Shallow Water Waves: Theory, Simulation and Experiments

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1 Shallow Water Waves:Theory, Simulation and Experiments p. 1/8 Shallow Water Waves: Theory, Simulation and Experiments Yuji Kodama Ohio State University Joint work with S. Chakravarty (UC at Colorado Springs) C-Y. Kao (Ohio State) Tutorial 5 at IMACS Athene GA, March 24, 29

2 Shallow Water Waves:Theory, Simulation and Experiments p. 2/8 Shallow Water Wave Equation The shallow water wave equation: (a) Φ =, for H < Z < H(X,Y,T) (b) Φ Z =, at the bottom Z = H (c) Φ T Φ 2 + gh =, H T + Φ H = Φ Z, at Z = H(X,Y,T) The non-dimensional variables: x = X L, y = Y L, z = Z H, t = φ = H A L gh Φ. gh T, η = H, L A { L = wavelength, H = depth A = amplitude, g = 9.8m/sec.

3 Shallow Water Waves:Theory, Simulation and Experiments p. 2/8 Shallow Water Wave Equation Small parameter (weak nonlinear and long waves): ǫ = A /H = (H /L ) 2. The velocity potential has the form, Then we have φ = cos( ǫ(z + 1) )ψ(x,y) = ψ ǫ 2 (z + 1)2 ψ + ǫ2 24 (z + 1)4 2 ψ +... ψ t + η + ǫ 2 ( ψ 2 ψ t ) = O(ǫ 2 ) η t + ψ + ǫ(η ψ + η ψ ψ) = O(ǫ 2 )

4 Shallow Water Waves:Theory, Simulation and Experiments p. 2/8 Shallow Water Wave Equation With the moving frame with slow variables: x = x t, t = 3 2 ǫt, y = ǫy, the variable u := 3 2 ψ x satisfies the KP equation in the leading order, x ( 4 u t + 3 u u + 6u x 3 x The dispersion relation is given by ) u =. y 2 ω = 1 4 k3 x k 2 y k x, v x = ω = 3 k x 4 (k2 x + k2 y kx 2 )<.

5 Shallow Water Waves:Theory, Simulation and Experiments p. 3/8 The KP Equation The KP equation: x ( 4 u ) t + 3 u x 3 + 6u u x }{{} KdV +3 2 u y 2 =. The solution u(x,y,t) in terms of the τ-function: u(x,y,t) = 2 2 ln τ(x,y,t). x2 The τ-function given by the Wronskian determinant: τ N = Wr(f 1,...,f N ).

6 Shallow Water Waves:Theory, Simulation and Experiments p. 3/8 The KP Equation The linearly independent set {f i (x,y,t) : i = 1,...,N}: f i y = 2 f i x }{{ 2, } Heat equation f i t = 3 f i x 3. Finite dimensional solutions: M f i (x,y,t) = a ij E j (x,y,t), i = 1,...,N < M, j=1 E j (x,y,t) = exp(k j x + kjy 2 kjt), 3 j = 1,...,M. with the ordering k 1 < k 2 < < k M.

7 Shallow Water Waves:Theory, Simulation and Experiments p. 3/8 The KP Equation Example 1: For N = 1, the function w = x ln τ 1 satisfies w y = 2w w x + 2 w x 2 (The Burgers equation). A shock solution is given by τ 1 = f 1 = E 1 + E 2, w = 1 2 (k 1 + k 2 ) (k 2 k 1 ) tanh 1 2 (θ 2 θ 1 ), where θ j = k j x + kj 2y k3 jt. Notice that { w k 1 k 2 x x

8 The KP Equation Example 1: One line-soliton solution with τ = E1 + E2. Ψ [1, 2] 1..5.?.5 4 y (1) K[1, 2] 2 [1, 2]? 4-2 x y (2)? x 3D figure of u = 2 w x, and the contour plot. The numbers (i) represent the dominant exponential term in the τ -function. We denote this [1, 2]-soliton. Shallow Water Waves:Theory, Simulation and Experiments p. 3/8

9 Shallow Water Waves:Theory, Simulation and Experiments p. 3/8 The KP Equation One-solton solution is given by a balance between two exponential terms, and in general it is expressed with the parameters {k i,k j }, u = A [i,j] sech 2 Θ [i,j], where the amplitude A[i,j] and the phase θ[i,j] are A [i,j] = 1 2 (k i k j ) 2, Θ [i,j] = 1 2 (θ j θ i ). The slope of the soliton in xy-plane is given by tan Ψ [i,j] = k i + k j.

