[PPT] - Approximate Analysis to the KdV-Burgers Equation Zhaosheng Feng PowerPoint Presentation

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Approximate Analysis to the KdV-Burgers Equation

Zhaosheng Feng

Department of Mathematics University of Texas-Pan American 1201 W. University Dr. Edinburg, Texas 78539, USA E-mail: zsfeng@utpa.edu October 26, 2013 Texas Analysis and Mathematical Physics Symposium—Rice University

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Outline

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Introduction Generalized KdV-Burgers Equation KdV-Burgers Equation Planar Polynomial Systems and Abel Equation

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 2 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Outline

1

Introduction Generalized KdV-Burgers Equation KdV-Burgers Equation Planar Polynomial Systems and Abel Equation

2

Qualitative Analysis Generalized Abel Equation Property of Our Operator Two Theorems

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 2 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Outline

1

Introduction Generalized KdV-Burgers Equation KdV-Burgers Equation Planar Polynomial Systems and Abel Equation

2

Qualitative Analysis Generalized Abel Equation Property of Our Operator Two Theorems

3

Approximate Solution 2D KdV-Burgers Equation Resultant Abel Equation Approximate Solution to 2D KdV-Burgers Equation

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 2 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Outline

1

Introduction Generalized KdV-Burgers Equation KdV-Burgers Equation Planar Polynomial Systems and Abel Equation

2

Qualitative Analysis Generalized Abel Equation Property of Our Operator Two Theorems

3

Approximate Solution 2D KdV-Burgers Equation Resultant Abel Equation Approximate Solution to 2D KdV-Burgers Equation

4

Conclusion

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 2 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Outline

1

Introduction Generalized KdV-Burgers Equation KdV-Burgers Equation Planar Polynomial Systems and Abel Equation

2

Qualitative Analysis Generalized Abel Equation Property of Our Operator Two Theorems

3

Approximate Solution 2D KdV-Burgers Equation Resultant Abel Equation Approximate Solution to 2D KdV-Burgers Equation

4

Conclusion

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Acknowledgement

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 2 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Generalized KdV-Burgers Equation Generalized Korteweg-de Vries-Burgers equation [1, 2] ut +

δuxx + β

p up

x

+ αux − µuxx = 0, (1) where u is a function of x and t, α, β and p > 0 are real constants, µ and δ are coefficients of dissipation and dispersion, respectively.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 3 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Generalized KdV-Burgers Equation Generalized Korteweg-de Vries-Burgers equation [1, 2] ut +

δuxx + β

p up

x

+ αux − µuxx = 0, (1) where u is a function of x and t, α, β and p > 0 are real constants, µ and δ are coefficients of dissipation and dispersion, respectively. The type of such problems arises in modeling waves generated by a wavemaker in a channel and waves incoming from deep water into nearshore zones.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 3 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Generalized KdV-Burgers Equation Generalized Korteweg-de Vries-Burgers equation [1, 2] ut +

δuxx + β

p up

x

+ αux − µuxx = 0, (1) where u is a function of x and t, α, β and p > 0 are real constants, µ and δ are coefficients of dissipation and dispersion, respectively. The type of such problems arises in modeling waves generated by a wavemaker in a channel and waves incoming from deep water into nearshore zones. The type of such models has the simplest form of wave equation in which nonlinearity (up)x, dispersion uxxx and dissipation uxx all occur.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 3 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Generalized KdV-Burgers Equation Generalized Korteweg-de Vries-Burgers equation [1, 2] ut +

δuxx + β

p up

x

+ αux − µuxx = 0, (1) where u is a function of x and t, α, β and p > 0 are real constants, µ and δ are coefficients of dissipation and dispersion, respectively. The type of such problems arises in modeling waves generated by a wavemaker in a channel and waves incoming from deep water into nearshore zones. The type of such models has the simplest form of wave equation in which nonlinearity (up)x, dispersion uxxx and dissipation uxx all occur. — —

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 3 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Generalized KdV-Burgers Equation Generalized Korteweg-de Vries-Burgers equation [1, 2] ut +

δuxx + β

p up

x

+ αux − µuxx = 0, (1) where u is a function of x and t, α, β and p > 0 are real constants, µ and δ are coefficients of dissipation and dispersion, respectively. The type of such problems arises in modeling waves generated by a wavemaker in a channel and waves incoming from deep water into nearshore zones. The type of such models has the simplest form of wave equation in which nonlinearity (up)x, dispersion uxxx and dissipation uxx all occur. — — [1] J.L. Bona, W.G. Pritchard and L.R. Scott, Philos. Trans. Roy. Soc. London Ser. A, 302 (1981), 457–510.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 3 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Generalized KdV-Burgers Equation Generalized Korteweg-de Vries-Burgers equation [1, 2] ut +

δuxx + β

p up

x

+ αux − µuxx = 0, (1) where u is a function of x and t, α, β and p > 0 are real constants, µ and δ are coefficients of dissipation and dispersion, respectively. The type of such problems arises in modeling waves generated by a wavemaker in a channel and waves incoming from deep water into nearshore zones. The type of such models has the simplest form of wave equation in which nonlinearity (up)x, dispersion uxxx and dissipation uxx all occur. — — [1] J.L. Bona, W.G. Pritchard and L.R. Scott, Philos. Trans. Roy. Soc. London Ser. A, 302 (1981), 457–510. [2] J.L.Bona, S.M. Sun and B.Y. Zhang, Dyn. Partial Differ. Equs. 3 (2006), 1–69.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 3 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Burgers Equation and KdV Equation Choices of δ = α = 0 and p = 2 lead (1) to the Burgers equation [3]: ut + αuux + βuxx = 0, (2)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 4 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Burgers Equation and KdV Equation Choices of δ = α = 0 and p = 2 lead (1) to the Burgers equation [3]: ut + αuux + βuxx = 0, (2) with the wave solution u(x, t) = 2k α + 2βk α tanh k(x − 2kt).

