On versions of the Kadomtsev-Petviashvili equation
Transcription
On versions of the Kadomtsev-Petviashvili equation
On versions of the Kadomtsev-Petviashvili equation Karima Khusnutdinova Department of Mathematical Sciences, Loughborough University, UK [email protected] in collaboration with C. Klein*, V.B. Matveev* and A.O. Smirnov** *Insitute de Mathématique de Bourgogne, Dijon, France **St. Petersburg Aerospace University, Russia Nonlinear Evolution Equations and Wave Phenomena March 25 - 28, 2013, Athens, USA Overview I Introduction: Kadomtsev-Petviashvili (KP) and cylindrical KP (cKP) equations. I Derivation of the elliptic cylindrical KP (ecKP) equation for long surface gravity waves. I Transformations between KP, cKP and ecKP equations. I New soliton solutions for surface waves. I Lax pair. I Concluding remarks. 1. Introduction: Euler equations for incompressible fluid Let (u, v , w ) be the three components of the velocity vector in the Cartesian coordinates (x, y , z), t - time, p - pressure (pa is the constant atmospheric pressure at the surface, and Γ is the coefficient of the surface tension), ρ - constant density, g - gravitational acceleration, z = 0 - the bottom, and z = h(x, y , t) - the free surface elevation. ρ(ut + uux + vuy + wuz ) = −px , ρ(vt + uvx + vvy + wvz ) = −py , ρ(wt + uwx + vwy + wwz ) = −pz − ρg , ux + vy + wz = 0, p|z=h(x,y ,t) = pa − Γ (1 + hy2 )hxx + (1 + hx2 )hyy − 2hx hy hxy , (1 + hx2 + hy2 )3/2 w |z=h(x,y ,t) = ht + uhx + vhy , w |z=0 = 0. (1) 1. Introduction: Kadomtsev-Petviashvili (KP) and cylindrical KP (cKP) equations The original KP equation (KP, 1970) (Ablowitz and Segur, 1979 for surface waves; Grimshaw, 1985 for internal waves with shear flow) (Uτ + 6UUξ + Uξξξ )ξ + 3α2 UYY = 0 (2) and cylindrical KP (cKP) equation (Johnson, 1980 for surface waves; Lipovskii, 1985 for internal waves) W 3α2 Wτ + 6WWχ + Wχχχ + (3) + 2 WVV = 0 2τ χ τ describe the evolution of nearly-plane and nearly-concentric waves, respectively. 1. Introduction: Kadomtsev-Petviashvili (KP) and cylindrical KP (cKP) equations Transformations between the KP and cKP equations were found by Johnson (1980) and rediscovered by Lipovskii, Matveev, Smirnov (1989). The map Y2 Y W (τ, χ, V ) → U(τ, ξ, Y ) := W τ, ξ + , 12α2 τ τ transforms any solution of the cKP equation (3) into a solution of the KP equation (2). Conversely, the map τV 2 U(τ, ξ, Y ) → W (τ, χ, V ) := U τ, χ − , τ V 12α2 transforms any solution of the KP equation (2) into a solution of the cKP equation (3). The transformation has been used to construct some special solutions of the cKP equation by Klein, Matveev, Smirnov (2007). 2. Derivation of the elliptic cylindrical KP equation We aim to consider waves with the nearly-elliptic front, and we write this set of equations in the elliptic cylindrical coordinate system: x = d cosh α cos β, y = d sinh α sin β, z = z, where d has the meaning of half a distance between the foci of the coordinate lines, and change the two horizontal components of the velocity vector: u → u cos β − v sin β, v → u sin β + v cos β. We nondimensionalise the variables λ t, x → λx, y → λy , z → h0 z, t → √ gh0 √ p p h0 gh0 u → gh0 u, v → gh0 v , w → w, λ h → h0 + hs η, p → pa + ρg (h0 − z) + ρgh0 p, which leads to the appearance of the two usual nondimensional parameters in the problem: the long wavelength parameter δ = hλ0 and the small amplitude parameter = hh0s , as well as a new nondimensional parameter γ = λd (ellipticity parameter). Scaling: u → u, v → v , w → w , p → p. 2. Derivation of the elliptic cylindrical KP equation Equation for the linear waves (in the long-wave approximation) is obtained for = δ = 0 as ηαα + ηββ . ηtt = 2 γ (sinh2 α + sin2 