10 The KP Equation Example 2: Y-type solution with τ1 = f1 = E1 + E2 + E3, 5 1 y ? 5 (3) (1)? 5? 5 (2) x? 1? 1? In each region, one of the exponential terms is dominant. Each line-soliton is given by the balance between two exponential terms, Ei and Ej, denoted as (i) and (j). Shallow Water Waves:Theory, Simulation and Experiments p. 3/8

11 Shallow Water Waves:Theory, Simulation and Experiments p. 3/8 The KP Equation Chord diagrams: One can express each soliton solution in a chord diagram (= permutation). For example, Y-type soliton with the parameters {k 1,k 2,k 3 } is given by π = The upper part represents [1, 3]-soliton in y >. The lower part represents [1, 2]- and [2, 3]-solitons in y <.

12 Shallow Water Waves:Theory, Simulation and Experiments p. 4/8 Grassmannian Gr(N,M) We first note: Span R {E j : j = 1,...,M} = R M. Span R {f i : i = 1,...,N} forms an N-dimensional subspace in R M, (f 1,...,f N ) = (E 1,...,E M )A T, where A-matrix is defined by A = a 11 a 1M a N1 a NM Each solution can be parametrized by the A-matrix.

13 Shallow Water Waves:Theory, Simulation and Experiments p. 4/8 Grassmannian Gr(N,M) Each point on Gr(N,M) can be expressed as f 1 f N = 1 i 1 < <i N M ξ(i 1,,i N ) E i1 E in, The coefficients ξ(i 1,,i N ) are N N minors of the A-matrix are called the Plücker coordinates which satisfy the Plücker relations. For example, ξ(i, j) for Gr(2, 4) satisfy ξ(1, 2)ξ(3, 4) ξ(1, 3)ξ(2, 4) + ξ(1, 4)ξ(2, 3) =. Note that the numbers of ξ(i 1,...,i N ) is ( M N), and dim Gr(N,M) = N(M N). So there are ( M N) N(M N) 1 numbers of the Plücker relations. ( 1 is for the norm.)

14 Shallow Water Waves:Theory, Simulation and Experiments p. 4/8 Grassmannian Gr(N,M) For H GL(N), (g 1,...,g N ) = (f 1,...,f N )H gives the same solution, i.e. τ(g) = H τ(f). This implies that the τ-function is identified as a point on the Grassmannian Gr(N,M), i.e. Gr(N,M) = M N M (R)/GL(N), with dim Gr(N,M) = NM N 2 = N(M N). H GL(N) gives a row reduction of the A-matrix. For example, a generic A can be written in the form (RREF), A =

15 Shallow Water Waves:Theory, Simulation and Experiments p. 4/8 Grassmannian Gr(N,M) We now say that the A-matrix is irreducible, if in each column, there is at least one nonzero element, in each raw, there is at least one more nonzero element in addition to the pivot. Example: For N = 2 and M = 4, there are only two types of irreducible A-matrices. ( ) ( ) 1 1, 1 1 Note that other cases can be expressed by a smaller Gr(N,M ) with either N < N or M < M..

16 Shallow Water Waves:Theory, Simulation and Experiments p. 4/8 Grassmannian Gr(N,M) We also say that the A-matrix is totally non-negative, if all the N N minors are non-negative. Example: For N = 2,M = 4, there are seven types of the A-matrices which satisfy both irreducibility and total non-negativity. For positive numbers a,b,c and d, they are ( ) ( ) ( ) 1 c d 1 b c 1 c 1 a b 1 a 1 a b ( 1 b 1 a ) ( 1 a c 1 b ) ( For the first one, either ad cb > or =. 1 a 1 b )

17 Shallow Water Waves:Theory, Simulation and Experiments p. 4/8 Grassmannian Gr(N,M) All the totally non-negative and irreducible matrices are parametrized by the permutation group S M : For example, the seven cases in Gr(2, 4) are given by (4312) (3421) (4321) (3412) (2413) (3142) (2143) The M-tuples of the diagrams represent the permutation, ( ) 1 2 M π = = (π(1),...,π(m)). π(1) π(2) π(m)

18 Shallow Water Waves:Theory, Simulation and Experiments p. 5/8 Classification Theorem Lemma: (Binet-Cauchy) The τ-function can be expanded as τ N = ξ(i 1,...,i N )E(i 1,...,i N ), 1 i 1 < <i N M where ξ(i 1,...,i N ) is the N N minor of the A-matrix, and E(i 1,...,i N ) is given by E(i 1,...,i N ) = (k ij k il ) E i1 E in, 1 j<l N Note that if the A-matrix is totally non-negative, then the τ-function is positive definite. This implies that the solution u(x, y, t) is non-singular.