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 4 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Burgers Equation and KdV Equation Choices of δ = α = 0 and p = 2 lead (1) to the Burgers equation [3]: ut + αuux + βuxx = 0, (2) with the wave solution u(x, t) = 2k α + 2βk α tanh k(x − 2kt). Choices of α = µ = 0 and p = 2 lead (1) to the KdV equation [4]: ut + αuux + suxxx = 0, (3)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 4 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Burgers Equation and KdV Equation Choices of δ = α = 0 and p = 2 lead (1) to the Burgers equation [3]: ut + αuux + βuxx = 0, (2) with the wave solution u(x, t) = 2k α + 2βk α tanh k(x − 2kt). Choices of α = µ = 0 and p = 2 lead (1) to the KdV equation [4]: ut + αuux + suxxx = 0, (3) with the soliton solution [5] u(x, t) = 12sk2 α sech2k(x − 4sk2t).

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 4 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Burgers Equation and KdV Equation Choices of δ = α = 0 and p = 2 lead (1) to the Burgers equation [3]: ut + αuux + βuxx = 0, (2) with the wave solution u(x, t) = 2k α + 2βk α tanh k(x − 2kt). Choices of α = µ = 0 and p = 2 lead (1) to the KdV equation [4]: ut + αuux + suxxx = 0, (3) with the soliton solution [5] u(x, t) = 12sk2 α sech2k(x − 4sk2t). [3] J.M. Burgers, Trans. Roy. Neth. Acad. Sci. 17 (1939), 1–53

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 4 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Burgers Equation and KdV Equation Choices of δ = α = 0 and p = 2 lead (1) to the Burgers equation [3]: ut + αuux + βuxx = 0, (2) with the wave solution u(x, t) = 2k α + 2βk α tanh k(x − 2kt). Choices of α = µ = 0 and p = 2 lead (1) to the KdV equation [4]: ut + αuux + suxxx = 0, (3) with the soliton solution [5] u(x, t) = 12sk2 α sech2k(x − 4sk2t). [3] J.M. Burgers, Trans. Roy. Neth. Acad. Sci. 17 (1939), 1–53 [4] D.J. Korteweg and G. de Vries, Phil. Mag. 39 (1895), 422–443.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 4 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Burgers Equation and KdV Equation Choices of δ = α = 0 and p = 2 lead (1) to the Burgers equation [3]: ut + αuux + βuxx = 0, (2) with the wave solution u(x, t) = 2k α + 2βk α tanh k(x − 2kt). Choices of α = µ = 0 and p = 2 lead (1) to the KdV equation [4]: ut + αuux + suxxx = 0, (3) with the soliton solution [5] u(x, t) = 12sk2 α sech2k(x − 4sk2t). [3] J.M. Burgers, Trans. Roy. Neth. Acad. Sci. 17 (1939), 1–53 [4] D.J. Korteweg and G. de Vries, Phil. Mag. 39 (1895), 422–443. [5] N.J. Zabusky and M.D. Kruskal, Phys. Rev. Lett. 15 (1965), 240–243.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 4 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Korteweg-de Vries-Burgers Equation Choices of α = 0 and p = 2 lead equation (1) to the standard form of the Korteweg-de Vries-Burgers equation [6]: ut + αuux + βuxx + suxxx = 0. (4)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 5 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Korteweg-de Vries-Burgers Equation Choices of α = 0 and p = 2 lead equation (1) to the standard form of the Korteweg-de Vries-Burgers equation [6]: ut + αuux + βuxx + suxxx = 0. (4) Solitary wave solutions of equation (4) are as follows [7, 8, 9]: u(x, t) = 3β2 25αssech2Ψ − 6β2 25αs tanh Ψ ± 6β2 25αs, (5)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 5 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Korteweg-de Vries-Burgers Equation Choices of α = 0 and p = 2 lead equation (1) to the standard form of the Korteweg-de Vries-Burgers equation [6]: ut + αuux + βuxx + suxxx = 0. (4) Solitary wave solutions of equation (4) are as follows [7, 8, 9]: u(x, t) = 3β2 25αssech2Ψ − 6β2 25αs tanh Ψ ± 6β2 25αs, (5) where Ψ = 1 2

− β

5sx ± 6β3 125s2 t

.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 5 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Korteweg-de Vries-Burgers Equation Choices of α = 0 and p = 2 lead equation (1) to the standard form of the Korteweg-de Vries-Burgers equation [6]: ut + αuux + βuxx + suxxx = 0. (4) Solitary wave solutions of equation (4) are as follows [7, 8, 9]: u(x, t) = 3β2 25αssech2Ψ − 6β2 25αs tanh Ψ ± 6β2 25αs, (5) where Ψ = 1 2

− β

5sx ± 6β3 125s2 t

.

— —

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 5 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Korteweg-de Vries-Burgers Equation Choices of α = 0 and p = 2 lead equation (1) to the standard form of the Korteweg-de Vries-Burgers equation [6]: ut + αuux + βuxx + suxxx = 0. (4) Solitary wave solutions of equation (4) are as follows [7, 8, 9]: u(x, t) = 3β2 25αssech2Ψ − 6β2 25αs tanh Ψ ± 6β2 25αs, (5) where Ψ = 1 2

− β

5sx ± 6β3 125s2 t

.

