New exact periodical solutions of mKP-1 equation via $\overline{\partial}$-dressing

TL;DR

This paper develops a general method using the $ar{ ext{d}}$-dressing technique to construct exact periodic solutions of the mKP-1 equation, including new classes of one- and two-periodic solutions with specific boundary conditions.

Contribution

It introduces a determinant formula for solutions and demonstrates how to satisfy reality and boundary conditions, expanding the set of known exact solutions for the mKP-1 equation.

Findings

Derived new classes of exact periodic solutions of mKP-1

Formulated a determinant-based calculation method for solutions

Interpreted certain solutions as eigenmodes analogous to standing waves

Abstract

We proposed general scheme for construction of exact real periodical solutions of mKP-1 equation via Zakharov-Manakov $\overline{\partial}$-dressing method, derived convenient determinant formula for calculation of such solutions and demonstrated how reality and boundary conditions for the field $u(x,y,t)$ can be satisfied. We calculated the new classes of exact periodical solutions of mKP-1 equation: 1. the class of nonsingular one-periodic solutions or nonlinear plane monochromatic waves; 2. the class of two-periodic solutions without imposition of any boundary condition; 3. the class of two-periodic solutions with integrable boundary condition $u(x,y,t)\mid_{y=0}=0$. We interpreted the third class of two-periodic solutions with integrable boundary condition obtained by the use of special nonlinear superpositions of two simple one-periodical waves as eigenmodes of oscillations of the field $u(x,y,t)$ in semi-plane $y\geq 0$, the analogs of standing waves on the string with fixed endpoints.