19 Shallow Water Waves:Theory, Simulation and Experiments p. 5/8 Classification Theorem Theorem 1: Let {e 1,...,e N } be the pivot indices, and let {g 1,...,g M N } be the non-pivot indices for an irreducible and totally non-negative A-matrix. Then the soliton solution associated with the A-matrix has (a) N line-solitons of [e n,j n ]-type for n = 1,...,N as y, (b) M N line-solitons of [i m,g m ]-type for m = 1,...,M N as y. [e, j ] y = j j [e k, j k] [e, j ] N+ N+ [e, j ] 1 1 (e 1,e 2,...,e N+ ) x = - Interaction Region x = [ i, g ] 1 1 y = - [ i, g ] s s [ i, g ] N- N-

20 Shallow Water Waves:Theory, Simulation and Experiments p. 5/8 Classification Theorem Theorem 2: Those [e n,j n ] and [i m,g m ] are expressed by a chord diagram which corresponds a derangement of the permutation group S M, i.e. ( ) e 1 e N g 1 g M N j 1 j N i 1 i M N Theorem 3: Conversely, for each chord diagram associated with the derangement, one can construct an A-matrix, and the corresponding τ-function gives the solution of the KP equation having line-solitons expressed by the chord diagram.

21 Shallow Water Waves:Theory, Simulation and Experiments p. 5/8 Classification Theorem Example: N = 3,M = 6 (7-dimensional solution). t = -3 t = t = [2,5] [1,4] [5,6] [3,6] [1,3] [2,4] a -b c A = 1 d e -f 1 g π = (451263)

22 Shallow Water Waves:Theory, Simulation and Experiments p. 6/8 Exact Solutions Example 1: O-type soliton solution. O - Type Soliton Solution k1 k2 k3 k4 π = (2143) E(1,4) t = -1 t = t = 1 [1,2] E(2,4) E(1,3) E(2,3) [3,4] A [1,2] + A [3,4] < u center < ( A [1,2] + A [3,4] ) 2.

23 Shallow Water Waves:Theory, Simulation and Experiments p. 6/8 Exact Solutions Example 2: P-type soliton solution. P - Type Soliton Solution k1 k2 k3 k4 π = (4321) t = -1 t = t = 1 [1,4] [2,3] ( A[1,4] A [2,3] ) 2 < ucenter < A [1,4] A [2,3].

24 Shallow Water Waves:Theory, Simulation and Experiments p. 6/8 Exact Solutions Example 3: (3142)-type soliton solution. 2 E(1,4) 2 [3,4] [1,3] E(1,3) E(3,4) [1,4]?1 E(2,3)?2?4?2 2 4?1 [1,2] [2,4]?2?4?2 2 4?1?2?4?2 2 4 (3142)-type [1,3] and [3,4]-solitons [1,2] and [2,4]-solitons Note that [1, 4] gives the maximum amplitude A = 1 2 (k 4 k 1 ) 2.

25 Shallow Water Waves:Theory, Simulation and Experiments p. 6/8 Exact Solutions Example 4: T-type soliton solution (i.e. (3412) = (13)(24)). 2 (1,4) (1,2) (3,4) (1,3) (2,3) T - type soliton [1,3] and [2,4]-solitons [1,3] and [2,4]-solitons Notice that the front half is the same as (3142)-type.

26 Shallow Water Waves:Theory, Simulation and Experiments p. 6/8 Exact Solutions Example 5: (3142)-type soliton solutions with different angles: k1 k2 k3 k4 A[1,3]=A[2,4] = 2 S[1,3] = - S[2,4]= 4 -k1 = k4 = 9/8 -k2 = k3 = 7/8 A[1,2]=A[3,4]=1/32 S[1,2]=-S[3,4]= 1/2 A[1,4] of Stem = 2.53 t = -1 t = t = 1 t = 2

27 Shallow Water Waves:Theory, Simulation and Experiments p. 6/8 Exact Solutions Example 6: (3142)-type soliton solutions with different angles: k1 k2 k4 A[1,3]=A[2,4] = 2 S[1,3] = -S[2,4]= 1 k3 -k1 = k4 = 3/2 -k2 = k3 = 1/2 A[1,2]=A[3,4]=.5 S[1,2]=-S[3,4]= 1/2 A[1,4] of Stem = 4.5 t = -1 t = t = 1 t = 2

