Bäcklund Transformations, Solitary Waves, Conoid Waves and Bessel Waves of the (2+1)-Dimensional Euler equation

In this paper, based on matrix and curve integration theory, we theoretically show the existence of Cartesian vector solutions u = b(t) + A(t)x for the general N -dimensional compressible Euler equations. Such solutions are global and can be explicitly expressed by an appropriate formulae. One merit of this approach is to transform analytically solving the Euler equations into algebraically constructing an appropriate matrix A(t). Once the required matrix A(t) is chosen, the solution u is directly obtained. Especially, we find an important solvable relation γ = 1 + 2/N between the dimension of equations and pressure parameter, which avoid additional independent constraints on the dimension N in existing literatures. Special cases of our results also include some interesting conclusions: (1) If the velocity field u is a linear transformation on x ∈ R N , then the pressure p is a relevant quadratic form. (2) The compressible Euler equations admit the Cartesian solutions if A(t) is an antisymmetric matrix. (3) The pressure p possesses radial symmetric form if A(t) is an antisymmetrically orthogonal matrix.

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“…Proof: We first prove how to get the solution (12) through solving equation (8). Substituting (11) into (8) produces…”

Section: Theorem 1 If We Define B As Part Of a Matrix Riccati Equationmentioning

confidence: 99%

“…Recently, Li found Lax pairs for 2D and 3D Euler equations (2) [5,6]. Lou et al proposed Backlund transformation, Darboux transformation, and exact solutions for the 2D Euler equations in vorticity form [7,8].…”

Section: Introductionmentioning

confidence: 99%

The Cartesian Vector Solutions for the N‐Dimensional Compressible Euler Equations

Fan

Yuen

2014

Stud Appl Math

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“…where T jk (t), j =1,2,3,4, k∈N 3 are functions to be determined. Then substituting (3.2) into (3.1) we get [1] +k [2] =k k [2] j T jk [1] T mk [2]…”

Section: Infinite Series Solution Of the Euler Equationsmentioning

confidence: 99%

Exact solutions for the Cauchy problem to the 3D spherically symmetric incompressible Navier-Stokes equations

Zhang

Qin

2018

Acta Mathematica Scientia

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“…Recently, Li found Lax pairs for 2D and 3D Euler equations [25,26]. Lou et al proposed Backlund transformation, Darboux transformation and exact solutions for the 2D Euler equations in vorticity form [27,28]. Yuen constructed a class of exact solutions with elliptical symmetry by using the new characteristic method [29].…”

Section: Introductionmentioning

confidence: 99%

“…(34) and(35) admit solutions(27) and(28).The solution procedure is now complete. Illustrative examples are given.…”

mentioning

confidence: 97%

The Analytical Solutions for the N‐Dimensional Damped Compressible Euler Equations

2016

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Employing matrix formulation and decomposition technique, we theoretically provide essential necessary and sufficient conditions for the existence of general analytical solutions for N -dimensional damped compressible Euler equations arising in fluid mechanics. We also investigate the effect of damping on the solutions, in terms of density and pressure. There are two merits of this approach: First, this kind of solutions can be expressed by an explicit formula u = b(t) + A(t)x and no additional constraint on the dimension of the damped compressible Euler equations is needed. Second, we transform analytically the process of solving the Euler equations into algebraic construction of an appropriate matrix A(t). Once the required matrix A(t) is chosen, the solution u is obtained directly. Here, we overcome the difficulty of solving matrix differential equations by utilizing decomposition and reduction techniques. In particular, we find two important solvable relations between the dimension of the Euler equations and the pressure parameter: γ = 1 − 2/N in the damped case and γ = 1 + 2/N for no damping. These two cases constitute a full range of solvable parameter 0 ≤ γ < +∞. Special cases of our results also include several interesting conclusions: (1) If the velocity field u is a linear transformation on the Euclidean spatial vector x ∈ R N , then the pressure p is a quadratic form of x.(2) The damped compressible Euler equations admit the Cartesian solutions if A(t) is an antisymmetric matrix. (3) The pressure p possesses radially symmetric forms if A(t) is an antisymmetrical orthogonal matrix.

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Bäcklund Transformations, Solitary Waves, Conoid Waves and Bessel Waves of the (2+1)-Dimensional Euler equation

Cited by 27 publications

References 45 publications

The Cartesian Vector Solutions for the N‐Dimensional Compressible Euler Equations

The Cartesian Vector Solutions for the N‐Dimensional Compressible Euler Equations

Exact solutions for the Cauchy problem to the 3D spherically symmetric incompressible Navier-Stokes equations

The Analytical Solutions for the N‐Dimensional Damped Compressible Euler Equations

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