# On the stability of periodic solutions of the generalized Benjamin-Bona-Mahony equation

@article{Johnson2010OnTS, title={On the stability of periodic solutions of the generalized Benjamin-Bona-Mahony equation}, author={Mathew A. Johnson}, journal={Physica D: Nonlinear Phenomena}, year={2010}, volume={239}, pages={1892-1908} }

## 22 Citations

### The regularized Benjamin–Ono and BBM equations: Well-posedness and nonlinear stability

- Mathematics
- 2011

### The Transverse Instability of Periodic Waves in Zakharov–Kuznetsov Type Equations

- Mathematics
- 2009

In this paper, we investigate the instability of one‐dimensionally stable periodic traveling wave solutions of the generalized Korteweg‐de Vries equation to long wavelength transverse perturbations…

### Modulational instability in nonlinear nonlocal equations of regularized long wave type

- Mathematics, Physics
- 2015

### Existence and orbital stability of periodic wave solutions for the nonlinear Schrodinger equation

- Mathematics
- 2012

In this paper, we study the existence and orbital stability of periodic wave solutions or the Schrodinger equation. The existence of periodic wave solution is obtained by using the phase portrait…

### Stability of Periodic Traveling Waves for Nonlinear Dispersive Equations

- MathematicsSIAM J. Math. Anal.
- 2015

It is shown that a local constrained minimizer for a suitable variational problem is nonlinearly stable to period preserving perturbations, provided that the associated linearized operator enjoys a Jordan block structure.

### Orbital stability of periodic traveling-wave solutions for the regularized Schamel equation

- Mathematics
- 2015

### Periodic traveling-wave solutions for regularized dispersive equations: sufficient conditions for orbital stability with applications

- MathematicsCommunications in Mathematical Sciences
- 2020

In this paper, we establish a new criterion for the orbital stability of periodic waves related to a general class of regularized dispersive equations. More specifically, we present sufficient…

### Rigorous Justification of the Whitham Modulation Equations for the Generalized Korteweg–de Vries Equation

- Mathematics
- 2009

In this paper, we consider the spectral stability of spatially periodic traveling wave solutions of the generalized Korteweg–de Vries equation to long‐wavelength perturbations. Specifically, we…

### Sufficient conditions for orbital stability of periodic traveling waves

- MathematicsJournal of Differential Equations
- 2019

## References

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### The Transverse Instability of Periodic Waves in Zakharov–Kuznetsov Type Equations

- Mathematics
- 2009

In this paper, we investigate the instability of one‐dimensionally stable periodic traveling wave solutions of the generalized Korteweg‐de Vries equation to long wavelength transverse perturbations…

### On the orbital ( in ) stability of spatially periodic stationary solutions of generalized Korteweg-de Vries equations

- Mathematics, Physics
- 2009

In this paper we generalize previous work on the stability of waves for infinite-dimensional Hamiltonian systems to include those cases for which the skew-symmetric operator J is singular. We assume…

### Nonlinear Stability of Periodic Travelling Wave Solutions for the Regularized Benjamin-Ono and BBM Equations

- Mathematics
- 2009

This paper has various goals: first, we develop a local and global well-posedness theory for the regularized Benjamin-Ono equation in the periodic setting, second, we show that the Cauchy problem for…

### Nonlinear Stability of Periodic Traveling Wave Solutions of the Generalized Korteweg-de Vries Equation

- MathematicsSIAM J. Math. Anal.
- 2009

It is shown that the orbital stability for a four-parameter family of periodic stationary traveling wave solutions to the generalized Korteweg–de Vries equation is equivalent to the solution being spectrally stable with respect to perturbations of the same period.

### Rigorous Justification of the Whitham Modulation Equations for the Generalized Korteweg–de Vries Equation

- Mathematics
- 2009

In this paper, we consider the spectral stability of spatially periodic traveling wave solutions of the generalized Korteweg–de Vries equation to long‐wavelength perturbations. Specifically, we…

### Transverse Instability of Periodic Traveling Waves in the Generalized Kadomtsev-Petviashvili Equation

- MathematicsSIAM J. Math. Anal.
- 2010

The spectral instability of periodic traveling wave solutions of the generalized Korteweg–de Vries equation to long wavelength transverse perturbations in the generalized Kadomtsev–Petviashvili equation is investigated and an orientation index is derived which yields sufficient conditions for such an instability to occur.

### The Modulational Instability for a Generalized Korteweg–de Vries Equation

- Mathematics
- 2008

We study the spectral stability of a family of periodic standing wave solutions to the generalized Korteweg–de Vries in a neighborhood of the origin in the spectral plane using what amounts to a…

### On the modulation equations and stability of periodic generalized Korteweg–de Vries waves via Bloch decompositions

- Mathematics
- 2010

### KdV cnoidal waves are spectrally stable

- Mathematics
- 2009

Going back to considerations of Benjamin (1974), there has been
significant interest in the question of stability for the stationary
periodic solutions of the Korteweg-deVries equation, the…