β) (4) Equation (4) reduces to the equation 1 1 ηtt − (ηrr + ηr + 2 ηββ ) = 0 (5) r r for the long linear waves in the polar cylindrical coordinates in the limit 1 α γe → r being finite. (6) α → ∞, γ → 0 with 2 The derivation of the cylindrical KP (cKP) equation (also known as the nearly-concentric KP equation or Johnson’s equation) is based on the existence of the exact reduction of the equation (5) to the equation 1 ηtt − (ηrr + ηr ) = 0. r Unlike (5), equation (4) does not have an exact reduction to the equation with no dependence on β. However, there is an asymptotic reduction. 2. Derivation of the elliptic cylindrical KP equation We introduce the variables: 2 6 δ (γ cosh α − t) , R = γ cosh α, ν = 2 sin β, δ2 δ4 3 3 3 5 5 η = 2 H, p = 2 P, u = 2 U, w = 4 W , v = 3 V , δ δ δ δ δ ζ= which generalise a change of variables for the cylindrical coordinates. Here, 2γ cosh α is the sum of the distances from a point on an ellipse to its foci. Thus, ζ is an asymptotic characteristic coordinate for the waves with the nearly-elliptic front, and it becomes a characteristic coordinate for the concentric waves in the limit (6). Notations: ∆= 4 1, δ2 A=γ 6 . δ4 2. Derivation of the elliptic cylindrical KP equation We now seek a solution of the system of equations and boundary conditions in the form of asymptotic multiple-scales expansions: H = H0 + ∆H1 + O(∆2 ), with similar expansions for U, V , W and P. We obtain the elliptic cylindrical KP equation (ecKP) « « „ „ 1 R ν2 2 2H0R + 3H0 H0ζ + − We H0ζζζ + 2 H − A H 0 0ζ 3 R − A2 R 2 − A2 ζ + 1 H0νν = 0, R 2 − A2 Γ where We = ρgh 2 (Weber number). 0 A scaling transformation brings the derived equation to the form τ a2 ν 2 3σ 2 H − H + 2 H = 0, Hτ + 6HHζ + Hζζζ + ζ 2 2 2 2 2 2 νν 2(τ − a ) 12σ (τ − a ) ζ τ −a where σ 2 = sign 1 3 − We . 3. Transformations between KP, cKP and ecKP equations The geometry of a wave with the nearly-elliptic front, considered simultaniously in the Cartesian and elliptic cylindrical coordinates, suggests the introduction of the sum and the difference of the distances from a point on the wave front to the two foci of the coordinate system d1 + d2 = 2γ cosh α, d1 − d2 = 2γ cos β, where the foci have the following Cartesian coordinates: F1 (−γ, 0) and F2 (γ, 0). We recall that the variables have been nondimensionalised, as discussed in section 2, and d γ= . λ Note that 21 (d1 + d2 ) − t corresponds, up to the scaling, to the asymptotic characteristic variable ζ. 3. Transformations between KP, cKP and ecKP equations Then, for the area satisfying asymptotic behaviour y y x−γ , x+γ → 0, we obtain the following 1 1 (d1 + d2 ) − t ∼ x − t + y 2 2 4 „ 1 1 + x +γ x −γ « . Next, for sufficiently large α and small β, our nondimensional variable 4 x = γ cosh α cos β ∼ δ6 R, and the previous asymptotics can be rewritten as 1 1 R (d1 + d2 ) − t ∼ ξ + Y 2 2 , 2 2 R − A2 where ξ = x − t, Y = 3 δ2 y 6 and A = γ δ4 . Similarly, A Y2 1 (d1 − d2 ) ∼ γ − . 2 2 R 2 − A2 This asymptotic behaviour of the geometrically meaningful objects motivates the change of variables: Rν 2 p 2 H0 (R, ζ, ν) = η(R, ζ − , R − A2 ν). 2 3. Transformations between KP, cKP and ecKP equations We write the KP equation in the form (Uτ + 6UUξ + Uξξξ )ξ + 3α2 UYY = 0, the cKP equation in the form 3α2 1 W + 2 WVV = 0, Wτ + 6WWχ + Wχχχ + 2τ τ χ and the ecKP equation as „ Hτ + 6HHζ + Hζζζ + τ a2 ν 2 H− Hζ 2 2 2 2(τ − a ) 12σ (τ 2 − a2 ) « + ζ 3σ 2 Hνν = 0. − a2 τ2 The map τV 2 , τV U(τ, ξ, Y ) → W (τ, χ, V ) := U τ, χ − 12α2 transforms any solution of the KP equation into a solution of the cKP equation, and the map τ ν2 p 2 2ν U(τ, ξ, Y ) → H(τ, ζ, ν) := U τ, ζ − , τ − a 12α2 transforms any solution of the KP eq. into a solution of the ecKP eq. 