— — [6] R.S. Johnson, J. Fluid Mech. 42 (1970), 49–60.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 5 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Korteweg-de Vries-Burgers Equation Choices of α = 0 and p = 2 lead equation (1) to the standard form of the Korteweg-de Vries-Burgers equation [6]: ut + αuux + βuxx + suxxx = 0. (4) Solitary wave solutions of equation (4) are as follows [7, 8, 9]: u(x, t) = 3β2 25αssech2Ψ − 6β2 25αs tanh Ψ ± 6β2 25αs, (5) where Ψ = 1 2

− β

5sx ± 6β3 125s2 t

.

— — [6] R.S. Johnson, J. Fluid Mech. 42 (1970), 49–60. [7] Z. Feng, J. Phys. A (Math. Gen.) 36 (2003), 8817–8827.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 5 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Korteweg-de Vries-Burgers Equation Choices of α = 0 and p = 2 lead equation (1) to the standard form of the Korteweg-de Vries-Burgers equation [6]: ut + αuux + βuxx + suxxx = 0. (4) Solitary wave solutions of equation (4) are as follows [7, 8, 9]: u(x, t) = 3β2 25αssech2Ψ − 6β2 25αs tanh Ψ ± 6β2 25αs, (5) where Ψ = 1 2

− β

5sx ± 6β3 125s2 t

.

— — [6] R.S. Johnson, J. Fluid Mech. 42 (1970), 49–60. [7] Z. Feng, J. Phys. A (Math. Gen.) 36 (2003), 8817–8827. [8] Z. Feng, Nonlinearity, 20 (2007), 343–356.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 5 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Korteweg-de Vries-Burgers Equation Choices of α = 0 and p = 2 lead equation (1) to the standard form of the Korteweg-de Vries-Burgers equation [6]: ut + αuux + βuxx + suxxx = 0. (4) Solitary wave solutions of equation (4) are as follows [7, 8, 9]: u(x, t) = 3β2 25αssech2Ψ − 6β2 25αs tanh Ψ ± 6β2 25αs, (5) where Ψ = 1 2

− β

5sx ± 6β3 125s2 t

.

— — [6] R.S. Johnson, J. Fluid Mech. 42 (1970), 49–60. [7] Z. Feng, J. Phys. A (Math. Gen.) 36 (2003), 8817–8827. [8] Z. Feng, Nonlinearity, 20 (2007), 343–356. [9] Z. Feng and S. Zheng, Z. angew. Math. Phys. 60 (2009), 756–773.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 5 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Figures of Wave Solutions

u1-Burgers-KdV u2-Burgers-KdV u3-Burgers u4-KdV Legend –2 –1 1 2 y –8 –6 –4 –2 2 4 6 8 x

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 6 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Planar Polynomial Systems and Abel Equation Consider planar polynomial systems of the form ˙ x = −y + p(x, y), ˙ y = x + q(x, y) (6)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 7 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Planar Polynomial Systems and Abel Equation Consider planar polynomial systems of the form ˙ x = −y + p(x, y), ˙ y = x + q(x, y) (6) with homogeneous polynomials p(x, y) and q(x, y) of degree k.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 7 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Planar Polynomial Systems and Abel Equation Consider planar polynomial systems of the form ˙ x = −y + p(x, y), ˙ y = x + q(x, y) (6) with homogeneous polynomials p(x, y) and q(x, y) of degree k. For the Poincar´ e center problem, setting x = r cos θ, y = r sin θ gives dr dθ = rkξ(θ) 1 + rk−1η(θ), (7) where ξ and η are polynomials in cos θ and sin θ of degree k + 1.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 7 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Planar Polynomial Systems and Abel Equation Consider planar polynomial systems of the form ˙ x = −y + p(x, y), ˙ y = x + q(x, y) (6) with homogeneous polynomials p(x, y) and q(x, y) of degree k. For the Poincar´ e center problem, setting x = r cos θ, y = r sin θ gives dr dθ = rkξ(θ) 1 + rk−1η(θ), (7) where ξ and η are polynomials in cos θ and sin θ of degree k + 1. Making the coordinate transformation ρ = rk−1 1 + rk−1η(θ),

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 7 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Planar Polynomial Systems and Abel Equation Consider planar polynomial systems of the form ˙ x = −y + p(x, y), ˙ y = x + q(x, y) (6) with homogeneous polynomials p(x, y) and q(x, y) of degree k. For the Poincar´ e center problem, setting x = r cos θ, y = r sin θ gives dr dθ = rkξ(θ) 1 + rk−1η(θ), (7) where ξ and η are polynomials in cos θ and sin θ of degree k + 1. Making the coordinate transformation ρ = rk−1 1 + rk−1η(θ), we get an Abel equation dρ dθ = a(θ)ρ2 + b(θ)ρ3, where a = (k − 1)ξ + η′ and b = (1 − k)ξη.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 7 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Traveling Wave Solution Assume that equation (1) has the traveling wave solution of the form u(x, t) = u(ξ), ξ = x − ct,

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 8 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Traveling Wave Solution Assume that equation (1) has the traveling wave solution of the form u(x, t) = u(ξ), ξ = x − ct, where c = 0 is the wave velocity. Then equation (1) becomes δu′′′ − µu′′ + (α − c)u′ + βup−1u′ = 0, (8)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 8 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Traveling Wave Solution Assume that equation (1) has the traveling wave solution of the form u(x, t) = u(ξ), ξ = x − ct, where c = 0 is the wave velocity. Then equation (1) becomes δu′′′ − µu′′ + (α − c)u′ + βup−1u′ = 0, (8) where u′ = du/dξ. Integrating equation (8) once gives u′′ − gu′ − eu − fup − d = 0, (9)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 8 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Traveling Wave Solution Assume that equation (1) has the traveling wave solution of the form u(x, t) = u(ξ), ξ = x − ct, where c = 0 is the wave velocity. Then equation (1) becomes δu′′′ − µu′′ + (α − c)u′ + βup−1u′ = 0, (8) where u′ = du/dξ. Integrating equation (8) once gives u′′ − gu′ − eu − fup − d = 0, (9) where e = c−α

