The existence of conservation laws for novel classes of nonlinear evolution equations (with linearlyx-dependent coefficients) solvable by the spectral transform is investigated. A remarkably explicit representation is moreover obtained for the conserved quantities of the “old” classes of nonlinear evolution equations (withx-independent coefficients; including the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, the nonlinear Schrodinger equation, etc.).

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Conservation laws for classes of nonlinear evolution equations solvable by the spectral transform

Exact inverse scattering solution of a non-linear evolution equation in a non-uniform medium

Solitons, negatons, positons, rational-like solutions and mixed solutions of a non-isospectral equation, the Korteweg–de Vries equation with loss and non-uniformity terms, are obtained through the Wronskian technique. The non-isospectral characteristics of the motion behaviours of some solutions are described with some figures made by using Mathematica. We also derive an infinite number of conservation laws of the equation.

Negatons, positons, rational-like solutions and conservation laws of the Korteweg?de Vries equation with loss and non-uniformity terms

It is demonstrated that the KdV equation with nonuniformities, ut+a(t)u+(b(x,t)u)x +c(t)uux+d(t)uxxx +e(x,t)=0, has the Painleve property if the compatibility condition among the coefficients of it holds: bt+(a−Lc)b+bbx +dbxxx =2ah+hL(d/c2)+(dh/dt)+ce +x[2a2+aL(d3/c4)+(da/dt) +L(d/c)L(d/c2)+(d/dt)L(d/c)], where L=(d/dt)lg and h(t) is an arbitrary function of t. The auto‐Backlund transformation and Lax pairs for this equation are obtained by truncating the Laurent expansion. Furthermore, assuming the compatibility condition, then the KdV equation with nonuniformities is transformable, via suitable variable transformations, to the standard KdV.

Painlevé property, auto-Bäcklund transformation, Lax pairs, and reduction to the standard form for the Korteweg-De Vries equation with nonuniformities

The nonisospectral Kadomtshev–Petviashvili (KP) equation is solved by the Hirota method and Wronskian technique. Exact solutions that possess soliton characters with nonisospectral properties are obtained. In addition, rational and mixed solutions are derived. We also obtain a new molecular equation that admits a solution in the Wronskian form.

Exact Solutions for the Nonisospectral Kadomtshev–Petviashvili Equation

N -soliton solution of the Korteweg-de Vries equation with loss and nonuniformity terms \begin{aligned} u_{t}+2{\alpha}u+(C_{0}+{\alpha}x)u_{x}+6uu_{x}+u_{xxx}=0, \end{aligned} are obtained. The solutions express multisolitons damping in time without radiation and mutually colliding in nonuniform media with a relaxation effect.

Hirota Ryogo

Papers

N-Soliton Solution of the K-dV Equation with Loss and Nonuniformity Terms