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The KP-II equation was derived by Kadomtsev and Petviashvili to explain stability of line solitary waves of shallow water. Using the Darboux transformations, we study linear stability of 2-line solitons whose line solitons interact…
This paper is concerned with a generalized type of Darboux transformations defined in terms of a twisted derivation $D$ satisfying $D(AB)=D(A)+\sigma(A)B$ where $\sigma$ is a homomorphism. Such twisted derivations include regular…
Soliton Solutions of Korteweg-de Vries (KdV) were constructed for given degenerate curves $y^2 = (x-c)P(x)^2$ in terms of hyperelliptic sigma functions and explicit Abelian integrals. Connection between sigma functions and tau function were…
For a generalized super KdV equation, three Darboux transformations and the corresponding B\"acklund transformations are constructed. The compatibility of these Darboux transformations leads to three discrete systems and their Lax…
An explicit solution formula for the matrix modified KdV equation is presented, which comprises the solutions given in Ref. 7 (S. Carillo, M. Lo Schiavo, and C. Schiebold. Matrix solitons solutions of the modified Korteweg-de Vries…
A nonlocal form of a two-layer fluid system is proposed by a simple symmetry reduction, then by applying multiple scale method to it a general nonlocal two place variable coefficient modified KdV (VCmKdV) equation with shifted space and…
We connect certain continuous motions of discrete planar curves resulting in semi-discrete potential Korteweg-de Vries (mKdV) equation with Darboux transformations of smooth planar curves. In doing so, we define infinitesimal Darboux…
We prove the nonsqueezing property of the coupled Korteweg-de Vries (KdV) equation. Relying on Gromov's nonsqueezing theorem for finite dimensional Hamiltonian systems, the argument is to approximate the solutions to the original infinite…
We study the recently-proposed hyperbolic approximation of the Korteweg-de Vries equation (KdV). We show that this approximation, which we call KdVH, possesses a rich variety of solutions, including solitary wave solutions that approximate…
We use a class of trial wave functions which are generalizations of gaussians to study single soliton approximate analytic solutions to the KdV equations. The variational parameters obey a Hamiltonian dynamics obtained from the Principle of…
The KdV-Sawada-Kotera equation has single-, two- and three-soliton solutions. However, it is not known yet whether it has N-soliton solutions for any N. Viewing it as a perturbed KdV equation, the asymptotic expansion of the solution is…
We calculate quantum mechanical scattering problems for multi-indexed extensions of soliton potential by Darboux transformations in terms of pseudo virtual wavefunctions. As an application, we calculate infinite set of higher integer KdV…
A noncommutative KdV-type equation is introduced extending the Baecklund chart in [S. Carillo, M. Lo Schiavo, and C. Schiebold, SIGMA 12 (2016)]. This equation, called meta-mKdV here, is linked by Cole-Hopf transformations to the two…
In this work we continue the description of soliton-like solutions of some slowly varying, subcritical gKdV equations. In this opportunity we describe, almost completely, the allowed behaviors: either the soliton is refracted, or it is…
Using bidifferential calculus, we derive a vectorial binary Darboux transformation for an integrable matrix version of the first negative flow of the Kaup-Newell hierarchy. A reduction from the latter system to an integrable matrix version…
We consider large-scale dynamics of non-equilibrium dense soliton gas for the Korteweg-de Vries (KdV) equation in the special "condensate" limit. We prove that in this limit the integro-differential kinetic equation for the spectral density…
The mKP equation with self-consistent sources (mKPESCS) is treated in the framework of the constrained mKP hierarchy. We introduce a new constrained mKP hierarchy which may be viewed as the stationary hierarchy of the mKP hierarchy with…
The method of Darboux transformation, which is applied on cnoidal wave solutions of the sine-Gordon equation, gives solitons moving on a cnoidal wave background. Interesting characteristics of the solution, i.e., the velocity of solitons…
All solutions of the Korteweg -- de Vries equation that are bounded on the real line are physically relevant, depending on the application area of interest. Usually, both analytical and numerical approaches consider solution profiles that…
The KP equation with self-consistent sources (KPESCS) is treated in the framework of the constrained KP equation. This offers a natural way to obtain the Lax representation for the KPESCS. Based on the conjugate Lax pairs, we construct the…