Three and n Dimensional Equations
2.6.2. Three and n Dimensional Equations
∂w ∂ 2 w
1. i
+ A|w| w = 0.
Three-dimensional Schr ¨odinger equation with a cubic nonlinearity . This is a special case of equation
2.6.2.2 with
f (u) = Au 2 .
1 ◦ . Suppose w(x, y, t) is a solution of the Schr ¨odinger equation in question. Then the functions
1 = C 1 w( C 1 x+C 2 , C 1 y+C 3 , C 1 z+C 4 , C 1 t+C 5 ), − i[λ 1 x+λ 2 y+λ 3 z+(λ 2 1 + λ 2 2 + λ 3 w 2
e ) t+C 6 ] w(x + 2λ 1 t, y + 2λ 2 t, z + 2λ 3 t, t), where C 1 , ...,C 6 , λ 1 , λ 2 , and λ 3 are arbitrary real constants, are also solutions of the equation. The
plus or minus signs in the expression for w 1 are chosen arbitrarily.
2 ◦ . There is an exact solution of the form
where f k = f k ( t), g k = g k ( t), and h k = h k ( t).
3 ◦ . Solution: w(x, y, z, t) = U (ξ 1 , ξ 2 , ξ 3 ) e i(k 1 x+k 2 y+k 3 z+at+b) , ξ 1 = x − 2k 1 t, ξ 2 = y − 2k 2 t, ξ 3 = z − 2k 3 t, where k 1 , k 2 , k 3 ,
a, and b are arbitrary constants, and the function U = U (ξ 1 , ξ 2 , ξ 3 ) is determined by the differential equation
2 + A|U | U − (k 1 + k 2 + k 3 + a)U = 0.
4 ◦ . “Three-dimensional” solution:
1 x+C 3 y+C 4 z+C 5 w(x, y, z, t) = √
C t+C 2 C 1 t+C 2 C 1 t+C 2 C 1 t+C 2 where C 1 , ...,C 5 are arbitrary constants, and the function u = u(ξ, η, ζ) is determined by the
differential equation
+ 2 + − iC 1 ξ + η + ζ + u + A|u| u = 0.
References : L. Gagnon and P. Winternitz (1988, 1989), N. H. Ibragimov (1995), A. M. Vinogradov and I. S. Krasil’shchik (1997).
∂w ∂ 2 w
2. i
+ f (|w|)w = 0.
∂t
∂x 2 ∂y 2 ∂z 2
Three-dimensional nonlinear Schr ¨odinger equation of general form . It admits translations in any of the independent variables.
1 ◦ . Suppose w(x, y, z, t) is a solution of the Schr ¨odinger equation in question. Then the function − i[λ 1 x+λ 2 y+λ 3 z+(λ 1 2 + 2 + w 2
e ) λ 2 λ 3 t+A] w(x + 2λ 1 t+C 1 , y + 2λ 2 t+C 2 , z + 2λ 3 t+C 3 , t+C 4 ), where
1 , A, C ...,C 4 , λ 1 , λ 2 , and λ 3 are arbitrary real constants, is also a solution of the equation.
2 ◦ . Exact solutions depending only on the radial variable r= 2 px + 2 y 2 + z and time t are determined by the equation
∂w
2 ∂w
+ f (|w|)w = 0,
which is a special case of equation 1.7.5.2 with n = 2.
w(x, y, z, t) = e i(At+B) u(x, y, z),
where
A and B are arbitrary real constants, and the function u = u(x, y, z) is determined by the stationary equation
∆ u + f (|u|)u − Au = 0.
4 ◦ . Axisymmetric solutions in cylindrical and spherical coordinates are determined by equations where the Laplace operator has the form
5 ◦ . “Three-dimensional” solution:
2 2 2 w = U (ξ, η, t), 2 ξ=y+ , η = (C − 1) x −2 Cxy + C z ,
where
C is an arbitrary constant (C ≠ 0), and the function U = U (ξ, η, t) is determined by the differential equation
6 ◦ . “Three-dimensional” solution: w = V (ξ, η, t), p ξ = Ax + By + Cz, η= ( Bx − Ay) 2 +( Cy − Bz) 2 +( Az − Cx) 2 ,
where
A, B, and C are arbitrary constants and the function V = V (ξ, η, t) is determined by the equation
References : L. Gagnon and P. Winternitz (1988, 1989), N. H. Ibragimov (1995).
∂w
∆w + |w| 2 w. ∂t This is an n-dimensional Schr¨odinger equation with a cubic nonlinearity. Conservation laws: | 2 w|
3. i
t + i∇ ⋅ ¯ w∇w − w∇ ¯ w x = 0,
w)∇ ¯ w − (∆ ¯ w + |w| w)∇w ¯ x = 0. The bar over a symbol denotes the complex conjugate. ❄☎❅
2 1 4 2 |∇ 2 w| −
2 | w| t + i∇ ⋅
Reference : A. M. Vinogradov and I. S. Krasil’shchik (1997).
