutions This suggests that solitary waves with any wavepacket position exist: But this suggestion is wrong. 4

5 1. Introduction Why? Because this suggestion ignores carrier-envelope coupling. This coupling is invisible when one pursues the usual multi-scale perturbation expansion to all orders, because it is an exponentially small effect. Thus exponential asymptotics (beyond all orders) is needed. When this carrier-envelope coupling is taken into account, one then finds that the wave envelope can only be at certain discrete positions 5

6 1. Introduction Previous exponential asymptotics theory for solitary-wave bifurcations was developed only for 1D: Yang and Akylas (1997): for fifth-order KdV Hwang, Akylas and Yang (2011): for 1D NLS with periodic potentials Hwang, Akylas and Yang (2012): for 1D NLS with nonlinear lattices But exponential asymptotics for 2D problems has remained a challenge. In this talk, we develop an exponential asymptotics theory for 2D problems (Nixon, Akylas and Yang, 2013) Note: Under certain special cases (such as symmetric potentials), other methods for solitary wave bifurcations may also be feasible: Dohnal, Pelinovsky, and Schneider (2009) Ilan and Weinstein (2010) Our goal is to develop a bifurcation theory for all potentials. 6

7 1. Introduction The problem we consider is the NLS equation with 2D periodic potentials: Question: in a general 2D lattice potential, what line solitons exist? line soliton 7

8 1. Introduction Experimental results: lattice focusing nonlinearity defocusing nonlinearity inside Bloch band Chen, et al 2004 Lou, Chen Yang 2007 Wang, Chen Yang 2007 These results show the existence of certain line solitons under appropriate nonlinearities. But how many line solitons can exist in this case? This question needs an analytical study. 8

9 Line solitons are of the form: We first consider line solitons bifurcating from edges of Bloch bands. Why? Because such solitons bifurcate into the gap, which is the most common case. 9

10 Multi-scale perturbation analysis and its failure Introducing variables: Then the multi-scale perturbation solution for line solitons is where 10

11 Pursuing this perturbation calculation to third order, we get the equation for envelope A(W) as 11

12 But this is wrong. 12

13 Determination of these growing tails requires the exponential asymptotics technique (since their amplitudes are exponentially small). This will be developed next. Since this is a 2D problem, certain key steps in the previous 1D theories no longer work, hence new approaches will be taken. 13

14 Exponential asymptotics theory 1. Solution in the wavenumber domain We first take the Fourier transform of the solution with respect to the slow variable W: 14

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17 2. Poles in the wavenumber plane Hence, 17

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21 3. Solution near the poles Near the poles, we introduce an inner variable In this region, the solution can be expanded as Plugging this expansion into the original integral equation, we find that 21

22 This inner integral equation can be solved exactly, and its solution is where Important properties of this solution: 22

23 By using original variables, we get the poles in solution U(x,y,K) as So residues of these poles are known. But, how to determine the constant C? Obviously, C cannot be determined from the inner integral equation itself (since it is homogeneous). Then, C has to be determined from solutions away from the poles (by the matching technique). 23

24 4. Solution away from the poles 24

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26 Our new strategy: give up the recurrence-relation approach and directly solve the outer equation numerically, How: by using the trapozoidal rule for the integrals, we get This is a linear inhomogeneous system which can be readily solved by the conjugate-gridient method. 26

27 Is this new approach more difficult? No, since directly computing this integral equation is actually easier than computing the recurrence relation. Our goal from the outer equation: From local analysis near the poles, we have got Using outer variables, we get the near-pole outer solution to behave as Thus, computation of the outer integral equation near the pole will yield the C value. 27

28 Numerical Example: We get : C=

29 Putting things together, we have obtained the poles of the solution U as Then performing the inverse Fourier transform and requiring the solution to decay far upstream (w << -- 1), 29

30 We get the wave profile of the solution far downstream (w >> 1) as 30

31 This growing-tail formula is very useful, and it can be used for many purposes, Determine how many line solitons exist and where they are located Construct multi-line-soliton bound states Even for the stability study. 31

32 Determination of line solitons: By requiring the growing tails to vanish, we find two line solitons for any rational slope, and they are located at: (called onsite and offsite) Thus, one of our main conclusions is that: from each edge of a Bloch band, TWO line solitons bifurcate out for each rational slope in any 2D lattice. 32

33 Numerical Example: asymmetric lattice, 33

34 3. Construction of multi-line-soliton bound states Determination of multi-line-soliton bound states: By matching the growing tails of the left packet with the decaying tails of the right packet: left packet right packet

35 3. Construction of multi-line-soliton bound states We get the matching conditoin By solving this algebraic equation, we find an infinite (countable) number of multi-line-soliton bound states. Furthermore, we can show analytically that the power curves of their solution families exhibit triple-branching structure. 35

36 3. Construction of multi-line-soliton bound states Numerical Example: 36

37 4. Line solitons bifurcated from interiors of Bloch bands Can line solitons bifurcate from interior points of Bloch bands? From point X, for example? Answer: Yes, but only for very special slopes (up to three, in general). Why? Because resonance with Bloch bands needs to be suppressed. 37

38 4. Line solitons bifurcated from interiors of Bloch bands Above the X point (defocusing nonlinearity) 38

39 4. Line solitons bifurcated from interiors of Bloch bands Below the X point (focusing nonlinearity) 39

40 4. Line solitons bifurcated from interiors of Bloch bands After resonance is suppressed at these special angles, the previous exponential asymptotics theory then applies, and we will find that For each of such special angles, TWO line solitons exist inside Bloch bands. 40

41 4. Line solitons bifurcated from interiors of Bloch bands Numerical Example: 41

42 Connection with previous results Experimental results: lattice from M point from X point All those results are special cases of our general theory. Furthermore, we have predicted that (1) For the middle two panels, there are two line solitons each, and their slope can be any rational. (2) For the last panel, only a couple of angles are 42 possible.

43 5. Summary We developed an exponential asymptotics theory for bifurcations of line solitons in general 2D periodic potentials Using this theory, we showed that for line solitons bifurcated from band edges, two line solitons exist for every rational slope. An infinite number of multi-line-soliton bound states are also constructed analytically For line solitons bifurcated from interior points of Bloch bands, we showed that such embedded line solitons exist, but only at special angles Open questions: exponential asymptotics theory for bifurcations of 2D localized solitons in lattices 43

44 Reference On this talk: S. Nixon, T.R. Akylas, and J. Yang, Stud. Appl. Math On previous 1D exponential asymptotics theories: T.S. Yang and T.R. Akylas, J. Fluid Mechanics 1997 G. Hwang, T.R. Akylas and J. Yang, Physica D 2011 T.R. Akylas, G. Hwang and J. Yang, Proc. Roy. Soc. A 2012 G. Hwang, T.R. Akylas and J. Yang, Stud. Appl. Math Thanks