δ , g = µ δ , f = − β pδ and d is an integration constant.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 8 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Traveling Wave Solution Assume that equation (1) has the traveling wave solution of the form u(x, t) = u(ξ), ξ = x − ct, where c = 0 is the wave velocity. Then equation (1) becomes δu′′′ − µu′′ + (α − c)u′ + βup−1u′ = 0, (8) where u′ = du/dξ. Integrating equation (8) once gives u′′ − gu′ − eu − fup − d = 0, (9) where e = c−α

δ , g = µ δ , f = − β pδ and d is an integration constant.

Assume that y = u and u′ = z, then equation (9) is equivalent to y′ = z, z′ = ey + gz + fyp + d. (10)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 8 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Global Structure of p = 2

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 9 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Transformed to Abel Equation It follows from system (10) that dz dy = ey + gz + fyp + d z . (11)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 10 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Transformed to Abel Equation It follows from system (10) that dz dy = ey + gz + fyp + d z . (11) Let z = r−1. Equation (11) reduces to dr dy = a(y)r2 + b(y)r3, (12)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 10 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Transformed to Abel Equation It follows from system (10) that dz dy = ey + gz + fyp + d z . (11) Let z = r−1. Equation (11) reduces to dr dy = a(y)r2 + b(y)r3, (12) where a(y) = −g and b(y) = −(ey + fyp + d).

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 10 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Transformed to Abel Equation It follows from system (10) that dz dy = ey + gz + fyp + d z . (11) Let z = r−1. Equation (11) reduces to dr dy = a(y)r2 + b(y)r3, (12) where a(y) = −g and b(y) = −(ey + fyp + d). Question: Under what condition one can determine the number of closed solutions of the Abel equation (12).

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 10 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Transformed to Abel Equation It follows from system (10) that dz dy = ey + gz + fyp + d z . (11) Let z = r−1. Equation (11) reduces to dr dy = a(y)r2 + b(y)r3, (12) where a(y) = −g and b(y) = −(ey + fyp + d). Question: Under what condition one can determine the number of closed solutions of the Abel equation (12). Open Problem: There have been two longstanding problems, called the Poincar´ e center-focus problem and the local Hilbert 16th problem. Both are closely related to the Bautin quantities and the Bautin ideal of the Abel equation.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 10 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Integral Form Consider the generalized Abel equation r′ = a(t)r2 + b(t)rn, r(t0) = c, t ∈ [t0, t1], n ≥ 3. (13)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 11 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Integral Form Consider the generalized Abel equation r′ = a(t)r2 + b(t)rn, r(t0) = c, t ∈ [t0, t1], n ≥ 3. (13) Dividing both sides of equation (13) by r2 gives r′ r2 = a(t) + b(t)rn−2. (14)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 11 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Integral Form Consider the generalized Abel equation r′ = a(t)r2 + b(t)rn, r(t0) = c, t ∈ [t0, t1], n ≥ 3. (13) Dividing both sides of equation (13) by r2 gives r′ r2 = a(t) + b(t)rn−2. (14) Integrating equation (14) from t0 to t yields r(t) = c 1 − cA(t) − c t

t0 b(τ)rn−2dτ

, (15) where A(t) = t

t0 a(τ)dτ.

KdV-Burgers Equation

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Department of Mathematics, University of Texas-Pan American, Edinburg, USA 11 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Integral Form Consider the generalized Abel equation r′ = a(t)r2 + b(t)rn, r(t0) = c, t ∈ [t0, t1], n ≥ 3. (13) Dividing both sides of equation (13) by r2 gives r′ r2 = a(t) + b(t)rn−2. (14) Integrating equation (14) from t0 to t yields r(t) = c 1 − cA(t) − c t

t0 b(τ)rn−2dτ

, (15) where A(t) = t

t0 a(τ)dτ.

Rewrite equation (15) as r(t) = c

1 + A(t) + r(t)

t

t0

b(τ)rn−2dτ

.

(16)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 11 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

A Nonlinear Operator Let C[0, 1] denote the Banach space of all continuous functions on the interval [0, 1] with the norm f = max0≤t≤1 |f(t)|. We define the

perator [10]:

Tc : C[0, 1] → C[0, 1],

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 12 / 26

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A Nonlinear Operator Let C[0, 1] denote the Banach space of all continuous functions on the interval [0, 1] with the norm f = max0≤t≤1 |f(t)|. We define the

perator [10]:

Tc : C[0, 1] → C[0, 1], Tc(f)(t)

def

= c 1 − cA(t) − c t

0 b(τ)f(τ)n−2dτ

,

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 12 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

A Nonlinear Operator Let C[0, 1] denote the Banach space of all continuous functions on the interval [0, 1] with the norm f = max0≤t≤1 |f(t)|. We define the

perator [10]:

Tc : C[0, 1] → C[0, 1], Tc(f)(t)

def

= c 1 − cA(t) − c t

0 b(τ)f(τ)n−2dτ

, for given a, b ∈ C[0, 1] and c ∈ R. Obviously, Tc is well defined on an arbitrary bounded set of C[0, 1] if c is suitably small. Let us first observe some useful properties of Tc.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 12 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

