The ∂-dressing method for the (2+1)-dimensional Jimbo-Miwa equation

The (2+1)-dimensional Jimbo-Miwa equation is analyzed by means of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove partial-differential With bar"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∂</mml:mi> </mml:mrow> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\bar {{\partial }}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dressing method. By means of the characteristic function and Green’s function of the Lax representation, the problem has been transformed into a new <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove partial-differential With bar"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∂</mml:mi> </mml:mrow> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\bar {{\partial }}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> problem. A solution is constructed based on solving the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove partial-differential With bar"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">∂</mml:mi> </mml:mrow> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\bar {{\partial }}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> problem with the help of Cauchy-Green formula and choosing the proper spectral transformation. Furthermore, we can obtain the solution formally of the Jimbo-Miwa equation when the time evolution of the spectral data is determined.