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1. Exact solutions of stochastic Burgers–Korteweg de Vries type equation with variable coefficients

   https://doi.org/10.1016/j.padiff.2024.100753  

   Adjibi, Kolade; Martinez, Allan; Mascorro, Miguel; Montes, Carlos; Oraby, Tamer; Sandoval, Rita; Suazo, Erwin (September 2024, Partial Differential Equations in Applied Mathematics)

   We will present exact solutions for three variations of the stochastic Korteweg de Vries–Burgers (KdV–Burgers) equation featuring variable coefficients. In each variant, white noise exhibits spatial uniformity, and the three categories include additive, multiplicative, and advection noise. Across all cases, the coefficients are timedependent functions. Our discovery indicates that solving certain deterministic counterparts of KdV–Burgers equations and composing the solution with a solution of stochastic differential equations leads to the exact solution of the stochastic Korteweg de Vries–Burgers (KdV–Burgers) equations. 

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2. Well-posedness and dynamics of solutions to the generalized KdV with low power nonlinearity

   https://doi.org/10.1088/1361-6544/ac93e1  

   Friedman, Isaac; Riaño, Oscar; Roudenko, Svetlana; Son, Diana; Yang, Kai (December 2022, Nonlinearity)

   Abstract We consider two types of the generalized Korteweg–de Vries equation, where the nonlinearity is given with or without absolute values, and, in particular, including the low powers of nonlinearity, an example of which is the Schamel equation. We first prove the local well-posedness of both equations in a weighted subspace of H 1 that includes functions with polynomial decay, extending the result of Linares et al (2019 Commun. Contemp. Math. 21 1850056) to fractional weights. We then investigate solutions numerically, confirming the well-posedness and extending it to a wider class of functions that includes exponential decay. We include a comparison of solutions to both types of equations, in particular, we investigate soliton resolution for the positive and negative data with different decay rates. Finally, we study the interaction of various solitary waves in both models, showing the formation of solitons, dispersive radiation and even breathers, all of which are easier to track in nonlinearities with lower power. 

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3. Multisolitons for the cubic NLS in 1-d and their stability

   https://doi.org/10.1007/s10240-024-00148-8  

   Koch, Herbert; Tataru, Daniel (December 2024, Publications mathématiques de l'IHÉS)

   For both the cubic Nonlinear Schrödinger Equation (NLS) as well as the modified Korteweg-de Vries (mKdV) equation in one space dimension we consider the set of N--soliton states, and their associated multisoliton solutions. We prove that (i) this set is a uniformly smooth manifold, and (ii) the$$\mathbf {M}_{N}$$ $M_N$states are uniformly stable in H^s for each s>-1/2. One main tool in our analysis is an iterated Bäcklund transform, which allows us to nonlinearly add a multisoliton to an existing soliton free state (the soliton addition map) or alternatively to remove a multisoliton from a multisoliton state (the soliton removal map). The properties and the regularity of these maps are extensively studied. 

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4. Whitham equations and phase shifts for the Korteweg–de Vries equation

   https://doi.org/10.1098/rspa.2020.0300  

   Ablowitz, Mark J.; Cole, Justin T.; Rumanov, Igor (August 2020, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   null (Ed.)

   The semi-classical Korteweg–de Vries equation for step-like data is considered with a small parameter in front of the highest derivative. Using perturbation analysis, Whitham theory is constructed to the higher order. This allows the order one phase and the complete leading-order solution to be obtained; the results are confirmed by extensive numerical calculations. 

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5. Interaction of linear modulated waves and unsteady dispersive hydrodynamic states with application to shallow water waves

   https://doi.org/10.1017/jfm.2019.534  

   Congy, T.; El, G. A.; Hoefer, M. A. (September 2019, Journal of Fluid Mechanics)

   A new type of wave–mean flow interaction is identified and studied in which a small-amplitude, linear, dispersive modulated wave propagates through an evolving, nonlinear, large-scale fluid state such as an expansion (rarefaction) wave or a dispersive shock wave (undular bore). The Korteweg–de Vries (KdV) equation is considered as a prototypical example of dynamic wavepacket–mean flow interaction. Modulation equations are derived for the coupling between linear wave modulations and a nonlinear mean flow. These equations admit a particular class of solutions that describe the transmission or trapping of a linear wavepacket by an unsteady hydrodynamic state. Two adiabatic invariants of motion are identified that determine the transmission, trapping conditions and show that wavepackets incident upon smooth expansion waves or compressive, rapidly oscillating dispersive shock waves exhibit so-called hydrodynamic reciprocity recently described in Maiden et al.  ( Phys. Rev. Lett. , vol. 120, 2018, 144101) in the context of hydrodynamic soliton tunnelling. The modulation theory results are in excellent agreement with direct numerical simulations of full KdV dynamics. The integrability of the KdV equation is not invoked so these results can be extended to other nonlinear dispersive fluid mechanic models. 

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