A Nonlinear Operator Let C[0, 1] denote the Banach space of all continuous functions on the interval [0, 1] with the norm f = max0≤t≤1 |f(t)|. We define the

perator [10]:

Tc : C[0, 1] → C[0, 1], Tc(f)(t)

def

= c 1 − cA(t) − c t

0 b(τ)f(τ)n−2dτ

, for given a, b ∈ C[0, 1] and c ∈ R. Obviously, Tc is well defined on an arbitrary bounded set of C[0, 1] if c is suitably small. Let us first observe some useful properties of Tc. — —

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 12 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

A Nonlinear Operator Let C[0, 1] denote the Banach space of all continuous functions on the interval [0, 1] with the norm f = max0≤t≤1 |f(t)|. We define the

perator [10]:

Tc : C[0, 1] → C[0, 1], Tc(f)(t)

def

= c 1 − cA(t) − c t

0 b(τ)f(τ)n−2dτ

, for given a, b ∈ C[0, 1] and c ∈ R. Obviously, Tc is well defined on an arbitrary bounded set of C[0, 1] if c is suitably small. Let us first observe some useful properties of Tc. — — [10] Z. Feng, Z. angew. Math. Phys. under review.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 12 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Property of Our Operator Lemma (1) For f ∈ C[0, 1] and c ∈ R with f ≤ M and |c| < c0

def

= (a + bMn−2)−1, Tc(f) is well defined and differentiable, and satisfies d dtTc(f)(t) = a(t)[Tc(f)(t)]2 + b(t)[Tc(f)(t)]2f(t)n−2.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 13 / 26

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Property of Our Operator Lemma (1) For f ∈ C[0, 1] and c ∈ R with f ≤ M and |c| < c0

def

= (a + bMn−2)−1, Tc(f) is well defined and differentiable, and satisfies d dtTc(f)(t) = a(t)[Tc(f)(t)]2 + b(t)[Tc(f)(t)]2f(t)n−2. Furthermore, we have an identity Tc(f)(t) − Tc(g)(t) = Tc(f)(t)Tc(g)(t) t b(τ)(f(τ)n−2 − g(τ)n−2)dτ, 0 ≤ t ≤ 1

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 13 / 26

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Property of Our Operator Lemma (1) For f ∈ C[0, 1] and c ∈ R with f ≤ M and |c| < c0

def

= (a + bMn−2)−1, Tc(f) is well defined and differentiable, and satisfies d dtTc(f)(t) = a(t)[Tc(f)(t)]2 + b(t)[Tc(f)(t)]2f(t)n−2. Furthermore, we have an identity Tc(f)(t) − Tc(g)(t) = Tc(f)(t)Tc(g)(t) t b(τ)(f(τ)n−2 − g(τ)n−2)dτ, 0 ≤ t ≤ 1 for arbitrary f, g ∈ C[0, 1] and c ∈ R with f, g ≤ M and |c| < c0.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 13 / 26

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Outline of the Proof Step 1: well-defined 1 − cA(t) − c t b(τ)f(τ)n−2dτ = 0 ⇒ |c| ≥ 1 |A(t)| + t

0 |b(τ)f(τ)n−2| dτ

≥ 1 a + bMn−2 .

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 14 / 26

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Outline of the Proof Step 1: well-defined 1 − cA(t) − c t b(τ)f(τ)n−2dτ = 0 ⇒ |c| ≥ 1 |A(t)| + t

0 |b(τ)f(τ)n−2| dτ

≥ 1 a + bMn−2 . Step 2: A direct calculation gives d dtTc(f)(t) = −c[−ca(t) − cb(t)f(t)n−2] (1 − cA(t) − c t

0 b(τ)f(τ)n−2dτ)2

= c2a(t) (1 − cA(t) − c t

0 b(τ)f(τ)n−2dτ)2 +

c2b(t)f(t)n−2 (1 − cA(t) − c t

0 b(τ)f(τ)n−2dτ)2

Tc(f)(t) − Tc(g)(t) = c H(f) · c H(g) · t b(τ)(f(τ)n−2 − g(τ)n−2)dτ

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 14 / 26

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Lemma 2 Lemma (2) Let c1 = (a + b + 1)−1. Then we have Tcf ≤ 1 if f ≤ 1 and |c| ≤ c1.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 15 / 26

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Lemma 2 Lemma (2) Let c1 = (a + b + 1)−1. Then we have Tcf ≤ 1 if f ≤ 1 and |c| ≤ c1. Outline of the Proof. If f ≤ 1 and |c| ≤ c1, then we have Tcf ≤ |c| 1 − |c| (a + bfn−2) ≤ |c| 1 − |c| (a + b) ≤ 1. The conclusion follows.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 15 / 26

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Lemma 3 Lemma (3) Let c2 = (

(n − 2)b + a + b + 1)−1. If |c| ≤ c2, then Tc is a

contraction mapping on the close unit ball B1 = {f ∈ C[0, 1]| f ≤ 1} of C[0, 1]. Outline of the Proof. It follows from Lemmas 1 and 2 that Tc(f)(t) − Tc(g)(t) ≤ Tc(f)Tc(g)bf n−2 − gn−2 = C(f − g)(f n−3 + f n−4g + · · · + fgn−4 + gn−3) ≤ (n − 2)cf − g, where c

def

=

|c|

1 − |c| (a + b) 2 b.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 16 / 26

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Theorem 1 Theorem (1) For given a, b ∈ C[0, 1] and c ∈ R with |c| ≤ (

(n − 2)b + a + b + 1)−1, the solution r(t, c) of equation (1)

with r(0, c) = c can be uniformly approximated by an iterated sequence {Tn

c (f)(t)}:

r(t, c) = lim

n→∞ Tn c (f)(t),

0 ≤ t ≤ 1, (17) that is, r(t, c) = c 1 − cA(t) − cn−1 t

b(t1)dt1 1−cA(t1)−cn−1 t1

b(t2)dt2 1−cA(t2)−cn−1 t2 0 ···

(18) for arbitrary f ∈ C[0, 1] with f ≤ 1. Furthermore, the following error estimate holds r(t, c) − Tn

c (f)(t) = O(c2n).

