Class Of Rational Solutions Research Articles

Partial-rogue waves that come from nowhere but leave with a trace in the Sasa-Satsuma equation

Partial-rogue waves, i.e., waves that “come from nowhere but leave with a trace”, are analytically predicted and numerically confirmed in the Sasa-Satsuma equation. We show that, among a class of rational solutions in this equation that can be expressed through determinants of 3-reduced Schur polynomials, partial-rogue waves would arise if these rational solutions are of certain orders, where the associated generalized Okamoto polynomials have real but not imaginary roots, or imaginary but not real roots. We further show that, at large negative time, these partial-rogue waves approach the constant-amplitude background, but at large positive time, they split into several fundamental rational solitons, whose numbers are determined by the number of real or imaginary roots in the underlying generalized Okamoto polynomial. Our asymptotic predictions are compared to true solutions, and excellent agreement is observed.

Dynamical structures of exact soliton solutions to Burgers’ equation via the bilinear approach

The Cole–Hopf transformation is used to search for exact multiple wave solutions of Burgers’ equation. Particularly, solitary wave solutions and their interaction solutions are explicitly given, and a fission phenomenon is found, which is tough to be found by other traditional methods. Different solutions are presented with different combinations of elementary functions and a necessary condition for fission is proposed. Based on the bilinear formalism and with the aid of symbolic computation, we also determine two classes of rational solutions, rogue wave solutions and rogue–kink wave solutions, using various ansätze. To exhibit dynamics, we make diverse graphical analyses for the presented solutions, which show that the solutions are reliable in both mathematical physics and engineering.

Exact rational solutions to a Boussinesq-like equation in (1+1)-dimensions

A Boussinesq-like nonlinear differential equation in ( 1 + 1 )-dimensions is introduced by using a generalized bilinear differential equation with the generalized bilinear derivatives D 3 , x and D 3 , t . A class of rational solutions, generated from polynomial solutions to the associated generalized bilinear equation, is constructed for the presented Boussinesq-like equation. It is conjectured that this class of rational solutions contain all such rational solutions to the new Boussinesq-like equation. More concretely, the conjecture says that if a polynomial f = f ( x , t ) in x and t solves f t t f − f t 2 + 3 f x x 2 = 0 , then the degree of f with respect to t must be less than or equal to 1.

On recursive solutions to simple allocation problems

We propose and axiomatically analyze a class of rational solutions to simple allocation problems where a policy-maker allocates an endowment $$E$$ among $$n$$ agents described by a characteristic vector c. We propose a class of recursive rules which mimic a decision process where the policy-maker initially starts with a reference allocation of $$E$$ in mind and then uses the data of the problem to recursively adjust his previous allocation decisions. We show that recursive rules uniquely satisfy rationality, c-continuity, and other-c monotonicity. We also show that a well-known member of this class, the Equal Gains rule, uniquely satisfies rationality, c-continuity, and equal treatment of equals.

Commutative differential operator and a class of solutions for the BKP equation on the singular rational curve

The technique of Krichever and Novikov was used to construct a new class of solutions for the BKP equation. This class contains the solitary waves as a special case, and also a class of rational solutions. The present methodology overcomes the difficulty associated with the Gelfand–Levitan equation of the BKP system, as pointed out by Hirota [J. Phys. Soc. Japan 58, 2705 (1989)]. An added advantage of the Krichever–Novikov approach is that it also gives a straightforward derivation of the BKP equation.

Solutions of certain integrable nonlinear PDE’s describing nonresonant N-wave interactions

The evolution equations that describe, in appropriately ‘‘coarse-grained’’ and ‘‘slow’’ variables, the evolution of the envelopes of N nonresonant dispersive waves, can be reduced to the ‘‘universal’’ form (∂/∂t+vn ∂/∂x) un(x,t) =un(x,t) ∑Nm=1βnmum(x,t). In this paper the special case with βnm=(vm−vn) βm, which is integrable by quadratures, is investigated. The subclass of localized solutions (i.e., vanishing as x→±∞) gives rise to a novel solitonic phenomenology. The class of solutions that are asymptotically finite contains a richer solitonic phenomenology, including those of novel type (which move with the speeds vn, and can have any shape) and more standard kinks (which can move with any speed, and have standard shapes). The class of rational solutions, and the integrable dynamical systems naturally associated with these solutions, are also investigated; these dynamical systems include, and extend, known integrable systems. In this paper the treatment is confined to 1+1 dimensions; at the end, together with some other generalizations, a partially solvable multidimensional extension is reported.