# Exploring Solutions of Nonlinear Rossby Waves in Shallow Water Equations

15 Aug 1996-Journal of the Physical Society of Japan (Physical Society of Japan)-Vol. 65, Iss: 8, pp 2717-2721

TL;DR: In this article, the authors derived analytical conditions for the existence of finite amplitude nonlinear wave solutions in shallow water equations, including solitary wave, periodic trigonometrical and decaying exponential types.

Abstract: We study shallow water equations to derive analytical conditions for the existence of finite amplitude nonlinear wave solutions. The bounded solutions obtained by us include solitary wave, periodic trigonometrical and decaying exponential types.

##### Citations

More filters

••

[...]

TL;DR: In this paper, strong generalized solvability, uniqueness, and blow-up of solutions of Cauchy problems are discussed. But they do not consider the problem of finding a global solution of the problem.

Abstract: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1. Statement of Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2. Elementary Definitions and Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3. Weak Generalized Solvability, Uniqueness, and Blow-Up of Solutions of Problem (1.1) . . . . 15 4. Strong Generalized Solvability, Uniqueness, and Blow-Up of Solutions of Problem (1.1) . . . . 32 5. Weak Generalized Solvability, Uniqueness, and Blow-Up of Solutions of Problem (1.2) . . . . 38 6. Strong Generalized Solvability, Uniqueness, and Blow-Up of Solutions of Problem (1.2) . . . . 51 7. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 8. On Some Problems of the Form (1.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 8.1. Strong generalized local solvability of problems (8.1)–(8.3) . . . . . . . . . . . . . . . . . 63 8.2. Blow-up of solutions of problem (8.1)–(8.3) . . . . . . . . . . . . . . . . . . . . . . . . . . 67 8.3. Breakdown of weakened solutions of problem (8.1) . . . . . . . . . . . . . . . . . . . . . . 72 9. On One Problem of the Form (1.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 9.1. Local solvability and uniqueness in the strong generalized sense. . . . . . . . . . . . . . . 77 9.2. Blow-up and global solvability of problem (9.1)–(9.2) . . . . . . . . . . . . . . . . . . . . 81 10. On the Blow-Up of Solutions of One Class of Quasilinear Dissipative Wave Equations of Pseudoparabolic Type with Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 10.1. Local solvability and blow-up a of strong generalized solution of Cauchy problem (10.1)– (10.2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 10.2. Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 11. On the Blow-Up of Solutions of the Benjamin–Bona–Mahony–Burgers Equation with Cubic Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 11.1. Local and global solvability of problem (11.1)–(11.3) in the strong generalized sense. . . 91 11.2. Global solvability and blow-up of the strong generalized solution of problem (1.1)–(1.3) . 93 11.3. Physical interpretation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 12. On the Blow-Up of Solutions of the Benjamin–Bona–Mahony–Burgers Equation with PseudoLaplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 12.1. Local solvability of problem (12.1)–(12.3) in the strong generalized sense . . . . . . . . . 96 12.2. Blow-Up of weakened solutions of problem (12.1)–(12.3) . . . . . . . . . . . . . . . . . . 101 12.3. Physical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 13. On Sufficient, Approximately Necessary Conditions for Blow-Up of Solutions of One Problem 103 13.1. Blow-up of strong generalized solutions of problem (13.1)–(13.2) . . . . . . . . . . . . . . 104 13.2. Global-in-time weakened solvability of problem (13.1)–(13.2) for small initial data . . . . 107 13.3. Physical interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

21 citations

••

[...]

TL;DR: In this article, nonlinear Rossby waves in a Boussinesq fluid model were studied by using the semi-geostrophic approximation and the method of travelling-wave solution.

Abstract: Nonlinear Rossby waves in a Boussinesq fluid model which includes both the vertical and horizontal components of Coriolis force are studied by using the semi-geostrophic approximation and the method of travelling-wave solution. Taylor series expansion has been employed to isolate the characteristics of the linear Rossby waves and identify the Rossby cnoidal and solitary waves. Qualitative analysis indicates that if the disturbances are independent of latitude, the effect of horizontal components of Coriolis force disappears.

••

[...]

TL;DR: In this paper, Taylor series expansion has been employed to isolate the characteristics of the linear Rossby waves and to identify the nonlinear shock and kink waves in a Boussinesq fluid model which includes both the vertical and horizontal components of Coriolis force.

Abstract: Nonlinear waves in a Boussinesq fluid model which includes both the vertical and horizontal components of Coriolis force are studied by using the semi-geostrophic approximation and the method of travelling-wave solution. Taylor series expansion has been employed to isolate the characteristics of the linear Rossby waves and to identify the nonlinear shock and kink waves. The KdV-Burgers and the compound KdV-Burgers equations are derived, their shock wave and kink wave solution are also obtained.

##### References

More filters

••

[...]

TL;DR: In this paper, a coupled pair of Korteweg-de-vries equations is used to describe the interaction of long-wave solitons propagating in shear flows.

Abstract: Interactions of long-wave solitons propagating in shear flows are described by a coupled pair of Korteweg-de Vries equations. The basic equation of motion for the analysis is the quasi-geostrophic forecast equation, and the interaction of two wave modes is studied. The solution for mode 1-mode 2 interaction of solitary waves in an asymmetric shear flow of a barotropic atmosphere with divergence is constructed. Streamline patterns for certain flows are obtained. An unsteady solitary wave solution for a modified Korteweg-de Vries equation is derived.

42 citations