KdV-Burgers Equation

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Department of Mathematics, University of Texas-Pan American, Edinburg, USA 17 / 26

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Theorem 2: Case of n = 3 Denote M = max

t∈[0,1] |a(t) ± b(t)|.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 18 / 26

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Theorem 2: Case of n = 3 Denote M = max

t∈[0,1] |a(t) ± b(t)|.

Theorem (2) Suppose a, b ∈ C[0, 1] and c ∈ R with |c| ≤ max{(

b + a + b + 1)−1, (2M)−1}.

Then, in formula (18), the following part is bounded b(t1) 1 − cA(t1) − c2 t1

b(t2)dt2 1−cA(t2)−c2 t2

0 ···

= 1 c · b(t1) · c 1 − cA(t1) − c2 t1

0 b(t2) · c 1−cA(t2)−c2 t2

0 ···dt2

.

KdV-Burgers Equation

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Department of Mathematics, University of Texas-Pan American, Edinburg, USA 18 / 26

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2D Korteweg-de Vries-Burgers Equation Consider the 2D Korteweg-de Vries-Burgers equation: (Ut + αUUx + βUxx + sUxxx)x + γUyy = 0, (19)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 19 / 26

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2D Korteweg-de Vries-Burgers Equation Consider the 2D Korteweg-de Vries-Burgers equation: (Ut + αUUx + βUxx + sUxxx)x + γUyy = 0, (19) where α, β, s, and γ are constants and αβsγ = 0.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 19 / 26

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2D Korteweg-de Vries-Burgers Equation Consider the 2D Korteweg-de Vries-Burgers equation: (Ut + αUUx + βUxx + sUxxx)x + γUyy = 0, (19) where α, β, s, and γ are constants and αβsγ = 0. Assume that equation (19) has an exact solution in the form U(x, y, t) = U(ξ), ξ = hx + ly − wt. (20)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 19 / 26

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2D Korteweg-de Vries-Burgers Equation Consider the 2D Korteweg-de Vries-Burgers equation: (Ut + αUUx + βUxx + sUxxx)x + γUyy = 0, (19) where α, β, s, and γ are constants and αβsγ = 0. Assume that equation (19) has an exact solution in the form U(x, y, t) = U(ξ), ξ = hx + ly − wt. (20) Substitution of (20) into equation (19) and performing integration twice yields U

′′(ξ) + λU ′(ξ) + aU2(ξ) + bU(ξ) + d = 0,

(21) where v = U(ξ) ∈ [v0, v1], λ = β

sh, a = α 2sh2 , b = γl2−wh sh4

and d = − C

sh4 .

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 19 / 26

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Resultant Abel Equation Let v = U(ξ) and y = U

′(ξ). Equation (21) becomes

dy dvy + λy + av2 + bv + d = 0. (22)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 20 / 26

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Resultant Abel Equation Let v = U(ξ) and y = U

′(ξ). Equation (21) becomes

dy dvy + λy + av2 + bv + d = 0. (22) Using z = 1

y yields

dz dv = λz2 + (av2 + bv + d)z3, z(v0) = 1 U

′(ξ0) = c.

(23)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 20 / 26

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Resultant Abel Equation Let v = U(ξ) and y = U

′(ξ). Equation (21) becomes

dy dvy + λy + av2 + bv + d = 0. (22) Using z = 1

y yields

dz dv = λz2 + (av2 + bv + d)z3, z(v0) = 1 U

′(ξ0) = c.

(23) Let η = v−v0

v1−v0 , then η ∈ [0, 1] and v = v0 + (v1 − v0)η.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 20 / 26

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Resultant Abel Equation Let v = U(ξ) and y = U

′(ξ). Equation (21) becomes

dy dvy + λy + av2 + bv + d = 0. (22) Using z = 1

y yields

dz dv = λz2 + (av2 + bv + d)z3, z(v0) = 1 U

′(ξ0) = c.

(23) Let η = v−v0

v1−v0 , then η ∈ [0, 1] and v = v0 + (v1 − v0)η. So equation (23)

reduces to r

′ = h(η)r2 + k(η)r3,

r(0) = c, (24)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 20 / 26

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Resultant Abel Equation Let v = U(ξ) and y = U

′(ξ). Equation (21) becomes

dy dvy + λy + av2 + bv + d = 0. (22) Using z = 1

y yields

dz dv = λz2 + (av2 + bv + d)z3, z(v0) = 1 U

′(ξ0) = c.

(23) Let η = v−v0

v1−v0 , then η ∈ [0, 1] and v = v0 + (v1 − v0)η. So equation (23)

reduces to r

′ = h(η)r2 + k(η)r3,

r(0) = c, (24) where h(η), k(η) ∈ C[0, 1], and h(η) = (v1 − v0)λ, k(η) = (v1 − v0)(av2 + bv + d).