28 Shallow Water Waves:Theory, Simulation and Experiments p. 6/8 Exact Solutions Example 7: (3142)-type soliton solutions with different angles: k2 k4 A[1,3]=A[2,4] = 2 S[1,3]=-S[2,4] = 2/3 k1 -k1 = k4 = 7/4 -k2 = k3 = 1/4 k3 A[1,2]=A[3,4] = 9/8 S[1,2]=-S[3,4] = 1/2 A[1,4] of Stem = 49/8 t = -1 t = t = 1 t = 2

29 Shallow Water Waves:Theory, Simulation and Experiments p. 6/8 Exact Solutions Example 8: O-type soliton solutions with different angles: k1 k2 k3 k4 -k1 = k4 = 9/4 -k2 = k3 = 1/4 A[1,2]=A[3,4]= 2 S[1,2]= -S[3,4]= 2/5 Max. of u = 5 at the intersection point t = -1 t = t = 1

30 Shallow Water Waves:Theory, Simulation and Experiments p. 7/8 Numerical Simulations Semi-infinite line-soliton: T= T= k1 k2 Here tanψ = k 1 + k 2 > at t =, and it is decreasing as y.

31 Shallow Water Waves:Theory, Simulation and Experiments p. 7/8 Numerical Simulations Semi-infinite line soliton: T= T= k1 k2 Note here that tanψ = k 1 + k + 2 = at t =.

32 Shallow Water Waves:Theory, Simulation and Experiments p. 7/8 Numerical Simulations Semi-infinite line soliton: T= T= k1 k2 Now tanψ = k 1 + k 2 <, and tan Ψ 2k 1 as y.

33 Shallow Water Waves:Theory, Simulation and Experiments p. 7/8 Numerical Simulations The initial wave profile: [i, j] - soliton Ψ[i,j] 2A = (k i - k j) 2 [i,j] 1_ 2 Ψ[m,n] x tan Ψ[m,n] = - tan Ψ[i,j] = k m + k n = - (k i + k j ) [m, n] - soliton

34 Shallow Water Waves:Theory, Simulation and Experiments p. 7/8 Numerical Simulations V-shape initial data of O-type: T= T=

35 Shallow Water Waves:Theory, Simulation and Experiments p. 7/8 Numerical Simulations (3142)-type: T= T=

36 Shallow Water Waves:Theory, Simulation and Experiments p. 7/8 Numerical Simulations (3142)-type with a cut: T= T=

37 Shallow Water Waves:Theory, Simulation and Experiments p. 7/8 Numerical Simulations P-type: T= T=

38 Shallow Water Waves:Theory, Simulation and Experiments p. 7/8 Numerical Simulations Need to open files again!!! Now enjoy the movies!!!

39 Shallow Water Waves:Theory, Simulation and Experiments p. 8/8 References Y. K., Young diagrams and N-soliton solutions of the KP equation, J. Phys. A: Math. Gen. 37 (24) S. Chakravarty and Y. K., Classification of the line-solitons of KPII, J. Phys. A: Math. Theor. 41 (28) (33pp). S. Chakravarty and Y. K., A generating function for the N-soliton solutions of the Kadomtsev-Petviashvili II equation, Contemporary Math. 471 (28) S. Chakravarty and Y. K., Soliton solutions of the KP equation and application to shallow water waves. arxiv:

40 Shallow Water Waves:Theory, Simulation and Experiments p. 8/8 References S. Chakravarty and Y. K., Combinatorics and geometry of soliton solutions of the KP equation, coming soon. C-Y. Kao and Y. K., Numerical study on the Initial value problems of the KP equation, coming soon. Y. K., M. Oikawa and H. Tsuji, Experiments of shallow water waves and Numerical simulations of the KP equation, coming soon. B.-F. Feng, Y. K. and K.-I. Maruno, Tow dimensional solitary wave solutions in the Boussinesq approximation of shallow water wave equation, coming soon.

KP web-solitons from wave patterns: an inverse problem

Journal of Physics: Conference Series OPEN ACCESS KP web-solitons from wave patterns: an inverse problem To cite this article: Sarbarish Chakravarty and Yuji Kodama 201 J. Phys.: Conf. Ser. 82 012007 View