4. New soliton solutions for surface waves We return to the original nondimensional variables x, y , t: x = γ cosh α cos β, y = γ sinh α sin β, r 4 a η = 1/3 (1 − 3We )1/3 H(τ, ζ, ν), γ 6 where „ Hτ + 6HHζ + Hζζζ + τ a2 ν 2 H− Hζ 2 2 2 2(τ − a ) 12σ (τ 2 − a2 ) σ 2 = sign (1 − 3We ) « + ζ and τ = R = a cosh α, 1/3 ζ= 6 a (γ cosh α − t), γ∆(1 − 3We )1/3 ν= 62/3 sin β. ∆1/2 |1 − 3We |1/6 Here, t is the physical time (nondimensional), We = We also have r γ γ = ∆, δ = ∆3/2 . a a Γ . ρgh02 3σ 2 Hνν = 0, − a2 τ2 4. New soliton solutions for surface waves The 1-soliton solution of the ecKP-II equation is explicitly written in the form H(τ, ζ, ν) = » „ «– p τ ν2 K2 K ζ− sech2 + L τ 2 − a2 ν − (K 2 + 3L2 )τ + δ0 , 2 2 12 where K , L, δ0 are arbitrary constants. The corresponding surface wave elevation η is plotted below for γ = 1, a = 2, ∆ = 1/2 and We = 0, δ0 = 0. Figure: Surface wave corresponding to the one-soliton solution of the ecKP-II equation with K = 1, L = 0 for t = 0 (top left), t = 0.25 (top right), t = 0.5 (bottom left), t = 1 (bottom right). 4. New soliton solutions for surface waves Figure: Surface wave corresponding to the one-soliton solution of the ecKP-II equation with K = 1, L = 0.1 for t = 0 (top left), t = 0.25 (top right), t = 0.5 (bottom left), t = 1 (bottom right). 4. New soliton solutions for surface waves Figure: Surface wave corresponding to the one-soliton solution of the ecKP-II equation with K = 1, L = −0.5 for t = 0 (top left), t = 0.25 (top right), t = 0.5 (bottom left), t = 2 (bottom right). 4. New soliton solutions for surface waves Figure: Surface wave corresponding to the one-soliton solution of the ecKP-II eq. with K = 1.5, L = 0 for t = 0 (left), t = 2 (right). Figure: Surface wave corresponding to the one-soliton solution of the ecKP-II eq. with K = 1.6, L = 0.1 for t = 0 (left), t = 2 (right). Figure: Surface wave corresponding to the exceptional one-soliton solution of the ecKP-II eq. with K = 1.5 and L ≈ 0.1 for t = 0 (left) and t = 1 (right). 4. New soliton solutions for surface waves Figure: Surface wave corresponding to a two-soliton solution of the ecKP-II equation with for t = 1 (top left), t = 2 (top right), t = 3 (bottom left), t = 4 (bottom right). 5. Lax pair The ecKP equation can be obtained as a compatibility condition of the following linear problem (Lax pair): p τ 2 − a2 ψζζ + “p τ 2 − a2 H(τ, ζ, ν) « τζ a2 ν 2 √ − √ + ψ, 12 τ 2 − a2 144σ 2 τ 2 − a2 „ « a2 ν 2 ψτ = −4ψζζζ − 6H(τ, ζ, ν) − ψζ 12σ 2 (τ 2 − a2 ) ! e ζ, ν) 3σ H(τ, a2 ζν − 3Hζ (τ, ζ, ν) + √ − ψ. 12σ(τ 2 − a2 )3/2 τ 2 − a2 σψν = When a = 0 we recover the Lax pair of the cKP equation (Dryuma, 1983). 6. Concluding remarks I Applications of cKP and ecKP? I Wave instabilities? (Previous results for KP and cKP by Zakharov (1975), and Ostrovsky and Shrira (1976).) I Other solutions of ecKP and Euler equations? (using the map from KP and recent results by Biondini, Chakravarty, and Kodama for the KP). I Other versions of the KP and other admissible maps, associated with the physical problem formulation? References 1. B.P. Kadomtsev, V.I. Petviashvili, Sov. Phys. Dokl. 15 (1970) 539-541. 2. M.J. Ablowitz and H. Segur, J. Fluid Mech. 92 (1979) 691-715. 3. R.S. Johnson, J. Fluid Mech. 97 (1980) 701-719. 4. V.S. Dryuma, Dokl. Akad. Nauk SSSR, 268 (1982) 15-17. 5. R. Grimshaw, Stud. Appl. Math. 73 (1985) 1- 33. 6. V.D. Lipovskii, Izv. Acad. Nauk SSSR, Ser. Fiz. Atm. Okeana 21 (1985) 864-871. 7. V.D. Lipovskii, V.B. Matveev, and A.O. Smirnov, J. Soviet Math. 46 (1989) 1609-1612. 8. C. Klein, V.B. Matveev, and A.O. Smirnov, Theor. Math. Phys. 152 (2007) 1132-1145. 9. K.R. Khusnutdinova, C. Klein, V.B. Matveev, A.O. Smirnov, Chaos 23, 013126 (2013) 13 pages.