KdV-Burgers Equation

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Department of Mathematics, University of Texas-Pan American, Edinburg, USA 20 / 26

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Solution to Equation (24) By virtue of Theorem 1, if |c| ≤ (

k + h + k + 1)−1, the

solution to equation (24) is r(η) = lim

n→+∞ Tn c (w)(η),

(25)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 21 / 26

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Solution to Equation (24) By virtue of Theorem 1, if |c| ≤ (

k + h + k + 1)−1, the

solution to equation (24) is r(η) = lim

n→+∞ Tn c (w)(η),

(25) where 0 ≤ η ≤ 1 for any w ∈ C[0, 1] with w ≤ 1, and Tc(w) = c 1 − cH(η) − c η

0 k(x)w(x)n−2dx

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 21 / 26

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Solution to Equation (24) By virtue of Theorem 1, if |c| ≤ (

k + h + k + 1)−1, the

solution to equation (24) is r(η) = lim

n→+∞ Tn c (w)(η),

(25) where 0 ≤ η ≤ 1 for any w ∈ C[0, 1] with w ≤ 1, and Tc(w) = c 1 − cH(η) − c η

0 k(x)w(x)n−2dx

where H(η) = η h(x) dx = η (v1 − v0)λ dx = (v1 − v0)λη, k(x) = (v1 − v0)

a(v0 + (v1 − v0)x)2 + b(v0 + (v1 − v0)x) + d
.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 21 / 26

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Approximate Solution to 2D-KdV-Burgers Equation Recall that r = 1

y, y = U

′(ξ), η = v−v0

v1−v0 and v = U(ξ). When conditions

f Theorem 1 are fulfilled, we have

1 U′(ξ) = c 1 − cA(ξ) − c2 ξ

b(t1)dt1 1−cA(t1)−c2 t1

b(t2)dt2 1−cA(t2)−c2 t2 0 ···

. (26)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 22 / 26

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Approximate Solution to 2D-KdV-Burgers Equation Recall that r = 1

y, y = U

′(ξ), η = v−v0

v1−v0 and v = U(ξ). When conditions

f Theorem 1 are fulfilled, we have

1 U′(ξ) = c 1 − cA(ξ) − c2 ξ

b(t1)dt1 1−cA(t1)−c2 t1

b(t2)dt2 1−cA(t2)−c2 t2 0 ···

. (26) When c is small, according to Theorem 2, the coefficient of c2 is

bounded. So we can drop the term containing c2 and get

U

′(ξ)

≈ 1 − c(v1 − v0)λη c = 1 − cλ(U(ξ) − v0) c .

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 22 / 26

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Approximate Solution to 2D-KdV-Burgers Equation Recall that r = 1

y, y = U

′(ξ), η = v−v0

v1−v0 and v = U(ξ). When conditions

f Theorem 1 are fulfilled, we have

1 U′(ξ) = c 1 − cA(ξ) − c2 ξ

b(t1)dt1 1−cA(t1)−c2 t1

b(t2)dt2 1−cA(t2)−c2 t2 0 ···

. (26) When c is small, according to Theorem 2, the coefficient of c2 is

bounded. So we can drop the term containing c2 and get

U

′(ξ)

≈ 1 − c(v1 − v0)λη c = 1 − cλ(U(ξ) − v0) c . That is, U

′(ξ) + λU(ξ) = 1

c + λv0. (27)

KdV-Burgers Equation

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Department of Mathematics, University of Texas-Pan American, Edinburg, USA 22 / 26

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Approximate Solution to 2D KdV-Burgers Equation Solving equation (27) gives U(x, y, t) =

1 c + λv0

λ + ce−λξ, ξ = hx + ly − wt where λ = β

sh.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 23 / 26

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Approximate Solution to 2D KdV-Burgers Equation Solving equation (27) gives U(x, y, t) =

1 c + λv0

λ + ce−λξ, ξ = hx + ly − wt where λ = β

sh.

If we take v0 = b

2a and choose c = −2a λ√ b2−4ad sufficiently small, when

λξ → +∞, we obtain U(x, y, t) ∼ b2 − 4ad −2a + b 2a. (28)

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 23 / 26

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Approximate Solution to 2D KdV-Burgers Equation Solving equation (27) gives U(x, y, t) =

1 c + λv0

λ + ce−λξ, ξ = hx + ly − wt where λ = β

sh.

If we take v0 = b

2a and choose c = −2a λ√ b2−4ad sufficiently small, when

λξ → +∞, we obtain U(x, y, t) ∼ b2 − 4ad −2a + b 2a. (28) It is remarkable that the approximate solution (28) is in agreement with main results described in [7, 8] by the Hardy’s theory and the theory of Lie symmetry.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 23 / 26

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Approximate Solution to 2D KdV-Burgers Equation Solving equation (27) gives U(x, y, t) =

1 c + λv0

λ + ce−λξ, ξ = hx + ly − wt where λ = β

sh.

If we take v0 = b

2a and choose c = −2a λ√ b2−4ad sufficiently small, when

λξ → +∞, we obtain U(x, y, t) ∼ b2 − 4ad −2a + b 2a. (28) It is remarkable that the approximate solution (28) is in agreement with main results described in [7, 8] by the Hardy’s theory and the theory of Lie symmetry. — —

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 23 / 26

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Approximate Solution to 2D KdV-Burgers Equation Solving equation (27) gives U(x, y, t) =

1 c + λv0

λ + ce−λξ, ξ = hx + ly − wt where λ = β

sh.

If we take v0 = b

2a and choose c = −2a λ√ b2−4ad sufficiently small, when

λξ → +∞, we obtain U(x, y, t) ∼ b2 − 4ad −2a + b 2a. (28) It is remarkable that the approximate solution (28) is in agreement with main results described in [7, 8] by the Hardy’s theory and the theory of Lie symmetry. — — [7] Z. Feng, J. Phys. A (Math. Gen.) 36 (2003), 8817–8827.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 23 / 26

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Approximate Solution to 2D KdV-Burgers Equation Solving equation (27) gives U(x, y, t) =

1 c + λv0

λ + ce−λξ, ξ = hx + ly − wt where λ = β

sh.