Chapter 3
Hyperbolic Equations with One Space Variable
Parts
» NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
» Equations of the Form ∂w = aw +f x, t, w, ∂w
» Equations of the Form ∂w m =a∂ w ∂w + bw k
» 1.13. Equations of the Form ∂w i =∂ f (w) ∂w + g(w)
» Equations of the Form ∂w λw =a∂ e ∂w + f (w)
» Other Equations Explicitly Independent of x and t
» Equations of the Form ∂w 2 = f (x, t) ∂ w +g x, t, w, ∂w
» Equations of the Form ∂w = aw ∂ 2 + f (x, t, w) ∂w + g(x, t, w)
» Equations of the Form ∂w =a ∂ w ∂w +f (x, t) ∂w +g(x, t, w)
» 6.15. Equations of the Form ∂w i
» Equations of the Form ∂w 2 = f (x, t, w) ∂ w +g x, t, w, ∂w
» Equations with Cubic Nonlinearities Involving Arbitrary Functions
» Equations of General Form Involving Arbitrary Functions of a Single Argument
» Equations of General Form Involving Arbitrary Functions of Two Arguments
» Equations of the Form ∂w =a∂ w n ∂w w k +b∂ ∂w
» Heat and Mass Transfer Equations in Quiescent or Moving Media with Chemical Reactions
» Equations of Mass Transfer in Quiescent or Moving Media with Chemical Reactions
» Equations of Heat and Mass Transfer in Anisotropic Media
» Other Equations with Three Space Variables
» Equations with n Space Variables
» Three and n Dimensional Equations
» Equations of the Form ∂ 2 = a∂ w 2 + f (x, t, w)
» Sine Gordon Equation and Other Equations with Trigonometric Nonlinearities
» Equations of the Form ∂ + f (w) ∂w =∂ g(w) ∂w
» Equations Involving Arbitrary Parameters of the Form
» Equations Involving Power Law Nonlinearities
» Heat and Mass Transfer Equations of the Form
» Heat and Mass Transfer Equations of the Form h i h i h i
» Heat and Mass Transfer Equations with Complicating Factors
» Khokhlov–Zabolotskaya Equation
» Equation of Unsteady Transonic Gas Flows
» Equations of the Form ∂ 2 w ∂ 2 w
» Equations of the Form ∂w =F w, ∂w ,∂ w
» Equations of the Form ∂w =F t, w, ∂w ,∂ w
» Equations of the Form ∂w =F x, w, ∂w ,∂ w
» Equations of the Form ∂w =F x, t, w, ∂w ,∂ w
» Equations with Two Independent Variables, Nonlinear in Two or More Highest Derivatives
» Cylindrical, Spherical, and Modified Korteweg–de Vries Equations
» Equations of the Form ∂w +a∂ w 3 +f w, ∂w =0
» Burgers–Korteweg–de Vries Equation and Other Equations
» Steady Hydrodynamic Boundary Layer Equations for a Newtonian Fluid
» Steady Boundary Layer Equations for Non-Newtonian Fluids
» Unsteady Boundary Layer Equations for a Newtonian Fluid
» Unsteady Boundary Layer Equations for Non-Newtonian Fluids
» Equations Involving Second-Order Mixed Derivatives
» Equations Involving Third-Order Mixed Derivatives
» Equations of the Form ∂w =a∂ w 4 +F x, t, w, ∂w
» Boussinesq Equation and Its Modifications
» Kadomtsev–Petviashvili Equation
» Stationary Hydrodynamic Equations (Navier–Stokes
» Nonstationary Hydrodynamic Equations (Navier–Stokes equations)
» Equations of the Form ∂w n = a∂ w n + f (w) ∂w
» Equations of the Form ∂w n = a∂ w n + f (x, t, w) ∂w + g(x, t, w)
» Equations of the Form ∂w n = a∂ w n + F x, t, w, ∂w
» General Form Equations Involving the First
» Equations Involving ∂ m w and ∂ w
» Contact Transformations. Legendre and Euler Transformations
» B ¨acklund Transformations. Differential Substitutions
» Exponential Self Similar Solutions. Equations Invariant Under Combined Translation and Scaling
» Solution of Functional Differential Equations by Differentiation
» Solution of Functional Differential Equations by Splitting
» Simplified Scheme for Constructing Generalized Separable
» Special Functional Separable Solutions
» Splitting Method. Reduction to a Functional Equation with
» Solutions of Some Nonlinear Functional Equations and Their Applications
» Clarkson–Kruskal Direct Method: a Special Form for Similarity Reduction
» Some Modifications and Generalizations
» Group Analysis Methods 1. Classical Method for Symmetry Reductions
» First Order Differential Constraints
» Second and Higher Order Differential Constraints
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» Movable Singularities of Solutions of Ordinary Differential
» Solutions of Partial Differential Equations with a Movable
» Examples of the Painlev ´e Test Applications
» Method Based on Linear Integral Equations
» Conservation Laws 1. Basic Definitions and Examples
» Conservation Laws. Some Examples
» Characteristic Lines. Hyperbolic Systems. Riemann Invariants
» Self Similar Continuous Solutions. Rarefaction Waves
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» Evolutionary Shocks. Lax Condition (Various Formulations)
» Solutions for the Riemann Problem
» Examples of Nonstrict Hyperbolic Systems
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