If we take v0 = b

2a and choose c = −2a λ√ b2−4ad sufficiently small, when

λξ → +∞, we obtain U(x, y, t) ∼ b2 − 4ad −2a + b 2a. (28) It is remarkable that the approximate solution (28) is in agreement with main results described in [7, 8] by the Hardy’s theory and the theory of Lie symmetry. — — [7] Z. Feng, J. Phys. A (Math. Gen.) 36 (2003), 8817–8827. [8] Z. Feng, Nonlinearity, 20 (2007), 343–356.

KdV-Burgers Equation

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Department of Mathematics, University of Texas-Pan American, Edinburg, USA 23 / 26

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Boundedness of Solutions Note that equation (26) can be rewritten as 1 U′(ξ) = c 1 − cA(ξ) − c2Φ(ξ), (29) where L ≤ Φ(ξ) ≤ R.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 24 / 26

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Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Boundedness of Solutions Note that equation (26) can be rewritten as 1 U′(ξ) = c 1 − cA(ξ) − c2Φ(ξ), (29) where L ≤ Φ(ξ) ≤ R. When Φ is a quadratic or cubic function or special function of U(ξ), we can analyze equation (29) qualitatively and numerically with

classifications. For instance, if Φ is quadratic, we take v0 = b

2a and

choose c =

−2a λ√ b2−4ad sufficiently small, we can obtain the solution of

the type u(x, y, t) = 3β2 + γ + c 25αs sech2ξ − 6β2 + γ + c 25αs tanh ξ ± 6β2 25αs + C0.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 24 / 26

SLIDE 88

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Boundedness of Solutions Note that equation (26) can be rewritten as 1 U′(ξ) = c 1 − cA(ξ) − c2Φ(ξ), (29) where L ≤ Φ(ξ) ≤ R. When Φ is a quadratic or cubic function or special function of U(ξ), we can analyze equation (29) qualitatively and numerically with

classifications. For instance, if Φ is quadratic, we take v0 = b

2a and

choose c =

−2a λ√ b2−4ad sufficiently small, we can obtain the solution of

the type u(x, y, t) = 3β2 + γ + c 25αs sech2ξ − 6β2 + γ + c 25αs tanh ξ ± 6β2 25αs + C0. When Φ is a function with the lower and upper bounds, we can also find bounds of solutions of equation (29) by the comparison principle, which match well with the phase analysis described in [7].

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 24 / 26

SLIDE 89

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Summary In this talk, we provided a connection between the Abel equation of the first kind, an ordinary differential equation that is cubic in the unknown function, and the Korteweg-de Vries-Burgers equation, a partial differential equation that describes the propagation of waves on liquid-filled elastic tubes. We presented an integral form of the Abel equation with the initial condition.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 25 / 26

SLIDE 90

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Summary In this talk, we provided a connection between the Abel equation of the first kind, an ordinary differential equation that is cubic in the unknown function, and the Korteweg-de Vries-Burgers equation, a partial differential equation that describes the propagation of waves on liquid-filled elastic tubes. We presented an integral form of the Abel equation with the initial condition. By virtue of the integral form and the Banach Contraction Mapping Principle we derived the asymptotic expansion of bounded solutions in the Banach space, and used the asymptotic formula to construct approximate solutions to the Korteweg-de Vries-Burgers equation.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 25 / 26

SLIDE 91

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Summary In this talk, we provided a connection between the Abel equation of the first kind, an ordinary differential equation that is cubic in the unknown function, and the Korteweg-de Vries-Burgers equation, a partial differential equation that describes the propagation of waves on liquid-filled elastic tubes. We presented an integral form of the Abel equation with the initial condition. By virtue of the integral form and the Banach Contraction Mapping Principle we derived the asymptotic expansion of bounded solutions in the Banach space, and used the asymptotic formula to construct approximate solutions to the Korteweg-de Vries-Burgers equation. As an example, we presented the asymptotic behavior of traveling wave solution for a 2D KdV-Burgers equation which agrees well with existing results in the literature.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 25 / 26

SLIDE 92

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Summary In this talk, we provided a connection between the Abel equation of the first kind, an ordinary differential equation that is cubic in the unknown function, and the Korteweg-de Vries-Burgers equation, a partial differential equation that describes the propagation of waves on liquid-filled elastic tubes. We presented an integral form of the Abel equation with the initial condition. By virtue of the integral form and the Banach Contraction Mapping Principle we derived the asymptotic expansion of bounded solutions in the Banach space, and used the asymptotic formula to construct approximate solutions to the Korteweg-de Vries-Burgers equation. As an example, we presented the asymptotic behavior of traveling wave solution for a 2D KdV-Burgers equation which agrees well with existing results in the literature. Under certain conditions, we can also study bounds of traveling wave solutions of KdV-Burgers type equations by the comparison principle.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 25 / 26

SLIDE 93

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Acknowledgement I would like to thank

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 26 / 26

SLIDE 94

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Acknowledgement I would like to thank Xiaoqian Gong

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 26 / 26

SLIDE 95

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Acknowledgement I would like to thank Xiaoqian Gong for discussions and help on computations.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 26 / 26

SLIDE 96

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement

Acknowledgement I would like to thank Xiaoqian Gong for discussions and help on computations. Thank you.

KdV-Burgers Equation

Z. Feng

Department of Mathematics, University of Texas-Pan American, Edinburg, USA 26 / 26